Observing the Jumping Laser Dogs A. Tufaile, T. A. Vanderelli, A. P. B. Tufaile………………………………………………………………………………1977 Interaction Energy between an Atomic Force Microscope Tip and a Charged Particle in Electrolyte W.-T. Lam, F. R. Zypman……………………………………………………………………………………………………1989 The Advection Diffusion-in-Secondary Saturation Movement Equation and Its Application to Concentration Gradient-Driven Saturation Kinetic Flow T. Nemaura…………………………………………………………………………………………………………………1998 Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods F. A. Alhendi, A. A. Alderremy……………………………………………………………………………………………2011 Numerical Solution of the Diffusion Equation with Restrictive Pade Approximation A. Boz, F. Gülsever…………………………………………………………………………………………………………2031 Gauss-Legendre Iterative Methods and Their Applications on Nonlinear Systems and BVP-ODEs Z. L. Liu, G. Q. Sun…………………………………………………………………………………………………………2038 Nano-Contact Problem with Surface Effects on Triangle Distribution Loading L. Y. Wang, W. Han, S. L. Wang, L. H. Wang, Y. P. Xin…………………………………….……………………………2047 Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals H. L. Gao……………………………………………………………………………………………………………………2061 An Alternating Direction Nonmonotone Approximate Newton Algorithm for Inverse Problems Z. H. Zhang, Z. S. Yu, X. Y. Gan…………………………………………………………..………………………………2069 High-Capacity Quantum Associative Memories M. C. Diamantini, C. A. Trugenberger……………………………………………………………………………………2079

The figure on the front cover is from the article published in Journal of Applied Mathematics and Physics, 2016, Vol. 4, No. 11, pp. 1989-1997 by Wai-Ting Lam and Fredy R. Zypman.

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Journal of Applied Mathematics and Physics, 2016, 4, 1977-1988 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Observing the Jumping Laser Dogs

Alberto Tufaile1*, Timm A. Vanderelli2, Adriana Pedrosa Biscaia Tufaile1

1Soft Matter Laboratory, Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, São Paulo, Brazil 2Ferrocell USA, 739 Route 259, Ligonier, PA, USA

How to cite this paper: Tufaile, A., Van- Abstract derelli, T.A. and Tufaile, A.P.B. (2016) Observing the Jumping Laser Dogs. Jour- Jumping sun dogs are rapid light flashes changing over clouds, with some of them nal of Applied Mathematics and Physics, 4, located close to the places of halo formation in thunder storms clouds. This paper 1977-1988. presents an outline of some aspects that are required for understanding the jumping http://dx.doi.org/10.4236/jamp.2016.411198 sun dogs, using some experiments with light scattering in complex fluids. In our Received: September 27, 2016 analogy, we have observed the jumping laser dogs, in which the ice crystals are re- Accepted: November 7, 2016 placed by needlelike structures of ferrofluid, the electric field in the atmosphere is Published: November 10, 2016 represented by an external magnetic field, and the laser beam scattered by the ferrof-

Copyright © 2016 by authors and luid structure has the same role of the sun as the source of light scattered by the ice Scientific Research Publishing Inc. crystals subjected to changing electric fields in thunderstorm clouds. This work is licensed under the Creative Commons Attribution International Keywords License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Diffracted Rays, Sun Dogs, Laser Dogs, Geometrical Theory of Diffraction Open Access

1. Introduction

In the past few years, some people have reported the occurrence of a bizarre phenomenon: the observation of rapid light flashes changing over clouds, with some of them located close to the places of halo formation in thunderstorms clouds, as it is shown in Figure 1, like the motion of a “giant light saber”. These phenomena were observed by accident in the sky, and because of these reports, some researchers have suggested the hypothesis that these phenomena could be related with jumping sun dogs [1] [2]. It has been known for sometime that the light scattering in the ice crystals in a cloud is related to the optical effects like halos and sun pillars, and this hypothesis holds that a lightning discharge in a thundercloud can temporally change the electric field above the cloud where charged ice crystals are reflecting sunlight, and these changes in the electric fields reorient the ice crystals to a new position that scatters sun light differently [3], creating a visual cloud streamer [4]. In this way, we consider that the system

DOI: 10.4236/jamp.2016.411198 November 10, 2016 A. Tufaile et al.

Figure 1. The jumping sun dog phenomenon is shown in these pictures. At the top, in the general view perspective, we can see the thunderstorms clouds with the light flash at the left side of the picture. The luminous white column is swung back and forth at the frequency approximately of 1 Hz during 1 minute, combining fast and slow motions, presented in the pictures at bottom. Pic- tures obtained from Youtube user QUADME13 using the camera of a smartphone in 2015, Greenwood, Indiana, USA.

present in the atmosphere acts as a complex fluid interacting with sun light, in which a mixture of solid phase (ice crystals) and gas phase (air/water vapour) [5] shows the evidence of unusual responses to the applied electric field, creating jumping sun dogs or crown flashes. Based on our previous papers, we are suggesting that some these jumping sun dogs are related with the parhelic circle. Like the jumping sun dogs, the parhelic circle is mainly white, and the absence of separated colors could indicate a different kind of light scattering, not only light refraction. The aim of this paper is thus to explore some patterns obtained by light scattering in a complex fluid and comparing them with the jumping sun dogs. In order to accomplish our goal, we have used a device known as Ferrocell, which consists in a Hele-Shaw cell containing a ferrofluid suspension of ferromagnetic nanoparticles that will be explained in Section 2. The scattering of light in the Ferrocell can create diffracted rays, and in a recent work, we have shown that the formation of sun dogs, sun pillars, and the parhelic circle could be related to these diffracted rays, using the Geometrical Theory of Diffraction (GTD) [6] [7] [8]. In Section 3, we present some properties of light polarization in the light scattering with the Ferrocell, which plays the role of the atmosphere in our analogy. We report the simulation of the jumping sun dogs using the Ferrocell in Section 4. Finally, we present our conclusions in Section 5.

2. Experimental Apparatus

This experiment is essentially the light scattering in ferrofluid, and our atmosphere is

1978 A. Tufaile et al.

the colloidal suspension of nanoparticles inside the Ferrocell. The ferrofluid is a stable colloidal dispersion using light mineral oil. In order to explore the effects of the light scattering in the Ferrocell subjected to an external magnetic field, we have used two different types of light source configurations, the setup 1, for the observation of polarized light effects, and the setup 2, for the case of laser beam scattering obtaining the jumping laser dogs, represented in Figure 2. In the setup 1, a white light source was generated using a LED panel light (12 W), il- luminating over the complete surface of a square Ferrocell (22 mm × 22 mm) homoge- neously, with a methacrylate diffuser and backlighting technique. Just for this case, the Ferrocell was placed between two crossed polarizers, as it is shown in Figure 2(a). The glass plates used in the Ferrocell in this case are microscope slides. The image is obtained directly from the system polarizers/Ferrocell. The setup 2 uses a circular Ferrocell made up of two 58.5 mm diameter glass plates, represented in Figure 2(b). For this case, the light source is a green diode laser (10 mW) with a wavelength of 532 nm. The beam width is around 2,00 mm. Once the beam reaches the Ferrocell, part is reflected, and part is transmitted, scattering the light. The image formed by reflection and transmission of light scattering is projected onto a screen, and a digital camera is used as a light detector of the patterns observed in the experiment.

Figure 2. In (a) the square Ferrocell1 used in setup 1 with backlighting technique between the polarizer 1 and the analyzer (polarizer 2). In (b), there is a picture of the circular Ferrocell used to simulate jumping sun dogs in setup 2.

1979 A. Tufaile et al.

We have used neodymium (Ne-Fe-B) magnets placed at some distance of the Ferro- cell in order to change the light patterns. For example, the value magnetic field of a cylindrical magnet was obtained with a Lakeshore gaussmeter model 475 DSP at room temperature of 27˚C in the plots is shown in Figure 3. Changing the position of the magnets, we can modify the magnetic field in each Ferrocell, and consequently change the structure of the magnetic nanoparticles.

3. Polarized Light and Ferrocell

Before getting into the jumping sun dog simulation using laser dogs, let us explain some features of the optical properties of the Ferrocell in the presence of the magnetic field, in order to understand how to control the nanoparticles with an external magnetic

Figure 3. Using the gaussmeter, we have obtained the values of the magnetic field of a cylindrical magnet with the distance for two distinct configurations. In (a) the magnetic field of pole configuration with the distance r, and in (b) the dipole configuration with the distance l. Each dashed line is a curve fitting that has the best fit to the experimental data, and they are explained in Section 3.

1980 A. Tufaile et al.

field. Basically, the magnetic nanoparticles arrange themselves in complex geometries in ferrofluids, such as labyrinth or needles, due to the long range dipolar interactions. One way to observe these complex geometries is using polarized light. Polarization measurements show that the static magnetic field can alter the structure of the nanoparticles inside the Ferrocell, changing the light polarization, as it is shown in Figure 4. We can see the main effect of a cylindrical magnet placed at the center of the Ferrocell, with the magnetic pole facing the Ferrocell in Figure 4(a). There are black fringes and bright regions, creating a pattern resembling a cross. The outstanding feature of this pattern is its stability. For example, in Figure 4(e), Figure 4(f) we have used a cubic magnet in the pole configuration, and we have observed a similar pattern of light polarization, like the pattern observed for the cylindrical magnet of Figure 4(a). Changing the position of the magnet, the pattern remains as a dark cross with four bright lobes, centered at the magnet. These patterns are strong evidence that the Ferrocell acts as one diffraction grating in the presence of the magnetic field. The basic explanation for this behavior is that the ferrofluid is a complex fluid, changing its structure for different values of applied magnetic field. Other structures were observed in ferrofluids, and these structures were reported in the literature more than a decade ago [9]. In a more technical way, these patterns of light polarization represent the Faraday effect in light transmission geometry. For the same value of the magnetic field, there are different values of light intensity in different points of the Ferrocell, indicating some sort of interaction between the structures of the ferrofluid. For example, in Figure 4(b), we can observe the quantitative aspect of light amplitude of a circular stripe chosen around the magnet placed at the center of the Ferrocell1 using the pole configuration. The rotation polarization is periodic inside this circular stripe, and Figure 4(c) shows the diagram of light amplitude. The plot of Figure 4(d) represents the light amplitude for different angle values, ranging from −90˚ to 90˚, projected in y axis. Using the information obtained from these patterns, we can explain the observed data with a phenomenological model. In order to obtain this model, we need to know the mathematical expression of the external magnetic field created by the magnet, and the equation representing needlelike structure. First, the calculation of the exact magnetic field of the magnet at any point in space has some mathematical complexities and involves the properties of the magnetic materials. Things are simpler using direct measures of the magnetic field discussed in the previous section, such as the equation obtained for the horizontal component of the magnetic field Bhor of a magnet in the pole configuration of Figure 3(a), which is given by: Dx(1− 2 ) Bhor = 52 , (1) 4π( xD22+ ) and for the case of dipole configuration of Figure 3(b): Dx Bhor = − 52 , (2) 4π( xD22+ )

1981 A. Tufaile et al.

Figure 4. Optical properties of the Ferrocell subjected to a magnetic field, with a magnetic pole facing the Ferrocell. In (a), the image of the light pattern obtained from the Faraday effect in the transmission geometry of the Ferrocell with a cylindrical magnet placed in the middle of the Fer- rocell. In (b), the plot of the light amplitude of the circular stripe in (a) was chosen in order to show the different values of the light intensity. The diagram of the light intensity of the circular stripe in (c). In (d), light amplitude from −90˚ to 90˚ showing the rotation polarization of the ferrocell, and it gives a hint of how the Ferrocell can interact with light for the same intensity of magnetic field for different positions. We have observed that this pattern is centered in the pole of the magnet (e), and the isoclines remain aligned with the axis of the cell, if the magnet is placed in a different position (f).

1982 A. Tufaile et al.

where D is a constant featured for this magnet and x is the distance from the center of the magnet to the tip of the magnetic field sensor, based on the diagrams in the plots of Figure 3. Second, the formation of the needlelike structures of Figure 5 is directed related with the magnetic field B around 200 G, and for the case of the circular stripe of Figure 4(a), Figure 4(b) obtained from the experiment, the direction of the polarization of the needles inside the circular region, with radius R and width a, is given by the coordinates

(xi, yi) and (xi+1, yi+1) for the angle θ:

xxii+1 cosθ = (R( Ra+ )) (3) yyii+1 sinθ

The neddles are oriented from R to R + a direction, and we have observed from the experiment that the light intensity has the maximum intensity for θ values equal 45˚, 135˚, 225˚, and 315˚. In addition to this, the pattern is black placing a cylindrical magnet in the pole configuration in the Ferrocell1 for θ values equal 0˚, 90˚, 180˚, and 270˚, because for these angle values the ferrofluid needles are aligned with one of the two polarizers. With Equations (1)-(3), we can derive the equation for light intensity through the system formed by the first polarizer, Ferrocell, and the second polarizer. The first and last polarizers are oriented at 90˚ with respect to each other, the Ferrocell has its polarization

Figure 5. Comparisons between light patterns and simulations using polarized light with the Ferrocell. In (a) the cubic magnet placed at pole configuration in the middle of the Ferrocell, and the simulation is presented in (b). The experiment in (c), for the case of the magnet in the dipole configuration, and in (d) the simulation of the light intensity through the Ferrocell in the dipole configuration.

1983 A. Tufaile et al.

aligned with the angle θ, with the needles shaped by the magnetic field given by Equa- tion (1). Therefore the second polarizer is rotated an angle (π/2 − θ) from the Ferrocell.

Considering the intensity after passing the first polarizer is I1, the intensity after passing

the Ferrocell is I2, is given by: Dr(1− 2 ) = 2 θ II21 52 cos . (4) 4π(rD22+ )

The intensity after the second polarizer for each point with coordinates (r, θ), I3, is given by:

2 π Dr(1− ) π =22 −=θ θθ− II32cos I1 52 cos cos, (5) 224π(rD22+ ) 

and rearranging the terms of Equation (5), the intensity after the second polarizer is: Dr(1− 2 ) θθ= 2 Ir31( , ) I 52 sin . (6) 16π(rD22+ )

where r ranges from 0 to infinity, and θ ranges from 0˚ to 360˚. Based on this model, we can simulate the light intensity for different configurations of external magnetic fields, as it is shown in Figure 5. This analysis is a very good way to understand the role of ferrofluid needles interacting with light, and we can extrapo- late that a system with ice crystals with same size, oriented by an external electric field could have similar effect in the light scattering. To sum up, we can see that a static magnetic field placed near the Ferrocell creates a structure inside the Ferrocell, which can be mapped with polarized map. In the next section we will explore some aspects of the light scattering in this structure.

4. The Jumping Laser Dogs

Performing some experiments [7] and obtaining similar light patterns to those observed in the atmospheric optics, we consider that it is important to note that the parhelic circle is an effect combining the properties of the geometric optics and wave optics at same time, involving the concept of diffracted ray [8]. In this way, the parhelic circle was explained in our previous papers [7] [8] as the manifestation of diffracted rays, as it is shown in Figure 6 and, according to GTD, when a light beam hits a needlelike structure of ice crystals obliquely, there is a cone of diffracted rays and the cross section of this cone is given by: −12 ikr uei( r) = Ku r e (7)

where K is the diffraction coefficient, ui is the incident field, and r is the distance between the ferrofluid and the screen, and k = 2π/λ is the wave number of the incident field with wavelength λ. This is similar to the case when the laser beam hits curved prisms and creates laser dogs [10]. Using the diagram of Figure 6(a), the z coordinate of the ferrofluid needle coordinate system (z, r, θ) coincides with the needle axis. The

1984 A. Tufaile et al.

Figure 6. The diagram of the laser scattering with the parlaseric circle with the laser dog pattern with nanoparticles forming a needle in (a). In (b), the laser spot and the laser dog with B = 0 gauss, and in (c) the parlaseric circle for B = 600 gauss. plane θ = 0 contains the incident beam. The angle ϕ defines the tilt between the perpendicular to the direction of the propagation of the incident beam and the needle. The light scattered emerges along the surface of the cone with apical angle 180˚ - 2ϕ and is viewed on the circle ue, sighting towards the apex of the cone. The same effect occurs when light hits the needlelike formation in ferrofluids [11]. We can observe this type of diffracted rays for a setup without any polarizer, as it is shown in Figure 6. Part of the green laser is passing through the Ferrocell forming a laser spot at right side of this picture, and part of light is reflected forming the laser dog at left side. Applying the magnetic field in the Ferrocell used in setup 2 induces the cre- ation of diffracted rays, represented by the curved lines of Figure 6(c) for 600 gauss, because the colloidal system of ferrofluids undergoes structural changes under external field leading to linear chains [9], as it was discussed in the previous section. Our simulation of jumping sundogs is shown in Figure 7. Basically, the laser beam is scattered by the needlelike structure in the Ferrocell (setup 2), while the magnetic field is displaced. The optical patterns observed for micron sized needles are a simultaneous manifestation of both scattering and diffraction. In order to measure the displacement of the diffracted rays obtained with the Ferrocell, the same laser beam was diffracted by a vertical soap film, forming a light pattern of a horizontal well defined dashed line. In Figure 7(a), for the magnetic field of B = 900 gauss, we have obtained a vertical diffracted ray, forming the simulation of the sun dog. Changing the orientation of the magnetic field, we can observe the motion of the sun dog in comparison to the horizontal dashed line. The motion of these patterns represents the jumping laser dogs, which are controlled by the motion of the external magnetic field.

1985 A. Tufaile et al.

Figure 7. The jumping laser dog dynamics for different orientations of the needlelike particles, changing the external magnetic field. Using the fixed light diffraction of a soap film as the reference, we have measure the following angles between the laser dog and the reference: (a) 90˚, (b) 120˚, (c) 135˚, (d) 180˚, (e) 225˚, and in (f) the laser dog is rotated 245˚.

We can observe that the light streak related to the jumping laser dog is different from that produced by the soap film. First, the laser dog can be controlled by the magnetic field, and second the absence of well defined spacing between fringes from multiple microscope particles, such as ice crystals or nanoparticles. According some authors, the absence of fringe pattern in ferrofluid is an integral sum of diffraction events from individual chains, and the observed pattern can be interpreted as a result of the combination of multiple diffraction. Considering that the scale of the ice crystals present in clouds are around hundreds of micrometers, it is plausible to think that the same type of light scattering is present in the systems composed by ferrofluid or ice crystal, and some computational approaches [12] use a mix of geometric optics and Fraunhofer diffraction to calculate the scattering of light by ice crystals present in clouds.

5. Conclusions

Using some simple concepts described in the geometrical theory of diffraction, we have obtained the jumping laser dogs with the motion of an external magnetic field, observing the patterns created by diffracted rays in the Ferrocell system. In this way, we have tested the hypothesis that particles with the same size of ice crystals in a thundercloud could be reoriented by an external field, scattering the light differently. Like most dis- ciplines involving natural phenomena, the study of atmospheric optics utilizes many branches of physics. The deeper one tries to understand them, the more soft matter physics, optics, ice crystals thermodynamics and other facets are required. In this paper, we have proposed an experiment in which a complex fluid subjected to external field scatters the light forming jumping laser dog, a phenomenon analogous to the jumping sun dogs. Using polarized light, we have observed how the magnetic nanoparticles arrange themselves in complex geometries in ferrofluids, such as needles, due to the long

1986 A. Tufaile et al.

range dipolar interactions. In our analogy, the ice crystals are represented by the needlelike structures of ferrofluid, and electric fields are represented by magnetic fields. For certain conditions of the wavelengths and sizes of these light scatterers, the angles of reflection and refraction are not defined by the angle of incidence, and part of the incoming light striking the scat- terer will give rise to a conical shell of diffracted rays, forming halos or partial halos. Using this comparison, we have inferred that the atmosphere could be associated with a complex fluid, in which a mixture of solid (ice crystals) and gas (air) exhibits unusual responses to applied electric field, creating jumping sun dogs or crown flashes.

Acknowledgements

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), FAPESP/CNPq #573560/2008-0.

References [1] Plait, P. (2011) Discover Magazine Blog. http://blogs.discovermagazine.com/badastronomy/2011/10/25/amazing-video-of-a-bizarre- twisting-dancing-cloud/#.V36ZzNIrJdh [2] Tufaile, A.P.B., Vanderelli, T., Amorim, R. and Tufaile, A. (2016) Simulating the Jumping Sun Dogs. Proceedings of 12th Conference on Light and Color in Nature, Granada, 31 May-3 June 2016, 24-25. https://drive.google.com/drive/folders/0BxhJRGXxaO-FUURIcnpIdEU4QjA [3] Foster, T.C. and Hallett, J. (2008) Enhanced Alignment of Plate Ice Crystals in a Non- Uniform Electric Field. Atmospheric Research, 90, 41-53. http://dx.doi.org/10.1016/j.atmosres.2008.02.017 [4] Vonnegut, B. (1965) Orientation of Ice Crystals in the Electric Field of a Thunderstorm. Weather, 20, 310-312. http://dx.doi.org/10.1002/j.1477-8696.1965.tb02740.x [5] Koop, T. (2013) Rare but Active. Nature, 498, 302. http://dx.doi.org/10.1038/nature12256 [6] Keller, J.B. (1962) Geometrical Optics Theory of Diffraction. Journal of the Optical Society of America, 52, 116-130. http://dx.doi.org/10.1364/JOSA.52.000116 [7] Tufaile, A. and Tufaile, A.P.B. (2015) Parhelic-Like Circle from Light Scattering in Plateau Borders. Physics Letters A, 379, 529-534. http://dx.doi.org/10.1016/j.physleta.2014.12.006 [8] Tufaile, A. and Tufaile, A.P.B. (2015) The Dynamics of Diffracted Rays in Foams. Physics Letters A, 379, 3059-3068. http://dx.doi.org/10.1016/j.physleta.2015.10.011 [9] Islam, M.F., Lin, K.H., Lacoste, D., Lubensky, T.C. and Yodt, A.G. (2003) Field-Induced Structures in Miscible Ferrofluids Suspensions with and without Latex Spheres. Physical Review E, 67, 021402. http://dx.doi.org/10.1103/PhysRevE.67.021402 [10] Conover, E. (2015) Researchers Create “Laser Dogs” with Soap Bubbles. Science. http://dx.doi.org/10.1126/science.aaa7816 [11] Laskar, J.M., Brojabasi, S., Raj, B. and Philip, J. (2012) Comparison of Light Scattering from Self Assembled Array of Nanoparticle Chains Cylinders. Optics Communications, 285, 1242-1247. http://dx.doi.org/10.1016/j.optcom.2011.11.103

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[12] Bi, L., Yang, P., Kattawar, G.W., Hu, Y. and Baum, B.A. (2011) Scattering and Absorption of Light by Ice Crystals: Solution by a New Physical-Geometric Optics Hybrid Method. Journal of Quantitative Spectroscopy & Radiative Transfer, 112, 1492-1508. http://dx.doi.org/10.1016/j.jqsrt.2011.02.015

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1988 Journal of Applied Mathematics and Physics, 2016, 4, 1989-1997 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Interaction Energy between an Atomic Force Microscope Tip and a Charged Particle in Electrolyte

Wai-Ting Lam, Fredy R. Zypman

Yeshiva University, New York, NY, USA

How to cite this paper: Lam, W.-T. and Abstract Zypman, F.R. (2016) Interaction Energy between an Atomic Force Microscope Tip A variational principle to the nonlinear Poisson-Boltzmann equation (PB) in three and a Charged Particle in Electrolyte. Jour- dimensions is used to first obtain solutions to the electrostatic potential surrounding nal of Applied Mathematics and Physics, 4, a pair of spherical colloidal particles, one of them modeling the tip of an Atomic 1989-1997. http://dx.doi.org/10.4236/jamp.2016.411199 Force Microscope. Specifically, we consider the PB action integral for the electrostatic potential produced by charged colloidal particles and propose an analytical ansatz Received: October 18, 2016 solution. This solution introduces the density and its corresponding electrostatic po- Accepted: November 7, 2016 tential parametrically. The PB action is then minimized with respect to the parame- Published: November 10, 2016 ter. Polynomial-exponential approximations for the parameters as functions of tip- Copyright © 2016 by authors and particle separation and boundary electrostatic potential are obtained. With that in- Scientific Research Publishing Inc. formation, tip-particle energy-separation curves are computed as well. Finally, based This work is licensed under the Creative on the shape of the energy-separation curves, we study the stability properties pre- Commons Attribution International License (CC BY 4.0). dicted by this theory. http://creativecommons.org/licenses/by/4.0/ Open Access Keywords

Atomic Force Microscopy, Colloids, Electrostatics in Liquids

1. Introduction

An open problem of current scientific and technological interest is the theoretical pre- diction of the force between an Atomic Force Microscope (AFM) probe and a charged particle, in particular when both are immersed in an electrolytic environment [1]. The main interest of this problem comes from the need to understand the electrostatics of biological matter, a problem in which water is inherently present [2]. In addition, AFM has become the de facto metrological tool to probe organic and inorganic matter from the micron down to the nanometer length scales [3]. The tip comprises the sensing

DOI: 10.4236/jamp.2016.411199 November 10, 2016 W.-T. Lam, F. R. Zypman

element of the AFM and, at its apex ranges in size from microns to nanometers, thus the ability of the AFM to probe those length-scales. When the AFM tip is immersed in an electrolyte, it can gain surface charge due to pH, and also can develop a diffuse charge layer due to the presence of ions in solution [4]. Thus we see a natural conceptual connection between AFM measurements in liquid and colloidal science, whereby the interest is in the interaction between colloidal particles and their corresponding stability. Therefore, although our interest is in AFM in liquid, the results obtained here are readily usable in colloidal systems. We focus here in a liquid system in which 1- 1000 nm particles are submerged in an ionic solution. Colloidal systems comprise one of the primary types of mixtures in chemistry. One of the central problems in colloids is their stability [5], that is, under known conditions, such as concentration; will the system coagulate or remain indefinitely stable? The stability of colloids is indeed known to depend on the presence of charge at their surfaces [6]; the electrical double layer controls electrostatic stabilization. When particles approach each other, the interaction leads to the rearrangement of charges in the ambient liquid, outside of the colloidal particles. For instance, these interactions could be determined by the surface charge on the particles and electrolyte concentration. The stability of colloidal systems is interesting conceptually, and critical for industrial applications. In chemistry, there are different types of colloidal systems such as solid-liquid dispersions (suspensions), liquid-liquid dispersions (emulsions) and gas-liquid dispersions (foams). Paints, milk, proteins as well as fog are some of the daily examples of colloids [7] [8]. Mathematically, the stability depends on the details of the pair-wise energy as a function of separation of colloidal particles [9]. The valleys of such function determine the separations of the possible equilibrium. One approach to obtain that energy is to first solve the Poisson-Boltzmann (PB) equation, whose solution gives the charge density and electrostatic potential in the liquid surrounding the colloidal particles [10]. In general, PB equation provides the distribution of the electric potential in solution with charged ions present. This distribution, in turn, provides information to determine how the electrostatic interactions will affect colloidal forces. PB is a nonlinear second-order partial differential equation which has an exact known solution only in one dimension. In higher dimensions, PB equation is commonly solved by numerical analysis. Alterna- tively, if the colloidal particle charge or voltage is not high, the PB equation can be linearized, in which case solutions for spherical [11] and cylindrical [12] geometries have also been obtained. For the particular geometry of sphere-plane, useful to study charge transfer in scanning tunneling microscopy and forces in atomic force microscopy, there have been analytical and numerical approaches to the nonlinear PB equation in three dimensions [13] [14] [15] [16]. Here, we present a method to tackle the full nonlinear PB equation in three-dimensions for interacting particles. The method is analytical, based on the choice of a parametric trial family of functions. While the method is approximate, its analytical nature should provide conceptual insight into the problem. The exact PB nonlinear equation in 3D is not amenable to analytical solutions even

1990 W.-T. Lam, F. R. Zypman

for a single colloidal particle in the electrolyte. We here consider the problem of two interacting particles by introducing an ansatz for the charge density function and corresponding electrostatic potential parametrically; the variational method is then used to minimize the PB functional with respect to the parameters.

2. Electrostatics Potential Produced by a Pair of Colloids

Figure 1 shows schematics of the system of interest. Two charged spherical colloidal particles of unit diameter are separated by a distance d. The PB equation is typically obtained by combining Poisson’s equation [17] and the Boltzmann factor [18] for the distribution of electrostatic energies at a given temperature. Thus Poisson’s equation gives the relationship between the electrical potential Φ ( R) and the charge density ρ ( R) at location R , ∇Φ2 ( RR) =−(4π ) ρ ( ) (1) with  is the dielectric constant of the surrounding fluid. On introducing the Boltz- mann distribution of ions, the non-linear PB equation is obtained,

2 ∇Φ( RR) =−(8πne)sinh ( eΦ( ) kB T ) (2) where n is the ion bulk concentration of electrolyte, T is the absolute temperature, e the ion charge magnitude of anions and cations, and kB is Boltzmann’s constant. Equation (2) is converted into dimensionless form [2] by defining the dimensionless e electrostatic potential ϕ (rR) = Φ( ) and the dimensionless position vector kTB 8πne2 rR= , kTB 

a = 1

Figure 1. Two colloidal particles (large, blue) separated by a distance d. The unit of length throughout the paper is the particles diameter, or the AFM tip diameter. The small red particles represent the ions dissolved in water and are treated as a continuum in the Poisson-Boltzmann approach. These ions could be different, we show them here with the same color for graphical simplicity.

1991 W.-T. Lam, F. R. Zypman

∇=−2ϕϕsinh (3)

where ϕ now represents the dimensionless electrostatic potential, and the Laplacian in Equation (3) is with respect to r . Since n has unites of inverse volume, and  is 8πne2 the absolute dielectric constant, has units of inverse area. kTB  Equation (3) can be derived from a variational principle, by applying Euler-Lagrange to the action

1 2 IV= ∇+ϕϕcosh( ) − 1 d (4) ∫Space 2

where V is volume. The minimum of I occurs for the function φ that satisfies the Eu- ler-Lagrange equation, which gives rise Equation (3). Taking z as the axis that joins the centers of the two colloidal particles, and due to the axial symmetry of the problem, we rewrite the action in cylindrical coordinates as

∞∞1 2 4π ∇+ϕcosh( ϕ) − 1 ηη d dz (5) ∫∫002

where η is the radial polar coordinate in the xy plane, while the angular polar integration is readily performed and gives 2π. The additional factor of 2 comes from integrat- ing z in half space and multiplying by 2 due to mirror symmetry. To make progress, we propose the following ansatz for the density and corresponding electrostatic potential which depends on the parameter k,

 22  kd 2211 d  ϕη( ,zz) = ϕ0 exp  − − +−η  z + +−η (6) 22222      

where ϕ0 is the Dirichlet boundary condition, d is the center-to-center separation between the two spherical colloids and k is a constant that can be interpreted as an inverse Debye length times the radius of the interacting particles. The intuitive justifica- tions for the functional form are 1) its exponential decay typical of ionic screening, 2)

that proper boundary conditions ϕη( , z) = ϕ0 are achieved at the surface of the colloids, and 3) that as d →∞, the electrostatic potential between the two colloids tends to zero. Figure 2 shows a contour plot of the potential around the colloidal particles for a choice of k. To emphasize, this choice of potential is dictated first by the fact that the potential must rapidly approach its bulk value away from the spheres, and second by the fact that the electrostatic potential must have a constant value at the surface of the colloidal particles (Dirichlet boundary conditions). To find the sought solution to the original PB equation (Equation (3)), we minimize

the PB action functional with respect to the parameter k. For fixed potential φ0 and

fixed separations d, we find the constant k (i.e. minimum point k = k(φ0, d)) for the proposed φ(x, y) that minimizes the action. As Figure 3 shows, for small separations, the data suggests that there is an approx-

imately linear relationship between the best constant kbest, and the separation d for each

1992 W.-T. Lam, F. R. Zypman

0 (in units of a) of units (in η η

−2

−4

−4 −2 0 2 4 Z (in units of a)

Figure 2. Contour plot of ϕη( , z) . The potential is constant ϕ0 at the surfaces of the particles, becomes spherical far away while decaying to zero.

Best k vs small separation d 1.2

0.8

best 0.6 K 0.4

0.2

0 0 1 2 3 4 5

Figure 3. Graphs of kbest as a function of small separation d, as ϕ0 is changed. It shows an approximately linear relationship between kbest and small separation d for each ϕ0 . The black line is drawn to show the average trend between kbest and ϕ0 .

1993 W.-T. Lam, F. R. Zypman

boundary condition φ0. It leads us to further investigate the relationship between the

linear relationship and the boundary condition φ0. We obtain polynomial approximations for the functions that relate the linear parameters (that are the slopes and η-intercepts) and the boundary conditions (as shown in Figure 4(a) and Figure 4(b)). Moreover, for large separations between the colloidal particles, the best constant k

converges to 0.1 for all of the boundary conditions φ0. This should be a universal feature regardless of the model used since at large separations we should obtain a simple superposition the potential around single spheres.

The functional forms for the kbest allow us to write a simple function as follows (Figure 5).

B(ϕ ) d A(ϕ )−0.1 kAbest =−+( (ϕ ) 0.1) e 0.1 (7)

where A(φ) is the polynomial approximation between the linear parameter-η-intercepts

and φ0, B(φ) is the polynomial approximation between the other linear parameter-slope

and φ0, and d is the center-to-center separation between the colloids.

3. Colloid Interaction Energy

Since the charge is distributed in the whole space that surrounds the colloidal particles, we have the energy as a function of separation d [13] 1 E( d) = drρ (dV) ( d) (8) ϕ 2 ∫space ϕϕ

where to recall ρ is density and φ is voltage, which are now known from the previous section. For each boundary condition, the integral in (8) is performed for the corresponding optimal value of k.

Slope vs. φ0 η intercept vs. φ0 30 9 25 8 7 20 6 0

0 5

φ 15 φ 4 10 3 5 2 1 0 0 -0.15 -0.1 -0.05 0 0.05 0 0.5 1 1.5 Slope η-intercept

(a) (b) Figure 4. (a) Polynomial approximation between the linear parameter-slope and the boundary

conditions ϕ0 . This is the slope of kbest vs d (Figure 3). (b) Polynomial approximation between

the η-intercept and the boundary condition ϕ0 .This is the η-intercept of kbest vs d (Figure 3).

1994 W.-T. Lam, F. R. Zypman

kbest 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 10 20 30 40 d

Figure 5. With Equation (7), curves of kbest as functions of separation d for

different boundary condition ϕ0 are sketched. While in Figure 3, we show

kbest vs d only for small d, here we show the whole range of d values, from small to large.

Equation (8) then provides the sought sphere-sphere energy-separation curves. The computed results are shown in Figure 6. Based on the shape of the curves, we now can draw conclusions regarding the stability properties predicted by this theory.

4. Conclusions

Figure 6 is the main quantitative result of this work and it shows that for all boundary conditions φ0 the particles attract each other at small separations. This is consistent with all the published experimental literature [19]. In addition, our results show that for large φ0 the energy decreases monotonically giving rise to repulsion at large separations.

While, for small φ0 there are plateaus that suggest the existence of secondary minima.

[15] For all values of φ0 there are local minima at distances larger than 30, but they cannot be expected to represent experimental behavior since they correspond to distance too large compared to the size of the particles. We also notice that the peak positions of the energy curves shift to larger distances as φ0 increases, as expected. Finally the behavior of the screening parameter as shown in Figure 5 shows a strong dependence with distance and reducing its value by more than 50%. This has been observed in experimental measurements [15]. For the AFM community, these results are useful in comparing experimental forces with the derivative of the curves in Figure 6. Although the results presented here are limited to particles of the same size, it is clear that the extension of two different radii amounts to adding a second parameter to Equation (6). With that modification, and by choosing the proper theoretical curve, one can infer the charge of the particle interacting with the AFM tip.

1995 W.-T. Lam, F. R. Zypman

Energy

100

80 1 60 2 3 40 4 5 20 6 7 d 10 20 30 40 50 8

Figure 6. The energy-separation curves for φ0 from 1 to 8.

Acknowledgements

Funding for this project comes from the National Science Foundation grant #CHE- 1508085.

References [1] Zypman, F.R. and Eppell, S.J. (2013) Electrostatic Force Curves in Finite-Size-Ion Electro- lytes. Langmuir, 29, 11908-11914. http://dx.doi.org/10.1021/la402344m [2] McLaughlin, S. (1989) The Electrostatic Properties of Membranes. Annual Review of Bio- physics and Biophysical Chemistry, 18, 113-136. http://dx.doi.org/10.1146/annurev.bb.18.060189.000553 [3] Maver, U., Velnar, T., Gaberšček, M., Planinšek, O. and Finšgar, M. (2016) Recent Progres- sive Use of Atomic Force Microscopy in Biomedical Applications. TrAC Trends in Analyt- ical Chemistry, 80, 96-111. http://dx.doi.org/10.1016/j.trac.2016.03.014 [4] Jarmusik, K.E., Eppell, S.J., Lacks, D.J. and Zypman, F.R. (2011) Obtaining Charge Distri- butions on Geometrically Generic Nanostructures Using Scanning Force Microscopy. Langmuir, 27, 1803-1810. http://dx.doi.org/10.1021/la104153p [5] Napper, D.H. (1970) Colloid Stability. Product R&D, 9, 467-477. [6] Israelachvili, J.N. (1997) Intermolecular and Surface Forces. Academic Press, London. [7] Birdi, K.S. (2016) Handbook of Surface and Colloid Chemistry. 4th Edition, Chapter 1, CRC Press, Taylor and Francis, London, New York, Boca Raton, 1-144. [8] Evans, F. and Wennerström, H. (1994) The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet. VCH Publishers, New York. [9] Verwey, J. and Overbeek, J.Th.G. (1984) Theory of Stability of Lyophobic Colloids. Elsevier Publishing Company, New York, 22-30. [10] Davis, M.E. and McCammon, J.A. (1990) Electrostatics in Biomolecular Structure and Dy- namics. Chemical Reviews, 90, 509-521. http://dx.doi.org/10.1021/cr00101a005 [11] Brenner, S.L. and Roberts, R.E. (1973) Variational Solution of the Poisson-Boltzmann Equ- ation for a Spherical Colloidal Particle. The Journal of Physical Chemistry, 77, 2367-2370.

1996 W.-T. Lam, F. R. Zypman

http://dx.doi.org/10.1021/j100639a001 [12] Bohinc, K., Iglic, A. and Slivnik, T. (2008) Linearized Poisson Boltzmann Theory in Cylin- drical Geometry. Elektrotechniškivestnik, 75, 82-84. [13] Henderson, D. and Chan, K.Y. (1992) Potential Distribution in the Solution Interface of a Scanning Tunneling Microscope. Journal of Electroanalytical Chemistry, 330, 395-406. http://dx.doi.org/10.1016/0022-0728(92)80320-4 [14] Chan, K.Y., Henderson, D. and Stenger, F. (1994) Nonlinear Poisson-Boltzmann Equation in a Model of a Scanning Tunneling Microscope. Numerical Methods for Partial Differen- tial Equations, 10, 689-702. [15] Pecina, O., Schmickler, W., Chan, K.Y., Henderson, D.J. (1995) A Model for the Effective Barrier Height Observed with a Scanning Tunneling Microscope. Journal of Electroanalyti- cal Chemistry, 396, 303-307. http://dx.doi.org/10.1016/0022-0728(95)03853-9 [16] Zypman, F. (2006) Exact Expressions for Colloidal Plane-Particle Interaction Forces and Energies with Applications to Atomic Force Microscopy. Journal of Physics: Condensed Matter, 18, 2795-2803. http://dx.doi.org/10.1088/0953-8984/18/10/005 [17] Jackson, D. (1998) Classical Electrodynamics. John Wiley, New York. [18] Hill, L. (1960) An Introduction to Statistical Thermodynamics. Addison Wesley, Reading. [19] Israelachvili, J.N. and Adams, G.E. (1978) Measurement of Forces between Two Mica Sur- faces in Aqueous Electrolyte Solutions in the Range 0 - 100 nm. Journal of the Chemical Society, Faraday Transactions, 74, 975-1001. http://dx.doi.org/10.1039/f19787400975

1997 Journal of Applied Mathematics and Physics, 2016, 4, 1998-2010 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

The Advection Diffusion-in-Secondary Saturation Movement Equation and Its Application to Concentration Gradient-Driven Saturation Kinetic Flow

Tafireyi Nemaura1,2

1Department of Clinical Pharmacology, University of Zimbabwe, Harare, Zimbabwe 2Department of Applied Mathematics, National University of Science and Technology, Bulawayo, Zimbabwe

How to cite this paper: Nemaura, T. (2016) Abstract The Advection Diffusion-in-Secondary Satu- ration Movement Equation and Its Applica- This work describes the deterministic interaction of a diffusing particle of efavirenz tion to Concentration Gradient-Driven Satu- through concentration gradient. Simulated pharmacokinetic data from patients on ration Kinetic Flow. Journal of Applied Mathe- efavirenz are used. The Fourier’s Equation is used to infer on transfer of movement matics and Physics, 4, 1998-2010. http://dx.doi.org/10.4236/jamp.2016.411200 between solution particles. The work investigates diffusion using Fick’s analogy, but in a different variable space. Two important movement fluxes of a solution particle Received: September 14, 2016 are derived an absorbing one identified as conductivity and a dispersing one identi- Accepted: November 11, 2016 fied as diffusivity. The Fourier’s Equation can be used to describe the process of Published: November 14, 2016 gain/loss of movement in formation of a solution particle in an individual. Copyright © 2016 by author and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative Commons Attribution International Partial Differential Equations, Conductivity Flux, Diffusivity Flux License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction

The derivation of an important equation by Fourier brought so many insights in the study of dynamics of flow through his law of thermal conduction [1]. It is an equation that has shaped the study of flow in liquids, solids and gases, for over two centuries. Additionally, Fourier proposed a method of solution to the equation using separation of variables [2]. The present work adopts similar construction arguments but differs in method of solution and variable space [3]. Fick used the same analogy of heat conduction in modelling diffusion [1]. Thus, Fick formulated what is now known as Fick’s Second Law of Diffusion. Fick’s work suggests that the diffusivity parameter is a con-

DOI: 10.4236/jamp.2016.411200 Novembe 14, 2016 T. Nemaura

stant [1] [4] [5]. The present work adopts a variable, concentration which enables the study of diffusivity/conductivity fluxes with the aid of movement gradient (observable from differences in concentrations) in a medium and time as independent variables instead of a location variable and time. It notes that the anti-solvation inducing measure (concentration) is also implicitly a function of time. The dependent variable is the secondary saturation movement which can be independently written as a function of time or concentration. Thus, the partial differential equation modelling movement derived from formulation reduces to an ordinary differential equation. Further, the work notes that “diffusivity’’ is not independent of concentration as proposed by Fick’s laws and proposes composition-dependent diffusion coefficients as suggested by Boltzmann, but differs in form [1] [4]. We propose a novel form of conductivity flux, a function of concentration, time, movement density and specific movement capacity. The motivation for modelling secondary saturation movement of solution particle in tracking diffusivity is that the variable increases with increasing concentration. However, the relationship of convection and advection with concentration is concave [6]. In addition, it is the only secondary movement measure that for a unique value of saturation movement there is a corresponding unique value of concentration. This results in less ambiguity in the formulation process. The derivation from the flow equation allows further understanding of diffusivity and conductivity fluxes of concentration in media. Conductivity flux in this work defines the zero sum inward acquiring “concentration” movement of an interacting solution particle in a volume space from a neighbouring solution particle or central volume space in a solution particle bridge. A solution particle bridge is a state of exchange of concentration material between two interacting solution particles. Diffusivity flux is the zero sum outward dispersing concentration movement generated from an interacting solution particle to a neighbouring solution particle or peripheral volume space in a solution particle bridge. A total movement flux is postulated to be comprised of four main entities. These components have been identified as advective, saturation, passive and convective. Advective diffusivity is the deceleration of the advective diffusivity flux. Advective conductivity is the acceleration of advective conductivity flux. The work considers definitions of important terms of advective movement flux. They are saturation kinetic advective conductivity and saturation kinetic advective diffusivity. Saturation kinetic advective conductivity flux is defined as the solution particle’s inward formulation velocity rate constant of the secondary saturation movement with respect to advection. Additionally, saturation kinetic advective diffusivity flux as the interacting solution particle’s outward formulation velocity of the secondary saturation movement of an interacting solution particle with respect to advection (rate of change of velocity rate constant of the volume (ml) that dissolves a mass of 1 μg of efavirenz with respect to advection). These two important parameters vary with the state and properties of the volumetric spaces (Central and Peripheral), the concentration of the solution particle and possibly the factors that affect absorption and elimination of solu-

1999 T. Nemaura

tion particle. Diffusivity and conductivity fluxes can be tracked using the saturation relation that governs kinetic solubility movement in individuals to infer on the probable resultant effects. The solution particle characterisation is tracked in an individual through time and concentration. The work characterizes conductivity and diffusivity fluxes by studying the secondary saturation movement dynamics in solution particle’s bridge. The calculations are done in the concentration-time space. This work can further be developed to construct similar arguments for the primary saturation [6]. However, the primary level is not apparent. The primary saturation is a component of potential. On the other hand, the secondary saturation is the initial realized potential of solution-particle formation movement. Furthermore, the primary system is a sub-system of the secondary system.

2. Methods

Simulated projected data on secondary saturation movement, time and concentration was obtained from pharmacokinetic projections made on patients on 600 mg dose of efavirenz considered in Nemaura (2014 & 2015). Partial and Ordinary Differential equations are used in the development of models. A statistical Package R is used to develop nonlinear regression models.

Derivation of Advection Kinetic Flow for the Secondary Saturation Movement

Let F( xt,/)( h) be the secondary saturation movement at time t and concentration x- (Gradient-driven transportation inducing measure) in the blood. The variable F( xt, ) , is a measurement of density of movement. There is derivation of the flow equation of F( xt, ) from two physical “laws’’, that are assumed to be valid: • The amount of movement required to raise the saturation in a solution particle by δ Fh(/ ) is sqδ F where, q is the concentration-time amount of movement in solution and s (Specific Movement Capacity) is a positive physical constant determined by material contained in the solution particle. Specific movement capacity is the movement per unit concentration-time amount of movement required to raise secondary saturation by 1 h . • The rate at which saturation movement crosses a solution particle is proportional to concentration, time and secondary saturation gradient of solution particle. The following proposition is made that the amount of advective conductivity flux of solu- μgh ⋅ tion particle κ  is the difference between κ ρ (that amount of advective ml conductivity flux which is generated by movement density ρ ( xt, ) (obtaining in

the process of solution formation)) and κ0 (an already existing constant (base) amount of advective conductivity flux (primarily obtaining in the independent of solute of x, solvent state)). The function of proportionality is called the amount of

advective conductivity flux and is denoted by κκ =ρ − κ0 . The functions κ ρ and

2000 T. Nemaura

κ0 are such that κρρ ∝ ( xt, ) and κρ00∝ ( yt, ) where y0 is the homogenous

concentration mix in the base solvent without solute x, thus κρρ = k( xt, ) ,

κ00= κρ ( yt, ) , and where k,κ are constants of direct proportionality. Considering an abitrary solution particle bridge (A) (see Figure 1) the following relationship is proposed for its density dependent advective conductivity flux, kρ ( xt, ) κρ ( yt, ) κˆ A ( xt,.) = − 0 (1) xt y0 t

ρ ( yt, ) In a homogenous mix, 0 = ω. Thus, the advective conductivity flux [advec- yt0 tive conductivity flux in the solvent of a solute of x, in the absence of the solute x (before mixing)] is constant and Equation (1) reduces to, kρ ( xt, ) κˆ A ( xt,.) = − κω (2) xt

Assume that secondary saturation is increasing from left to right. The saturation gra- ∂F dient at the right end is ( x+ δ xt, ) , so the rate at which movement crosses the right ∂x ∂F end is −+κδˆ A ( xt,,) ( x xt) . Similarly, the rate at which movement crosses the left ∂x ∂F end is −κˆ A ( xt,,) ( xt) . Since movement flows from regions of higher concentration ∂x to regions of lower concentration, movement thus should be entering the infinitesmal bridge through the right end and exiting the infinitesimal bridge through the left end. So in an infinitesimal time interval (δt) , the net amount of movement that enters the medium (solution particle bridge), is the amount that enters through the right end minus the amount that departs through the left end, which is given by,

AA∂∂FF −κˆ ( xt,) ( x +− δκ xt ,) ˆ ( xt ,) ( xt ,.) δ t (3) ∂∂xx

Considering a small time interval (δt) the movement in the solution particle bridge ∂F changes by ( xt, )δ t. The concentration-time amount of movement in the solution ∂t ρ ( xt, ) particle bridge is δ x . Thus by the first physical law, the net amount of movement xt

Figure 1. Infinitesmal bridge (medium) between solution particles at time t.

2001 T. Nemaura

∂F ρ ( xt, ) ∂F which increase by ( xt, )δ t is sδδ x( xt, ) t. Assuming that the solution ∂t xt∂ t particle bridge is not generating or destroying movement itself. The following conclusion is made that it is equal to the amount of movement that entered the solution particle bridge in the interval time δt , that is

ρ ( xt, ) ∂FAA ∂∂ FF sδ x( xt,) δκ t=−ˆ ( xt ,) ( x +− δκ xt ,) ˆ ( xt ,) ( xt ,.) δ t (4) xt∂ t  ∂∂x x

Dividing both sides by δδxt× and taking limits δδxt,0→ gives, ρ ( xt, ) ∂∂FF2 s( xt,) = −κˆ A ( xt ,) ( xt ,.) (5) xt∂ t ∂x2 Further, division of Equation (5) both sides by, ρ ( xt, ) s . xt The following result is immediate, ∂∂FF2 ( xt,) = −κˆ A ( xt ,) ( xt ,,) (6) ∂t s ∂x2 where, xt κκˆ AA( xt,) = ˆ ( xt,,) s sρ ( xt, )

is the specific secondary saturation advective conductivity flux (/h) associated with a unit amount of movement in a solution bridge. This reduces to, 2 ∂∂FFκ ,s ( xt,) =−−( MρI ( xt,)) ( xt ,,) (7) ∂t ∂x2 where, κω xt k ρκ ,s ( xt, ) = = D( xt,) xt and M= . I sρ ( xt, ) s

The terms, κω ρκ ,s ( xt,,,) D( xt) = and M , I sρ ( xt, )

are specific secondary saturation base-advection diffusivity flux (/h), secondary satura- ml tion advection diffusivity 2 , and specific secondary saturation initial or rest μgh⋅ advection conductivity flux (/h) respectively. The following relation from Equations (6) and (7) is established between advection conductivity flux and advection diffusivity flux,

auxilliary advection conductivity flux  Asκ , κρˆsI( xt,) −=− M ( xt,,) (8)  advection diffusivity flux

A where M = κˆs (0, 0) .

2002 T. Nemaura

A It is important to note that κˆs ( xt, ) is the advection conductivity flux.

3. Results 3.1. Applications to Deteministic Saturation Advection Kinetic Flow of Concentration of Efavirenz and Numerical Analysis

There is use of the relationship developed from [6] [7] for some patient P. The first being the concentration-time profile of patient P [7]. Additionally, the secondary saturation movement-concentration profile [6]. Separate projectiles are followed for secondary saturation using time and concentration for the single patient. These two relations (Figure 2) are used to come up with the following fitted system of Equation(s) for patient P. λ e−−λλbatt−∈ e , t [ 0, 24] , ( i)  r ( ) = F( xt, )  ux (9)  , xc∈[ 0,] , ( ii) vx+ max

(a)

(b)

Figure 2. (a) The x-(concentration) projectile of F( xt, ) in patient P. (b) The t-(time) projectile of F( xt, ) in patient P.

2003 T. Nemaura

where, x is the concentration and t is time. These two variables track solution particle

dynamics. Furthermore, λλλrab,,,u and v are constants to be found. The time (t) variable does not necessarily stop at 24 hours. This is only considered because it is the dosing interval of the drug efavirenz. Equation (9) estimates secondary saturation movement in terms of time and concentration independently (Table 1 and Table 2). Following from Equation (9), Equation (6) assumes the form of, ddFF2 ( xt,) = −κˆ A ( x ,,tx) ( t), (10) dt s dx2 and Equation (8) becomes, 2 ddFFκ ,s ( xt,,) =−−(MρI ( xt, )) ( xt). (11) dt dx2 The following conditions holds for secondary saturation advective diffusivity D ( xt, ) at boundary points of loss/formation of the solution particle in the central volume space. limD( xt ,) = −∞ , ( xt,)→( 0,0)

κ ,s limρI ( xt ,) = 0, ( xt,)→( 0,0)

limD( xt ,) =∞> for some t 0, ( xt,)→( 0, t)

κ ,s limρI ( xt ,) = M for some t> 0. ( xt,)→( 0, t)

A It is noted that for κˆs ( xt, ) the following holds at boundary points of advective A conductivity flux: κˆs (0, 0) := M , value at rest of saturation conductivity flux. The part CC shows the advective diffusivity solution particle space (Central volume space) and CE shows the diffusivity in the complement space of the solution particle (neighbourhood volume space-Peripheral volume space) (Figure 3). A solution particle

Table 1. Parameter estimates in modelling saturation movement rate with respect to t (Model 9(i)).

Parameters Estimate Std Error t Value Pr(> t )

−16 λr 0.623106 0.012214 51.02 <×2 10

−15 λb 0.022465 0.001203 18.68 5.35× 10

−16 λa 0.439258 0.017354 25.31 <×2 10

Table 2. Parameter estimates in modelling saturation movement rate with respect to x (Model 9(ii)).

Parameters Estimate Std Error t Value Pr(> t )

u 0.801936 0.05934 135.1 <×2 10−16

v 5.624198 0.126684 44.4 <×2 10−16

2004 T. Nemaura

Figure 3. Time matched plot of advection diffusivity of saturation movement in the locality of two neighbouring volume spaces. is affected by two neighbourhoods, one that is in the central volume space (CC) and the other is related to the complement (CE) immediate peripheral volume space. Move- ment in these two spaces varies according to the characterisation of these two spaces. The advective diffusivity flux’s deceleration within the central compartment (that volume space which the solution particle is taken from) is given by DCC ( xt, ) . While, the advective diffusivity flux’s deceleration in the complement and is defined by DCE ( xt, ) .

They are equal only DCC ( xt,) = DCE ( xt ,0) = . Thus,

DCC ( xt,) <⇔ 0 DCE ( xt ,) > 0, Case( i) ,  DCC ( xt,0) =⇔= DCE ( xt ,0,) Caseii,( ) (12)  DCC ( xt,) >⇔ 0 DCE ( xt ,) < 0, Case( iii) .

A positive value for advective diffusivity in the volume space (CC) implies that the movement is “concentrating” within CC space. However, a negative value shows movement is “concentrating” in the complement space (CE).

3.2. Saturation Movement: Advection-Base Diffusivity and Conductivity Fluxes

The diffusivity flux is an important parameter which is further investigated to see its dynamics in terms of the four entities that formed the basis of a solution particle. The existence of negative values for advection diffusivity shows presence of negative advective diffusivity flux (Figure 3). The following relations describe the expected advective diffusivity flux in the locality of the central volume space:

2005 T. Nemaura

D( xt,) <⇒ 0 ρκ ,s (t) <0, Case( i) ,  CC I CC  κ ,s DCC ( xt,) =⇒= 0 ρI (t) 0, Case( ii) , (13)  CC D( xt,) >⇒ 0 ρκ ,s (t) >0, Case( iii) ,  CC I CC

where, ρκ ,s is the advective diffusivity flux relative to the central volume space. I CC Considering Equation (13), we deduce that for case (i), the advective diffusivity flux κ ρ ,s is in the direction of peripheral volume space from the central volume space. ( I CC ) The advective diffusivity flux is zero for case (ii) the movement is inferred to be equal in both spaces, and in case (iii) advective diffusivity flux is positive implying movement is in the direction of central volume space from the peripheral volume space. The form of advection diffusivity flux has allowed for the postulation that diffusivity flux is actually a movement entity in equilibrium which comprises four entities. The principal parameter being the advection diffusivity flux. This is motivated by the same behaviour that has been observed in solution particle concentration in relation to saturation [6]. There is a mutualistic relationship postulated that enables us to study equilibrium of movement diffusivity flux. It is noted that the characterisation of saturation movement behaves similarly to concentration dynamics in-vivo. The reference movement with respect to saturation is advection, it is investigated and the constituent behaviour is suggested by Equation (14) (Table 3).

saturation diffusivity flux convection diffusivity flux  κ ,s −−nt dt ft ρI (t) = wte+b( e1 −+) . (14)      gt+ base-advection diffusivity flux passive diffusivity flux

The characterisation of the four main diffusivity flux entities are shown and magnitudes of effects (Figure 4). The region CC represents that which affects the central volume space and CE that which affects the immediate peripheral volume space. From Equation (8), we can infer that auxilliary conductivity flux consists of four main components and affects the movement into-solution particle and diffusivity flux affects

Table 3. Parameter estimates in modelling saturation movement diffusivity fluxes in Equation (14).

Parameters Estimate Std Error t Value Pr(> t )

w −37.240855 1.320478 −28.203 <×2 10−16

n 0.416443 0.007034 59.207 <×2 10−16

b 87.672322 12.141991 7.221 7.42× 10−7

d 0.157279 0.002074 75.816 <×2 10−16

f 99.406672 13.520844 7.352 5.74× 10−7

g 1.979299 0.179995 10.996 1.12× 10−9

w—residence rate of the convective diffusivity flux, n—declining rate of the convective diffusivity flux, b—maximum passive diffusivity flux rate constant, d—declining rate of the passive diffusivity flux, f—maximum saturation diffusivity flux rate constant and, g—time at which the saturation diffusivity flux was half of f.

2006 T. Nemaura

Figure 4. Components of diffusivity flux in the locality of two neighbouring volume spaces. the movement out-of-solution particle. Diffusivity flux is symmetrical to conductivity flux in solution particle formation, saturation conductivity flux auxilliary advection    conductivity   flux convection conductivity   flux  A −−nt dt − ft κˆs ( x, t) − M = −wte +−b( e1 −) + , (15)  gt+ passive conductivity flux where, wnbd,,,, f , g are constants with respect to conductivity flux as in Equation (14) and M = 5.12193 . It is noted that auxilliary conductivity flux is translated conductivity flux by a constant (−M ) . One notices a range of interpretations that can be deduced from the numerics done for the saturation dynamics of a solution particle (Figure 5). The advective conductivity flux is negatively correlated to advective diffusivity flux. Furthermore, auxilliary advective conductivity flux is negatively correlated to advective diffusivity flux thus advective diffusivity (see Equation (8)). The striking observation is how all the other variables relate similarly to the two variables that of concentration (x) and saturation movement (F( xt, )) (Figure 5) ( corr( x, F( x , t)) = 0.95 ).

4. Discussion

This work proposes possible markers in the transportation of a drug which are conductivity and diffusivity. The partial differential equation derived has been shown to model the advection component of diffusivity and conductivity. A new development is inferred where the movement diffusivity flux is shown to be a function of four primary components. The primary components have been identified as

2007 T. Nemaura

Asκ , Figure 5. Summary scatter time-matched plot of the relationship between the following variables txF,,( xt ,,) κρˆsI( xt ,,) ( xt ,) and D( xt, ) for patient P informed by Equations (9), (10) and (11).

advection, saturation, convection and passive. The total movement diffusivity flux in the system is zero for any given concentration at a given time. The two movement diffusivity fluxes convection and passive are inferred to exert their influence in the peripheral volume system. However, saturation and advection predominantly exerts influence in the central volume space. The increase in advective diffusivity flux shows a corresponding reduction of advective conductivity, this is postulated from the negative correlation that these two parameters share in this system. It has been shown that, not only concentration affect diffusivity and conductivity, but also the state of the neighbourhood. Additionally, there are potential factors that could be involved in the processes of transportation, absorption and elimination of the drug. The advective diffusivity of the drug has partly negative values. Furthermore, it is not a constant and the regions of negative advective diffusivity are relatively very high as compared to positive diffusivity in the volume spaces. Initially, the negative values of advective diffusivity indicate that the peripheral volume space has relatively higher levels of flow. The drug is flowing more in the peripheral space. Thus the drug is quickly available to the peripheral space initially and furthermore is “concentrating” in this

2008 T. Nemaura

space [8]. This could potentially have beneficial or negative effects depending on what is being transported and the individual’s reaction to it. This result led to the proposal of the potential effect on CNS adverse drug reactions. There is increased transportation in the peripheral regions in the initial stages. Other researchers have found that the ma- jority of efavirenz-induced CNS adverse drug reactions appear early even after a single dose [9]. Diffusivity is one parameter that could potentially be affected with the presence of the residual drug in the system (from previous dose) in the volume spaces. This work shows that transportation consistently alters in the human body and there are differences in the two neighbouring volume spaces. A relatively controlled state is achieved during the terminal phase. The solution particle is “concentrating” in the central volume space. Inference on in-vivo release can be made using models developed. The interpretation of kinetic conductivity developed here allowed study of diffusivity. The work made use of data informed directly from the developed PK/PD models [6] [7]. Other researchers have proposed models to deal with complications involving concentration, time and location dependence on diffusivity. As a result, Fick’s formalisation has been thought to be less effective [10]. However, this work recast Fick’s law with the aid of secondary saturation transportation that has been found to be positively correlated to concentration. In place of the location variable it uses concentration and retains the time variable. This system enables the study of characterisation of diffusion and conduction. It proposes a new space that can be used to further our understanding of the phenomenological approach of the Fick’s law [11]. Diffusivity and conductivity of drugs in patients finds its use in studying drug release kinetics. This is important to the successful design of polymeric delivery systems bear- ing in mind that in-vivo release models are laborious and costly [12]. Using the new variable space proposed experiments can be developed to aid our understanding of diffusion across a range of applications [4] [11]. The results proposed here have implications in the way gradient induced diffusion could be handled mathematically. Currently, diffusivity and conductivity are investigated using the location and time as independent variables. This work proposes a different approach in investigating them in-vivo. Diffusivity is symmetrical to conductivity in solution particle formation. These two processes occur concurrently. The pro-gradient-driven-movement entities of a solution particle with respect to conductivity and diffusivity are convective and passive. While, anti-gradient-driven-movement entities are saturation and advection. This analysis is inferred from the characterisation of a solution particle in Nemaura (2015) [6]. The derivation of the phenomenon observed, points to advection as a significant primary entity. With the aid of the characterisation of a saturation kinetic flow (strictly positively correlated relation to concentration), one can use this to infer the behaviour of the solution particle in relation to its neghbourhood through advective diffusivity. This work shows the existence of a model which is deterministic and describes the behaviour of how a solution particle gain/lose movement relative to its local neighbourhood. The

2009 T. Nemaura

gain of movement is synonymous with conductivity and loss of movement is synonymous with diffusivity. The two advection movement fluxes are symmetric about half the initial value of advection conductivity (M/2) and also all the other subsequent corresponding components in the 24 h.

Acknowledgements

The author would like to thank the following; C. Nhachi, C. Masimirembwa, and G. Kadzirange, AIBST and The College of Health Sciences, University of Zimbabwe.

References [1] Mehrer, H. and Stolwijk, N.A. (2009) Heroes and Highlights in the History of Diffusion. Diffusion Fundamentals, 11, 1-32. [2] Narasimhan, T.N. (1999) Fourier’s Heat Conduction Equation: History, Influence and Con- nections. Reviews of Geophysics, 37, 151-172. http://dx.doi.org/10.1029/1998RG900006 [3] Feldman, J. (2007) The Heat Equation (One Space Dimension). http://www.math.ubc.ca/feldman/m267/heat1d.pdf [4] Crank, J. (1975) The Mathematics of Diffusion. 2nd Edition, Oxford University Press, Ox- ford. [5] De Kee, D., Liu, Q. and Hinestroza, J. (2005) Viscoelastic (Non-Fickian) Diffusion. The Canadian Journal of Chemical Engineering, 83, 913-929. http://dx.doi.org/10.1002/cjce.5450830601 [6] Nemaura, T. (2015) Modeling Transportation of Efavirenz: Inference on Possibility of Mixed Modes of Transportation and Kinetic Solubility. Frontiers in Pharmacology http://dx.doi.org/10.3389/fphar.2015.00121 [7] Nemaura, T. (2014) Projections of Pharmacokinetic Parameter Estimates from Middose Plasma Concentrations in Individuals on Efavirenz; a Novel Approach. African Journal of Pharmacy and Pharmacology, 8, 929-952. [8] Argyrakis, P., Chumak, A.A., Maragakis, M. and Tsakiris, N. (2009) Negative Diffusion Coefficient in a Two-Dimensional Lattice-Gas System with Attractive Nearest-Neighbor Interactions. Physical Review B, 80, 1-7. http://dx.doi.org/10.1103/PhysRevB.80.104203 [9] Apostolova, N., Funes, H.A., Blas-Garcia, A., Galindo, M.J., Alvarez, A. and Esplugues, J.V. (2015) Efavirenz and the CNS: What We Already Know and Questions That Need to Be Answered. Journal of Antimicrobial Chemotherapy, 70, 2693-708. http://dx.doi.org/10.1093/jac/dkv183 [10] Petropoulos, J.H., Sanopoulou, M. and Papadokostaki, K.G. (2009) Beyond Fick: How Best to Deal with Non-Fickian Behavior in a Fickian Spirit. Diffusion Fundamentals, 11, 1-21. [11] Philibert, J. (2005) One and a Half Century of Diffusion: Fick, Einstein, Before and Beyond. Diffusion Fundamentals, 2, 1-10. [12] Fu, Y. and Kao, W. J. (2010) Drug Release Kinetics and Transport Mechanisms of Non- Degradable and Degradable Polymeric Delivery Systems. Expert Opinion on Drug Delivery, 7, 429-444. http://dx.doi.org/10.1517/17425241003602259

2010 Journal of Applied Mathematics and Physics, 2016, 4, 2011-2030 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

Fatheah Ahmad Alhendi1, Aisha Abdullah Alderremy2

1Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia 2Department of Mathematics, King Khaild University, Abha, Kingdom of Saudi Arabia

How to cite this paper: Alhendi, F.A. and Abstract Alderremy, A.A. (2016) Numerical Solu- tions of Three-Dimensional Coupled Burg- In this paper, we found the numerical solution of three-dimensional coupled Burg- ers’ Equations by Using Some Numerical ers’ Equations by using more efficient methods: Laplace Adomian decomposition Methods. Journal of Applied Mathematics method, Laplace transform homotopy perturbation method, variational iteration and Physics, 4, 2011-2030. http://dx.doi.org/10.4236/jamp.2016.411201 method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and Received: October 6, 2016 accuracy of these methods and they reduce the size of computation without the re- Accepted: November 11, 2016 strictive assumption to handle nonlinear terms and it gives the solutions rapidly. Published: November 14, 2016

Copyright © 2016 by authors and Keywords Scientific Research Publishing Inc. This work is licensed under the Creative Three-Dimensional Coupled Burgers’ Equations, Laplace Transform, Commons Attribution International Adomian Decomposition, Homotopy Perturbation, Variational Iteration Method License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access

1. Introduction

The Burgers Equation was first presented by Bateman [1] and treated later by J. M. Burgers (1895-1981) then it is widely named as Burgers’ Equation [2]. Burgers’ Equa- tion is nonlinear partial differential equation of second order which is used in various fields of physical phenomena such as boundary layer behaviour, shock weave formation, turbulence, the weather problem, mass transport, traffic flow and acoustic transmission [3] [4]. In addition, coupled Burgers’ Equations has played an important role in many physical applications such as hydrodynamic turblence, vorticity transport, skock wave, dispersion in porous media and wave processes. In particular, the three-dimentional coupled Burgers’ Equations are important in large scale structure formation in the un-

DOI: 10.4236/jamp.2016.411201 November 14, 2016 F. A. Alhendi, A. A. Alderremy

iverse [5] [6]. In order to make a great application for burgers’ Equations, many researchers have been interested in solving it by various techniques. Analytic solution of one-dimensional Burgers’ Equation is got by many standard methods such as Backland transformation method, differential transformation method and tanh-coth method [6], while an analytical solution of two-dimensional coupled Burgers’ Equations is first presented by Fletcher using the Hopf-Cole transformation [7] and an analytical solution of three-dimensional coupled Burgers’ Equations is derived by Srivastava et al using the variable of separable and Hopf-Cole transformation [6]. The finite difference, finite element, spectral methods, Adomian decomposition method, the variational iteration method, homotopy perturbation method HPM and Eulerian-Lagrangian method, etc. gave an numerical solution of one- and two-dimentional Burgers’ Equations [3] [8]- [15]. Shukla et al are proposed a numerical solutions of three-dimensional coupled viscous Burgers’ Equations by using a modified cubic B-spline differential quadrature method [5]. The motive of this paper is to find the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficiently methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, the variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. We consider three-dimensional couple Burgers’ Equations as the following [6]:

∂u ∂ u ∂ u ∂ u1  ∂∂∂222 uuu + + + = ++ uvw 222, ∂∂∂t x y ∂ zR∂∂∂xyz ∂v ∂ v ∂ v ∂ v1  ∂∂∂222 vvv + ++ = ++ uvw 222, (1) ∂∂∂t x y ∂ zR∂∂∂xyz ∂w ∂∂ ww ∂ w1  ∂∂∂222 www + ++ = ++ uvw 222. ∂∂∂t x y ∂ zR∂∂∂xyz

with the initial conditions:

u01= u( xyz,,,0) = g( xyz ,,) ,

v02= v( xyz,,,0) = g( xyz ,,) ,( xyz ,,)∈Ω (2)

w03= wxyz( ,,,0) = g( xyz ,,) .

and the boundary conditions:

u( xyzt,,,) = f1 ( xyzt ,,,) ,

v( xyzt,,,) = f2 ( xyzt ,,,) , xyz ,,∈Γ , t > 0 (3)

w( xyzt,,,) = f3 ( xyzt ,,,) .

where Ω={( xyz,,) | a1≤ x ≤ b 12 , a ≤ y ≤ b 23 , a ≤ z ≤ b 3} and Γ is its boundary,

u( xyzt,,,) and v( xyzt,,,) are the velocity components to be determined, g1, g2, g3,

f1, f2 and f3 are known functions and R is the Reynolds number.

2012 F. A. Alhendi, A. A. Alderremy

This paper is organized into five sections. Each method is in one section. We showed an overview of these methods then explained methodology and finally illustrated the methods by using examples. It is clear to see that numerical methods are reasonably in good covenant with the exact solution.

2. Laplace Adomian Decomposition Method

The Laplace transform LT is an integral transform discovered by Pierre-Simon Laplace. LT is a very powerful technique for solving ordinary and partial differential Equations, which transforms the original differential equation into an elementary algebraic expression [16]. Definition: the Laplace transform of ft( ) where t ≥ 0 , denoted by ft( ) = Fs( ) , is given by:

∞ − ft( ) = Fs( ) = est ft( ) d. t ∫0

Adomian decomposition method ADM is proposed in 1980 by Grorge Adomian. ADM has been encountered much attention in recent years in applied mathematics in general and in solving Burgers’ Equations in particular. A wide class of linear and nonlinear, ordinary and partial differential Equations solved easily and more accurately via ADM. It has successfully used to handle most type of physical models of partial differential Equations without dependence on linearization or any restrictive assumptions that may change physical behavior of the models under study [3] [17] [18] [19]. ADM consists of decomposing the unknown functions of any equations into a sums of an infinite number of components defined by the decomposition series:

∞

uxx( 12,,, xni) = ∑ u( xx12 ,,, x n) (4) i=0 where uin( xx12,,, x) , i≥ 0 are to be determined in recursive manner. The nonlinear term Nu( ) can be expressed by an infinite series of Adomian polynomial Am which is given as:

∞

Nu( ) = ∑ Aui ( 012,,, uu) , (5) i=0 where Adomian polynomial Am using the form

k ∞ 1d i Ami=  Nuλ . (6) λ k ∑ k ! d i=0 λ=0

There is a growing interest of researchers has been to study the Adomian decomposition method ADM [3] [20] [21] [22] [23]. In this work, we will use Laplace transform- Adomian decomposition method (LT-ADM) introduced by Khuri khuri [24]. Some time it is known as Laplace Adomian decomposition method (LADM). This numerical technique explains how the Laplace transform may be used to approximate the solutions of the nonlinear partial differential equations (PDEs) including Burgers’ Equa- tions with the decomposition method [11] [24] [25] [26].

2013 F. A. Alhendi, A. A. Alderremy

2.1. Methodology of LT-ADM for Three-Dimensional Couple Burgers’ Equations

We consider the system (1) and apply LT on both side of it: ∂u ∂ u ∂ u ∂ u1  ∂∂∂222 uuu +uvw + + =222 ++ , ∂∂∂t xy ∂ z R∂∂∂xyz ∂v ∂ v ∂ v ∂ v1  ∂∂∂222 vvv +uvw + + =222 ++ , (7) ∂∂∂t xy ∂ z R∂∂∂xyz ∂w ∂ w ∂ w ∂ w1  ∂∂∂222 www +uvw + + =222 ++ . ∂∂∂t xy ∂ z R∂∂∂xyz

We can write (7) as: ∂u ∂ u ∂ u ∂ u1  ∂∂∂222 uuu =− + + + ++  uvw 222, ∂t ∂∂ x y ∂ zR∂∂∂xyz ∂v ∂ v ∂ v ∂ v1  ∂∂∂222 vvv =− + + + ++  uvw 222, (8) ∂t ∂∂ x y ∂ zR∂∂∂xyz ∂w ∂ w ∂ w ∂ w1  ∂∂22 ww∂2w =−+ + + ++  uvw 222 . ∂t ∂∂ x y ∂ zR∂∂xy∂z

By using (2) we get: ∂u ∂ u ∂ u1  ∂∂∂222 uuu −= − + + + + + su( x,,, y z s) u0   u v w 222, ∂∂x y ∂ zR∂∂∂xyz ∂v ∂ v ∂ v1  ∂∂∂222 vvv −= − + + + + + sv( x,,, y z s) v0   u v w 222, (9) ∂∂x y ∂ zR∂∂∂xyz ∂w ∂ w ∂ w1  ∂∂22 ww∂2w −=− + + + + sw( x,,, y z s) w0   u v w 2 22+ . ∂∂x y ∂ zR∂∂x yz∂

Using inverse Laplace transform on both sides of (9), we have 222 −1 11∂u ∂ u ∂ u ∂∂∂ uuu u( xyzt,,,) =+ u0  −u + v + w +222 ++ , s∂∂ x y ∂ zR∂∂∂xyz 222 −1 11∂v ∂ v ∂ v ∂∂∂ vvv v( xyzt,,,) =+ v0  −u + v + w +222 ++ , (10) s∂∂ x y ∂ zR∂∂∂xyz 222 −1 1 ∂w ∂w ∂ w1  ∂∂∂ www w( xyzt,,,) =+− w0 u +vw + +222 ++ . sx∂ ∂∂y zR∂∂∂xyz

from (4) we can write the solutions as: ∞

u( xyzt,,,) = ∑ un ( xyzt ,,,) , n=0 ∞

v( xyzt,,,) = ∑ vn ( xyzt ,,,) , (11) n=0 ∞

w( xyzt,,,) = ∑ wn ( xyzt ,,,) . n=0

2014 F. A. Alhendi, A. A. Alderremy

Now, we assume the nonlinear terms as: ∂∂uu∞∞ ∂ u ∞ u= ∑∑ A(1,nn) ( u), v= A(2, ) ( uv,,) w= ∑ A(3, n) ( uw,,) ∂∂xynn=00= ∂ z n= 0 ∂v∞ ∂∂ vv ∞∞ u= ∑ B(1,n) ( uv,,) v= ∑∑ B(2, nn) ( v), w= B(3, ) ( vw,,) (12) ∂xn=0 ∂∂ yz nn= 00= ∂∂ww∞∞ ∂ w ∞ u= ∑∑ C(1,nn) ( uw,,) v= C(2, ) ( uw,,) w= ∑ C(3, n) ( w). ∂∂xynn=00= ∂ z n= 0 where Ain( ,,) Bin( ,) and C( ib,) ; i := 1, 2, 3 are the Adomian polynomial given as (6). From (11) and (12) into (10), we have:

∞ ∞∞ ∞ ∞ −1 11 ∑unn( xyzt,,,) = u0 + −∑∑∑AAA(1,nnn) +(2, ) +(3, ) +∆ ∑u , n=0 sRnn=00= n= 0 n= 0 ∞ ∞∞ ∞ ∞ −1 11 ∑vnn( xyzt,,,) =+ v0  −∑∑∑BBB(1,nnn) +(2, ) +(3, ) +∆ ∑v , (13) n=0 sRnn=00= n= 0 n= 0 ∞ ∞∞ ∞ ∞ −1 1 1 ∑ wn ( xyzt,,,) = w0 + −∑∑∑CCC(1,nnn) +(2, ) +(3, ) +∆ ∑ wn . n=0 s nn=00= n= 0R n= 0 ∂∂∂222 where ∆= + + . Then, using (2), (6), (12) into (13), we have ∂∂∂xyz222

uu=0123 ++++ u u u ,

vv=0123 + v + v + v +…, (14)

ww=0123 ++++ w w w .

2.2. Example

Consider (1) if the exact solution is [5]: − −2 1+ cos( xyz) sin( ) sin( ) e t u* xyzt,,, = , ( ) −t R 1++x sin( xyz) sin( ) sin( ) e − −2 sin( x) cos( yz) sin( ) e t v* xyzt,,, = , ( ) −t (15) R 1++x sin( xyz) sin( ) sin( ) e − −2 sin( xyz) sin( ) cos( ) e t w* xyzt,,, = . ( ) −t R 1++x sin( xyz) sin( ) sin( ) e To solve this example by LT-ADM, we follow the methodology which discussed in subsection (2.1). The accuracy of LT-ADM for the three-dimensional coupled Burgers’ Equations agrees very well with the exact solution and absolute errors are very small for the current choice of x, y, z and t. The results are shown in Tables 1-3 for R = 100, x = 0.1, y = 0.02 and z = 0.03.

3. Laplace Transform Homotopy Perturbation Method

Homotopy perturbation method HPM was first proposed by He. HPM is combination of traditional perturbation method and homotopy method. The important advantage of HPM is that the nonlinear term can be easily handled. It is easy to calculate the solution

2015 F. A. Alhendi, A. A. Alderremy

Table 1. The absolute error (AEs) of u( xyzt,,,) by LT-ADM for example 2.2.

t u* ( xyzt,,,) u( xyzt,,,) uu* −

0 −0.01819168001 −0.01819168001 0

− 0.002 −0.01819166033 −0.01819167942 1.909× 10 8

− 0.004 −0.01819164064 −0.01819167883 3.819× 10 8

− 0.006 −0.01819162103 −0.01819167823 5.720× 10 8

0.008 −0.01819160143 −0.01819167764 7.621× 10−8

0.010 −0.01819158189 −0.01819167705 9.516× 10−8

Table 2. The AEs of v( xyzt,,,) by LT-ADM for example 2.2.

t v* ( xyzt,,,) v( xyzt,,,) vv* −

0 −0.00005443257070 −0.00005443257070 0

0.002 −0.00005432382028 −0.00005442930556 1.0548528× 10−7

0.004 −0.00005421528708 −0.00005442604171 2.1075463× 10−7

0.006 −0.00005410697072 −0.00005442277913 3.1580841× 10−7

0.008 −0.00005399887076 −0.00005441951783 4.2064707× 10−7

0.010 −0.00005389098678 −0.00005441625782 5.2527104× 10−7

Table 3. The AEs of w( xyzt,,,) by LT-ADM for example 2.2.

t w* ( xyzt,,,) w( xyzt,,,) ww* −

0 −0.00003628233106 −0.00003628233106 0

− 0.002 −0.00003620984286 −0.00003628015467 7.031181× 10 8

− 0.004 −0.00003613749944 −0.00003627797914 1.4047970× 10 7

0.006 −0.00003606530058 −0.00003627580447 2.1050389× 10−7

0.008 −0.00003599324596 −0.00003627363065 2.8038469× 10−7

0.010 −0.00003592133528 −0.00003627145770 3.5012242× 10−7

with this method. Linear or nonlinear ODEs and PDEs are studied successfully by using LT-HPM [17], [27] [28] [29] [30] [31]. To figure out how HPM works [32], consider n-dimensional Burgers’ equation

2016 F. A. Alhendi, A. A. Alderremy

n ∂∂uuii +∑uji =∆=µ ui, 1, 2, , n (16) ∂∂txj=1 j

We construct the following homotopy:

∂∂uu n ∂u (ik, ) ( i,0) ∂ui (ik, ) (1−p)  − + pu +∑ jk,,−∆µ uik =0. (17) ∂∂ ∂( ) ∂ ( ) tt tj=1 xj or

n ∂∂uu(ik, ) ( i,0)  ∂ u(i,0) ∂ u(ik, ) − +pu + −∆µ u =0. (18) ∂∂ ∂∑ ( jk,,) ∂ (ik) tt tj=1 xj

∂∂22 ∂ 2 = = where ∆=22 + + + 2, in1, 2, , , k 1, 2, , p ∈[0,1] is an embedding xx12 xn parameter while uu(1,0) ,( 2,0) ,, u(n,0) are initial approximations of (16). Assume the solution of (16) has the form

∞ =  = uij∑ pu(i,) ( x, t) , i , j 1, 2, , n . (19) =0

Now, substituting ui from Equation (19) in Equation (18) and comparing coefficients of terms with identical powers of p, we get:

∂∂uu(ii,0) ( ,0) p0 :0−= ∂∂tt ∂∂uu 1 (ii,0) ( ,0) pu:,+( ji,0) −∆µ u( ,0) ∂∂txj ∂u 2 (i,1) pu: ( ji,1) −∆µ u( ,1) (20) ∂x j ∂u 3 (i,2) pu:,( ji,2) −∆µ u( ,2) ∂x j 

The solution is

uxtij( ,.) = u(i,0) +++ u( ii ,1) u( ,2 )  (21)

In [33], Aminikhah presented LT-HPM to solve nonlinear Blasius' viscous flow equation. In [34], some application examples of LT-HPM for nonlinear ODEs with Dirich- let, mixed, and Neumann boundary conditions were presented. It has used to solve PDEs then to solve Burgers’ Equations [12] [14] [30] [31] [34]. Now, we are going to study LT-HPM for (1) as following.

3.1. Methodology of LT-HPM for Three-Dimensional Couple Burgers’ Equations

We consider the system (1) and apply HPM on it. We construct the following homotopy:

2017 F. A. Alhendi, A. A. Alderremy

∂U∂u0  ∂∂∂ UUU ∂ U1 (1−p) − + p + UVW + + −∆ U =0, ∂∂t t ∂ t ∂ x ∂ y ∂ zR

∂V∂v0  ∂∂∂ VVV ∂ V1 (1−p) − + p + UVW + + −∆ V =0, (22) ∂∂t t ∂ t ∂ x ∂ y ∂ zR

∂W∂w0  ∂∂∂ WWW ∂ W1 (1−p) − + p + UVW + + −∆ W =0. ∂∂t t ∂ t ∂ x ∂ y ∂ zR

where uv00, and w0 are the initial approximation values of (1), and UV, and W have the following forms:

∞∞ ∞ ii i U= ∑∑∑ puii( xyzt,,,) , V= pv( xyzt,,,) , W= pw i( xyzt,,,) . (23) ii=00= i= 0

we can write (22) as:

∂U∂∂uu∂∂UU ∂ U1 =00 −p − pU + V + W −∆ U , ∂∂t t ∂ t ∂ x ∂ y ∂ zR ∂V∂∂vv∂∂VV ∂ V1 =00 −p − pU + V + W −∆ V , (24) ∂∂t t ∂ t ∂ x ∂ y ∂ zR ∂W∂∂ww∂∂WW ∂ W1 =00 −p + pU + V + W −∆ W . ∂∂t t ∂ t ∂ x ∂ y ∂ zR

Applying LT on both side of (24)

∂U∂∂uu∂∂UU ∂ U1 = 00 −p − pU + V + W −∆ U, ∂t ∂∂ t t ∂∂ x y ∂ zR ∂V∂∂vv∂∂VV ∂ V1 = 00 −p − pU + V + W −∆ V, (25) ∂t ∂∂ t t ∂∂ x y ∂ zR ∂W∂∂ww∂∂WW ∂ W1 = 00 −p − pU + V + W −∆ W. ∂t ∂∂ t t ∂∂ x y ∂ zR

By applying inverse of LT on (25), we get U( xyzt,,,)  −1 11 ∂∂uu00∂∂UU ∂ U  =u0 +  − p − pU + V + W −∆ U, s∂∂ t t ∂∂ x y ∂ zR V( xyzt,,,)  −1 11 ∂∂vv00∂∂VV ∂ V  (26) =v0 +  − p − pU + V + W −∆ V, s∂∂ t t ∂∂ x y ∂ zR W( xyzt,,,)  −1 1  ∂w0 ∂w0 ∂∂WW ∂ W1  =w0 +  −p − pU + V + W −∆ W. s ∂t∂ t ∂∂ x y ∂ zR

By substituting U, V and W from (23) in (26) and comparing coefficients of terms with identical powers of p:

2018 F. A. Alhendi, A. A. Alderremy

  −1 1  ∂u0 u00( xyz,,) = u +  ,  st∂   01− 1  ∂v0  p: v00( xyz ,,) = v +  ,  st∂  1  ∂w w xyz,, =−1 w + 0 ,  00( )     st∂   −1 11∂∂∂uuu000 ∂ u 0 u1( xyzt,,,) =  −−u00 − v − w 0 +∆u0 ,  s∂∂∂ t xy ∂ zR   1 −1 11∂∂∂vvv000 ∂ v 0  p : v1( xyzt,,,) =  − −u00 − v − w 0 +∆ v 0),  s∂∂∂ t x y ∂ zR (27)  11∂∂∂www ∂ w  =−1 −000 − − − 0 +∆ w1( xyzt,,,)   u00 v w 0 w 0).  s∂∂∂ t x y ∂ zR   −1 11∂∂uu11∂∂uu00 ∂ u 1 ∂ u 0 u2 ( xyz,,,,t) =  −−−−− uuvvww01 01 0 − 1 +∆ u 1  s∂∂∂∂ x x y y ∂ z ∂ zR   2  −1 11∂∂vv11∂∂vv00 ∂ v 1 ∂ v 0 p : v2( xyzt,,,) =  −−−−−u01 u v 01 v w 0 − w 1 +∆ v 1,  s∂∂∂∂ x x y y ∂ z ∂ zR  1 ∂w ∂∂ww∂∂ww ∂ w1  =−1 − 11−−−−00 1 − 0 +∆ w20( xyzt,,,)  u uvvww1 01 0 1 w 1.  s ∂∂∂∂x x y y ∂ z ∂ zR 

Finally, the approximate solutions are:

uu=012 +++ u u ,

vv=012 +++ v v , (28)

ww=012 +++ w w .

3.2. Example

To solve (1) by LT-HPM, we follow the methodology which discussed in subsection (3.1). The results in Tables 4-6 show that LT-HPM is more effective and high accuracy when compared with the exact solutions for R = 100, x = 0.1, y = 0.02 and z = 0.03.

Table 4. The AEs of u( xyzt,,,) by LT-HPM for example 3.2.

∗ t u( xyzt,,,) u( xyzt,,,) uu* −

0 −0.01819168001 −0.01819168001 0

0.002 −0.01819166033 −0.01819168060 2.027× 10−8

0.004 −0.01819164064 −0.01819168119 4.055× 10−8

0.006 −0.01819162103 −0.01819168179 6.076× 10−8

0.008 −0.01819160143 −0.01819168238 8.095× 10−8

0.010 −0.01819158189 −0.01819168297 1.0108× 10−7

2019 F. A. Alhendi, A. A. Alderremy

Table 5. The AEs of v( xyzt,,,) by LT-HPM for example 3.2.

∗ t v( xyzt,,,) v( xyzt,,,) vv* −

0 −0.00005443257070 −0.00005443257070 0

0.002 −0.00005432382028 −0.00005443583647 1.1201619× 10−7

− 0.004 −0.00005421528708 −0.00005443910222 2.2381514× 10 7

0.006 −0.00005410697072 −0.00005444236796 3.3539724× 10−7

0.008 −0.00005399887076 −0.00005444563368 4.4676292× 10−7

− 0.010 −0.00005389098678 −0.00005444889938 5.5791260× 10 7

Table 6. The AEs of w( xyzt,,,) by LT-HPM for example 3.2.

∗ t w( xyzt,,,) w( xyzt,,,) ww* −

0 −0.00003628233106 −0.00003628233106 0

− 0.002 −0.00003620984286 −0.00003628450787 7.466501× 10 8

0.004 −0.00003613749944 −0.00003628668465 1.4918521× 10−7

0.006 −0.00003606530058 −0.00003628886141 2.2356083× 10−7

− 0.008 −0.00003599324596 −0.00003629103816 2.9779220× 10 7

0.010 −0.00003592133528 −0.00003629321487 3.7187959× 10−7

4. The Variational Iteration Method

The variational iteration method (VIM) was proposed by Ji-Huan He in 1997 [35] [36] [37] [38]. The VIM has been applied successfully for the most important problems in physically important phenomena including Burgers’ Equation [13] [39] [40] [41] [42] The VIM solve linear or nonlinear ODEs and PDEs without needing small parameter or linearization and by few iterations lead to high accurate solutions. To explain the basic concepts of the VIM, we consider n-dim of burgers’ Equation (16). We can write the correction functional for it as [32]:

t n ∂∂uuii uunni+1 = +λξ( )  +∑uj −∆µ u id. ξ (29) ∫0 ∂∂ξ j=1 x j

where in= 1, 2, , , u= ux( i ,ξ ) , λi are the general Lagrangian multipliers which

can be find via variational theory, ui are restricted variation which means δui = 0 . The solution is given by

uxtij( ,) = lim u( xtj ,) , j= 1, 2, , n . (30) n→∞ (in, )

2020 F. A. Alhendi, A. A. Alderremy

4.1. Methodology of VIM for Three-Dimensional Couple Burgers’ Equations

Consider the system (1), we can construct a the following correction functional:

t ∂∂∂uuu ∂ u 1 = +λξ + + + −∆ unn+11( xyzt,,,) u ∫ u v w u d, 0 ∂ξ ∂∂x y ∂ zR t ∂∂∂vvv ∂ v 1 =+λξ + + + −∆ vnn+12( xyzt,,,) v ∫ u v w v d, (31) 0 ∂ξ ∂∂x y ∂ zR t ∂∂∂www ∂ w 1 = +λξ + + + −∆ wnn+13( xyzt,,,) w ∫ u v w w d. 0 ∂∂∂ξ x y ∂ zR

λλλ123= = = −1. Then, the iteration formulae are given as:

t ∂∂∂uuu ∂ u1 = − + + + −∆ ξ unn+1 ( xyzt,,,) u∫  u v w u d, 0 ∂ξ ∂∂x y ∂ zR

t ∂v ∂∂ vv ∂ v1 =− + + + −∆ ξ vnn+1 ( xyzt,,,) v∫  u v w v d, (32) 0 ∂ξ ∂∂x y ∂ zR

t ∂∂∂www ∂ w1 = − + + + −∆ ξ wnn+1 ( xyzt,,,) w∫  u v w w d. 0 ∂ξ ∂∂x y ∂ zR

4.2. Example

To solve (1) by VIM, we follow the methodology which discussed in subsection (4.1). The results in Tables 7-9 are shown that the efficiency and accuracy of the VIM. It reduces the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

5. Variational Iteration Decomposition Method

The variational iteration decomposition method (VIDM) is technique combination of two the most powerful mathematical methods for solving a large class of differential Equations, namely variational iteration method and Adomian decomposition method. In 2007 VIDM has been used to solve quadratic Riccati differential Equation problems

Table 7. The AEs of u( xyzt,,,) by VIM for example 4.2.

∗ t u( xyzt,,,) u( xyzt,,,) uu* −

− 0 −0.01819168001 −0.01819168085 8.4× 10 10

− 0.002 −0.01819166033 −0.01819168057 2.024× 10 8

0.004 −0.01819164064 −0.01819167882 3.818× 10−8

0.006 −0.01819162103 −0.01819167984 5.881× 10−8

0.008 −0.01819160143 −0.01819167803 7.660× 10−8

0.010 −0.01819158189 −0.01819167741 9.552× 10−8

2021 F. A. Alhendi, A. A. Alderremy

Table 8. The AEs of v( xyzt,,,) by VIM for example 4.2.

∗ t v( xyzt,,,) v( xyzt,,,) vv* −

− 0 −0.00005443257070 −0.00005443257572 5.02× 10 12

− 0.002 −0.00005432382028 0.00005442930575 1.0548547× 10 7

− 0.004 −0.00005421528708 0.00005442603014 2.1074306× 10 7

0.006 −0.00005410697072 0.00005442276920 3.1579848× 10−7

0.008 −0.00005399887076 0.00005441950206 4.2063130× 10−7

0.010 −0.00005389098678 0.00005441623773 5.2525095× 10−7

Table 9. The AEs of w( xyzt,,,) by VIM for example 4.2.

∗ t w( xyzt,,,) w( xyzt,,,) ww* −

0 −0.00003628233106 −0.00003628232124 9.82× 10−12

− 0.002 −0.00003620984286 −0.00003628012665 7.028379× 10 8

0.004 −0.00003613749944 −0.00003627797169 1.4047225× 10−7

0.006 −0.00003606530058 −0.00003627578326 2.1048268× 10−7

− 0.008 −0.00003599324596 −0.00003627365724 2.8041128× 10 7

0.010 −0.00003592133528 −0.00003627142962 3.5009434× 10−7

[43]. Noor et al. [44] [45] used this method for solving eighth-order boundary value problems, sixth-order boundary value problems and higher dimensional initial boundary value problems. Grover and Tomer solved twelfth order boundary value problems by using VIDM. In 2013, the fractional Riccati differential Equation is solved by VIM by using Adomian polynomials for nonlinear terms [46]. To illustrate the general concept of VIDM by using (5), (6) in (29), hence, we have the correction functional for (16) as:

t ∂u n uu= +λi +A −∆ µξ u d. (33) nni+1 ∫0 ∑ j i ∂ξ j=1

We are solved three-dimensional coupled Burgers’ Equations by using VIDM as following.

5.1. Methodology of VIDM for Three-Dimensional Couple Burgers’ Equations

We consider the system (1). Next, by using (11) and (12) in (32), we obtain the iterative scheme to find the approximate solutions by VIDM as following:

2022 F. A. Alhendi, A. A. Alderremy

∞∞ ∞ t ∂u 1 u( xyzt,,,) = u − +A( u) + A( uv,) + A( uw,) −∆ u d,ξ nn+1 ∫0 ∑∑(1,nn) (2, ) ∑(3, n) ∂ξ nn=00= n= 0 R ∞ ∞∞ t ∂v 1 vxyztv( ,,,) =− + BuvBvBvwv( ,) +( ) +( ,) −∆ d,ξ nn+1 ∫0 ∑(1,n) ∑∑(2, nn) (3, ) (34) ∂ξ n=0 nn= 00= R ∞∞∞ t ∂w 1 w( xyzt,,,) =−+ w C( uw, ) + C( vw, ) +Cw( ) −∆ wd.ξ nn+1 ∫0 ∑∑∑(1,nn) (2, ) (3,n) ∂ξ nnn=000= = R

5.2. Example

To solve (1) by VIDM by using (34). The numerical results in Tables 10-12 show that VIDM is an effective and powerful method to find better results.

Table 10. The AEs of u( xyzt,,,) by VIDM for example 5.2.

∗ t u( xyzt,,,) u( xyzt,,,) uu* −

0 −0.01819168001 −0.01819168001 0

− 0.002 −0.01819166033 −0.01819167942 1.909× 10 8

0.004 −0.01819164064 −0.01819167883 3.819× 10−8

0.006 −0.01819162103 −0.01819167823 5.720× 10−8

− 0.008 −0.01819160143 −0.01819167764 7.621× 10 8

0.010 −0.01819158189 −0.01819167705 9.516× 10−8

Table 11. The AEs of v( xyzt,,,) by VIDM for example 5.2.

∗ t v( xyzt,,,) v( xyzt,,,) vv* −

0 −0.00005443257070 −0.00005443257070 0

0.002 −0.00005432382028 −0.00005442930556 1.0548528× 10−7

− 0.004 −0.00005421528708 −0.00005442604171 2.1075463× 10 7

0.006 −0.00005410697072 −0.00005442277913 3.1580841× 10−7

0.008 −0.00005399887076 −0.00005441951783 4.2064707× 10−7

− 0.010 −0.00005389098678 −0.00005441625782 5.2527104× 10 7

Table 12. The AEs of w( xyzt,,,) by VIDM for example 5.2.

∗ t w( xyzt,,,) w( xyzt,,,) ww* −

0 −0.00003628233106 −0.00003628233106 0

0.002 −0.00003620984286 −0.00003628015467 7.031181× 10−7

0.004 −0.00003613749944 −0.00003627797914 1.4047970× 10−7

− 0.006 −0.00003606530058 −0.00003627580447 2.1050389× 10 7

0.008 −0.00003599324596 −0.00003627363065 2.8038469× 10−7

0.010 −0.00003592133528 −0.00003627145770 3.5012242× 10−7

2023 F. A. Alhendi, A. A. Alderremy

6. Variational Iteration Homotopy Perturbation Method

The variational iteration homotopy perturbation method (VIHPM) is combination of two well-known methods, namely variational iteration method and homotopy perturbation method. VIHPM has been applied in [32] [47]-[52] for solving a large class of differential Equations. To illustrate the concept of VIHPM [32], we consider (16) and assume the solution of (16) has the form ∞ =  = = vi∑ pu(i,) ( xji, t) v , i , j 1, 2, , n , =0 ∞ (35) =  = = vj∑ pu( j,) ( xjj, t) v , i , j 1, 2, , n . =0 from (35), (16) can be written as: n ∂∂vvii +∑ vji =∆=µ vi, 1, 2, , n . (36) ∂∂txj=1 j

we can from the correction functional for(36) we can write

t n ∂vi vi+10= vp +λξ i( )  −∑ v ji +∆µ vd. ξ (37) ∫0 ∂ j=1 x j

where in= 1, 2, , , v= vx( i ,ξ ) , from (35) in (37) and by comparing the coefficients of like powers of p, we get

0 p: u(i,0) ( xj ,0) = fx ij( ,0) , ∂ux,ξ 1 t (i,0) ( j ) p:, u( xtji) =λξ( )  − u( x j, ξ) +∆µu( xj, ξ) d, ξ (ij,1) ∫0 ( ,0) ∂x ( i,0) j ∂ux,ξ 2 t (i,1) ( j ) p:, u( xtji) =λξ( )  − u( x j, ξ) +∆µu( xj, ξ) d, ξ (38) (ij,2 ) ∫0 ( ,1) ∂x ( i,1) j ∂ux,ξ 3 t (i,2) ( j ) p:, u( xtji) =λξ( )  − u( x j, ξ) +∆µu( xj,d ξ) ξ, (ij,3) ∫0 ( ,2) ∂x ( i,2) j  The approximate solutions are give by

uxtij( ,.) = u(i,0) +++ u( ii ,1) u( ,2 )  (39)

we used this method to solve three-dimensional coupled Burgers’ Equations as followed.

6.1. Methodology of VIHPM for Three-Dimensional Couple Burgers’ Equations

we consider the correction functional (31) with λλλ123= = = −1 by assuming that nn n ii i U= ∑∑∑ puii( xyzt,,,) , V= pv( xyzt,,,) , W= pw i( xyzt,,,) . (40) ii=00= i= 0

2024 F. A. Alhendi, A. A. Alderremy

by VIHPM, we have

2t 22∂ u01++ pu p u 2 +=−−++ u 0 p( u 01 pu p u 2 + ) ( u 01 ++ pu p u 2 +)dξ ∫0 ∂x

t 22∂ −p −+( v01 pv + p v 2 +) ( u 0 + pu 1 + p u 2 +)dξ ∫0 ∂y

t 22∂ −−+p( w0 pw 1 + p w 2 +) ( u 01 ++ pu p u 2 +)dξ ∫0 ∂z

t 1 2 −p ∆+( u01 pu + p u 2 +)d,ξ ∫0 R (41) 2t 22∂ v01++ pv p v 2 +=−−++ v 0 p( u 01 pu p u 2 +) ( v 01 ++ pv p v 2+)dξ ∫0 ∂x

t 22∂ −−++p( v01 pv p v 2 +) ( v 01 ++ pv p v 2 +)dξ ∫0 ∂y

t 22∂ −−+p( w0 pw 1 + p w 2 +) ( v 01 ++ pv p v 2 +)dξ ∫0 ∂z

t 1 2 −p ∆+( v01 pv + p v 2 +)d,ξ ∫0 R

2t 22∂ w01++ pw p w 2 +=−−++ w 0 p( u 01 pu p u 2 + ) ( w 01 ++ pw p w 2 +)dξ ∫0 ∂x

t 22∂ −p −+( v01 pv + p v 2 +) ( w 0 + pw 1 + p w 2 +)dξ ∫0 ∂y (42) t 22∂ −−++p( w01 pw p w 2 +) ( w 01 ++ pw p w 2 +)dξ ∫0 ∂z

t 1 2 −p ∆+( w01 pw + p w 2 +)d.ξ ∫0 R by comparing the coefficients of like power of p, we get

u00( xyz,,) = u , 0  p: v00( xyz , ,) = v , from initial conditions  w00( xyz,,) = w .

 t ∂∂uu ∂ u1 u( xyzt,,,) = u00 + v + w 0 −∆ u d,ξ  1 ∫0 00 0 0  ∂∂x y ∂ zR   t ∂∂vv ∂ v1 1 =00 + + 0 −∆ ξ p: v1( xyzt ,,,) ∫  u00 v w 0 v 0d,  0 ∂∂x y ∂ zR  t ∂∂ww∂w 1  =00 ++0 −∆ ξ (43) w1( xyzt,,,) ∫  u00 v w 0 w0 d.  0 ∂∂xy∂zR

 t ∂u∂∂uu ∂∂ uu ∂ u 1 u( xyzt,,,) = u1 ++++ u00 v v 11 w0 + w −∆ u d,ξ  2 ∫0 0110 1 0 1  ∂∂∂∂x x y y ∂ z ∂ zR   t ∂∂vv∂v ∂∂ vv ∂ v1 2 =11 ++++0 00 +1 −∆ξ p: v2( xyzt ,,,) ∫  u01 u v 01 v w 1 w 0 v 1d,  0 ∂∂∂∂x x y y ∂ z ∂ zR  t ∂ww∂∂ww∂∂ w ∂ w1  = 00++++11 0 + 1 −∆ξ w21( xyzt,,,) ∫  u uvvww01 0 1 0 w 1d.  0 ∂x ∂∂∂ x y y ∂ z ∂ zR 

2025 F. A. Alhendi, A. A. Alderremy

The approximate solutions are given by

u( xyzt,,,) = u012 +++ u u ,

v( xyzt,,,) = v012 +++ v v , (44)

w( xyzt,,,) = w012 +++ w w .

6.2. Example

To solve (1) by VIHPM, we follow the methodology discussed in subsection (6.1). The accuracy of VIHPM for the three-dimensional coupled Burgers’ Equations agrees very well with the exact solution and absolute errors are very small for the current choice of x, y, z and t. The result are shown in Tables 13-15 for R = 100, x = 0.1, y = 0.02 and z = 0.03.

7. Conclusion

In this work, these previous methods mentioned above have been successfully used for finding the solution of three-dimensional coupled Burgers’ Equations. The numerical

Table 13. The AEs of u( xyzt,,,) by VIHPM for example 6.2.

∗ t u( xyzt,,,) u( xyzt,,,) uu* −

0 −0.01819168001 −0.01819168001 0

0.002 −0.01819166033 −0.01819167942 1.909× 10−8

− 0.004 −0.01819164064 −0.01819167883 3.819× 10 8

0.006 −0.01819162103 −0.01819167823 5.720× 10−8

0.008 −0.01819160143 −0.01819167764 7.621× 10−8

− 0.010 −0.01819158189 −0.01819167705 9.516× 10 8

Table 14. The AEs of v( xyzt,,,) by VIHPM for example 6.2.

∗ t v( xyzt,,,) v( xyzt,,,) vv* −

0 −0.00005443257070 −0.00005443257070 0

− 0.002 −0.00005432382028 −0.00005442930491 1.0548463× 10 7

− 0.004 −0.00005421528708 −0.00005442603912 2.1075204× 10 7

0.006 −0.00005410697072 −0.00005442277329 3.1580257× 10−7

− 0.008 −0.00005399887076 −0.00005441950745 4.2063669× 10 7

− 0.010 −0.00005389098678 −0.00005441624161 5.2525483× 10 7

2026 F. A. Alhendi, A. A. Alderremy

Table 15. The AEs of w( xyzt,,,) by VIHPM for example 6.2.

∗ t w( xyzt,,,) w( xyzt,,,) ww* −

0 −0.00003628233106 −0.00003628233106 0

0.002 −0.00003620984286 −0.00003628015423 7.031137× 10−8

0.004 −0.00003613749944 −0.00003627797737 1.4047793× 10−7

− 0.006 −0.00003606530058 −0.00003627580049 2.1049991× 10 7

0.008 −0.00003599324596 −0.00003627362357 2.8037761× 10−7

0.010 −0.00003592133528 −0.00003627144664 3.5011136× 10−7 results are obtained for approximation and compared with the exact solutions and the results show that we achieve an excellent approximation to the actual solution of the equations by using only two iterations. The results show that these methods are powerful mathematical tools to solving a three-dimensional coupled Burgers’ Equation. In our work, we use the Maple to calculate approximate solutions in our systems by using those very efficient methods.

Acknowledgements

This paper was funded by King Abdulaziz City for Science and Technology (KACST) in Saudi Arabia. The authors therefore thank them for their full collaboration.

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2030 Journal of Applied Mathematics and Physics, 2016, 4, 2031-2037 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Numerical Solution of the Diffusion Equation with Restrictive Pade Approximation

Ahmet Boz*, Fevziye Gülsever

Departments of Mathematics, Dumlupinar University, Kütahya, Turkey

How to cite this paper: Boz, A. and Abstract Gülsever, F. (2016) Numerical Solution of the Diffusion Equation with Restrictive Pade The problem of solving the linear diffusion equation by a method related to the Re- Approximation. Journal of Applied Mathe- strictive Pade Approximation (RPA) is considered. The advantage is that it has the matics and Physics, 4, 2031-2037. exact value at certain r. This method will exhibit several advantages for example http://dx.doi.org/10.4236/jamp.2016.411202 highly accurate, fast and with good results, etc. The absolutely error is still very small. Received: October 10, 2016 The obtained results are compared with the exact solution and the other methods. Accepted: November 12, 2016 The numerical results are in agreement with the exact solution. Published: November 15, 2016 Keywords Copyright © 2016 by authors and Scientific Research Publishing Inc. Restrictive Pade Approximation (RPA), Diffusion Equation, Finite Difference This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ 1. Introduction Open Access

In this paper, we apply a new implicit method of high accuracy and the number of linear systems which to be solved are smaller than that for many famous known implicit methods of small step length. Therefore, our required machine time is less than that for the other implicit methods. Restrictive Pade Approximation (RPA) for parabolic Partial Differential Equation (PDE) and Partial Difference Equations is a new technique done by İsmail and Elbar- bary [1]. In addition, they studied numerical solution of the Convection Diffusion Equ- ation [2]. RPA for hyperbolic PDE is done by İsmail and Younes [3] [4] [5]. Restrictive Taylor approximation solution for the parabolic PDE is studied by İsmail and Elbarbary [6]. Schrodinger and Singularly perturbed parabolic PDE studied by İsmail and Elbee- tar [7] [8]. G. Gurarslan [9] studied numerical modelling of linear and nonlinear diffusion equations by compact finite difference method. In this work, we consider the following one dimensional diffusion equation;

ut= u xx + fu( ),0 << x L (1)

DOI: 10.4236/jamp.2016.411202 November 15, 2016 A. Boz, F. Gülsever

u= Duu,0<< x L tx( ( ) )x (2) Subject to the initial condition ux( ,0) = f( x) , 0 << x L (3)

and boundary conditions

u(0, t) = gx0 ( ) , t > 0 (4)

u(1, t) = gx1 ( ) , t > 0 (5)

The functions fu( ) are linear source functions. The function “ Du( ) ” is the diffusion term that plays a crucial role in a wide range of applications in diffusion process. [10] [11] [12] The diffusion term Du( ) appears in several forms. Some of the well known diffusion proses are the fast and the slow diffusion process where the diffusion term is of the form Du( ) = un where n < 0 and n > 0 respectively.

2. Method 2.1. Restrictive Pade Approximation (RPA)

The Restrictive Pade Approximation (RPA) of function fx( ) is a particular type of rational functions, it can be written in the form [13] [14];

M i α Mi+ ∑∑αεiixx+ RPA[ M+=α N] ( x) ii=01= (6) fx( ) + N i 1 ∑i=1bxi

where the positive integer α doesn’t exceed the degree of the numerator N, α = 01( ) N , −+α =MN++1 f( x) RPA[ M N] fx( ) ( x) 0.( x ) (7)

Let fx( ) have a Maclaurin Series; ∞ i f( x) = ∑ cxi (8) i=0 From Equations (6)-(8) we get, [1]

∞ NNα i i  i iM+ M++ N1 ∑cxi 10+ ∑ bx i  −− ∑∑ ax iiε x =( x ) (9) i=0  i= 1  ii= 01=

The varishing of the first (MN++1) power of x on the left hand side of (9) implies a system of (MN++1) equations.

r ar=+==> c r∑ c rii− br, 0( 1) M( bi 0 if i M) i=1 N (10) cMNs+−+==>∑ cb MNisi +−−ε Ns − ( c i 0 if i 0) i=1

Hence we can determine the coefficient, ai and bi as a function of εαi ,i = 01( ) ,

where the parameters εi are to be determined, such that [15]; =+=αα f( xii) RPA[ M N] fx( ) ( x), i 11( ) (11)

Note: εαi =0,i = 1( 1) gives the classical Pade Approximation (RPA) of the form;

2032 A. Boz, F. Gülsever

M i ∑ βi x PA[ M N] ( x) = i=0 . (12) fx( ) + N γ i 1 ∑ i=1 i x

The local truncation error form the RPA can be summarized by the following theorem. +h THEOREM: If the function fx( ) has an (n +1) derivative, then for every ar- gument x there exist a number η in the smallest interval I containing the set of points

{xxx012,,, , xα , x} , such that;

πα +1 ( x) (α +1) RMN,,αα( x) =−+= f( x) RPA[ Mαη N ] (RMN,, ( )) (13) fx( ) (α +1!)

where ∏=αα+1xxxxx( − 12)( −)( xx −) and RMN,,α is the local truncation error for the RPA [1].

2.2. Restrictive Pade Approximation of the Exponential Matrix

The exponential matrix exp(rA) can be formally defined by the convergent power series, rr23 ∞ rn exp(rA) =++ I rA A23 + A += ∑ AAIr , 0 = (14) 2! 3! n=0 n! where A is ( NN−×11) ( −) matrix.

In the case of Restrictive Pade Approximation of single function the term εi in Eq- uation (4) can be reduced to the square matrix εi in the case of Restrictive Pade approximation of the exponential matrix, where

ε1,i 0  ε 2,i  ε = i    0  ε Ni−1, for example,

−1  11  RPA[11] ( r) =+− Iεε11 A r I ++  A r . (15) exp(rA)  22 

3. Method of Solution

We consider the diffussion for fu( ) = − u [16]. Thus, we use the equation;

UUt= xx − U (16)

Subject to the initial condition; Ux( ,0) = sin x , 0 << x 1 (17) and boundary conditions; Ut(0,) = 0, t > 0 (18) Ut(1,) = 0, t > 0

2033 A. Boz, F. Gülsever

The exact solution of the Equation (16) is given by: U( xt,) = e−2t sin x (19)

Condiser the diffusion Equation (16) with the initial and boundary condition. The open rectangular domain is covered by a rectangular grid with spacing h and k in the x and t

directions respectively. The grid point ( xt, ) denoted by (ih, jk ) and U( ih, jk) = Uij, where iN= 01( ) j is non-negaive integer. The exact solution of grid representation of (6) is given by [3];

2 Uij,1+ =exp( kD( x − 1)) U ij, (20)

2 The approximation of the partical derivatives Dx at the grid point (ih, jk ) will take the usual form:

2 1 Dx =( Ui+−1,,,1 j −+2 UU ij i j) (21) h2 and according to central finite difference for mulation

UU=−+i+−112 U ii U (22) The result of making this approximation is to replace (20) by the following equation k Ujj+1 = exp( rA) U , r = (23) h2 j T where U= ( U1,j, U 2, j ,, U Nj−1, ) , Nh = 1 and −+22hh22 1 − 0 2 22 1−h −+ 22 hh 1 − 1−h2 −+ 22 hh 22 1 −   A = (24)   2 22 1−h −+ 22 hh 1 − −22 −+ 0 1hh 22 ( NN−×11) ( −)

We use the −1 11    RPA[11] ( r) =+− Iεε A r ⋅ I ++  A r  (25) exp(rA) 22 

Equation to approximate the exponential matrix in Equation (23), then the approximate solution of grid representation of Equation (16) can take the form.

1122 2 εεεii−−(1h) rU −+1,1j +−−+( i(11h)) r U i,1j+ +−−ii( 1h) rU ++1,1j 22 (26) 11 =+−22 +−−+ ++− 2 εεεii(1h) rU −+1, j( i(11h)) r U i, jii( 1h) rU 1, j 22

The constrat matrix ε1 must determined by using the only one fact that the exact εε=  solution is given at first level. First choice of 1,ij was the tridiagonal form:

εii,1−+= εε ii , = ii ,1 = ε i and εij, = 0 . The second choise of ε1 is the diagonal matrix

2034 A. Boz, F. Gülsever

ε=  ε ε= εε = 1,, ij : ii i ,0 ij , otherwise. Other implicit methods can be derived if we use the possible Restrictive Pade Ap- proximation RPA [M/N], with non-negative integers M and N.

4. Findings

The accuracy of Restrictive Pade Approximation method are compared in tables for various values of the time t. Tables give exact value, approximate value for compact finite difference method, approximate value for Restrictive Taylor Approximation, Re- strictive Pade Approximation and absolute error for ε = 0.0032408523. Comparison of the RPA results with RTA method for k = 0.0001, N = 6, r = 0.0036 given below in tables. The absolute error (AE) is give by the following formula: AE= Exact value− Approximate value .

We tabulated all AE values at Table 1, Table 2 and Table 3.

Table 1. Absolute error (AE) for RPA at t = 0.01.

t x Exact RPA RTA AE [Present] 1/6 0.162611 0.162611 0.162611 1.02E−10 2/6 0.320715 0.320715 0.320715 1.01E−10 3/6 0.469932 0.469932 0.469932 1.01E−11 0.01 4/6 0.606125 0.606125 0.606125 2.06E−11 5/6 0.725520 0.725520 0.725520 1.12E−10 1 0.824808 0.824808 0.824808 1.15E−10

Table 2. Absolute error (AE) for RPA at t = 0.1.

t x Exact RPA RTA AE [Present] 1/6 0.135824 0.135824 0.135824 1.26E−9 2/6 0.267884 0.267884 0.267884 1.16E−9 3/6 0.392520 0.392520 0.392520 2.01E−11 0.1 4/6 0.506278 0.506278 0.506278 1.25E−10 5/6 0.606005 0.606005 0.606005 1.21E−9 1 0.688938 0.688938 0.688938 1.66E−9

Table 3. Absolute error (AE) for RPA at t = 1.

t x Exact RPA RTA AE [Present]

1/6 0.022451 0.022451 0.022451 2.54E−10 2/6 0.044280 0.044280 0.044280 2.12E−10 3/6 0.064883 0.064883 0.064883 4.25E−12 1.00 4/6 0.083687 0.083687 0.083687 1.92E−11 5/6 0.100172 0.100172 0.100172 2.04E−10 1 0.113880 0.113880 0.113880 2.52E−10

2035 A. Boz, F. Gülsever

5. Discussion & Conclusion

In this article, a numerical algorithm was applied in the one dimensional diffusion equation. Computed results were compared with other paper results in Table 1, Table 2 and Table 3. Especially, we compared it with Restrictive Taylor approximation method, because these methods have same properties. But as we have seen from the computational results, Restrictive Pade approximation method has more efficient results than restrictive Taylor approximation method. The proposed method results are quite satis- factory.

References [1] İsmail, H.N.A. and Elbarbary, E.M.E. (1998) Restrictive Pade Approximation and Parabolic Partial Difference Equations. International Journal of Computer Mathematics, 66, 343-351. http://dx.doi.org/10.1080/00207169808804645 [2] İsmail, H.N.A. and Elbarbary, E.M.E. (1999) Highly Accurate Method for the Convection- Diffusion Equations. International Journal of Computer Mathematics, 72, 271-280. http://dx.doi.org/10.1080/00207169908804850 [3] İsmail, H.N.A., Elbarbary, E.M.E. and Hassan, A.Y (2000) Highly Accurate Method for Solving Initial Boundary Value Problem for First Order Hyperbolic Differential Equation. International Journal of Computer Mathematics, 77, 251-261. http://dx.doi.org/10.1080/00207160108805063 [4] Ismail, H.N.A. and Hassan, A.Y. (2000) Restrictive Pade Approximation for Solving First Order Hyperbolic Partial Differential Equation. Journal of Institute of Mathematics and Computer Sciences, India, 11, 63-71. [5] Ismail, H.N.A. and Hassan, A.Y. (2000) Restrictive Pade Approximation for Solving First Order Hyperbolic Partial Differntial Equation in Two Space Dimensions. Journal of Ma- thematics and Computer Science, India, 11, 159-166. [6] Ismail, H.N.A. and Elbarbary, E.M.E. (2001) Restrictive Taylor Expansion and Parabolic Partial Differential Equations. International Journal of Computer Mathematics, 78, 73-82. http://dx.doi.org/10.1080/00207160108805097 [7] Ismail, H.N.A. and Elbeeatar, A.A. (2001) Restrictive Pade Approximation for the Solution of Singularity Perturbed Parabolic PDE. Journal of Mathematics and Computer Science, India, 12, 153-161. [8] Ismail, H.N.A. and Elbeetar, A.A. (2002) Restrictive Pade Approximation for the Solution of Schrodinger Equation. International Journal of Computer Mathematics, 79, 603-613. http://dx.doi.org/10.1080/00207160210951 [9] Gurarslan, G. (2010) Numerical Modelling of Linear and Nonlinear Diffusion Equations by Compact Finite Difference Method. Applied Mathematics and Computation, 216, 2472- 2478. http://dx.doi.org/10.1016/j.amc.2010.03.093 [10] Gurarslan, G. and Sari, M. (2011) Numerical Solutions of Linear and Nonlinear Diffusion Equation by a Differential Quadrature Method (DQM). International Journal for Numeri- cal Methods in Engineering, 27, 69-77. http://dx.doi.org/10.1002/cnm.1292 [11] Meral, G. and Tezer-Sezgin, M. (2009) Differential Quadrature Solution of Nonlinear Reac- tion-Diffusion Equation with Relaxation-Type Time Integration. International Journal of Computer Mathematics, 86, 451-463. http://dx.doi.org/10.1080/00207160701600127 [12] Meral, G. and Tezer-Sezgin, M. (2011) The Differential Quadrature Solution of Nonlinear

2036 A. Boz, F. Gülsever

Reaction-Diffusion and Wave Equations Using Several Time-Integration Schemes. Interna- tional Journal for Numerical Methods in Biomedical Engineering, 27, 461-632. http://dx.doi.org/10.1002/cnm.1305 [13] Ismail, H.N.A. and Elsayed, E.M. (1995) Restrictive Pade Approximation. Journal of Faculty of Education, Ain Shams University, Special Issue for the 1st National Conference of Ma- thematical & Physical Sciences and Applications, Cairo, 63-76. [14] Ismail, H.N.A. and Elsayed, E.M. (1995) Differential and Integral Properties of Pade and Restrictive Pade Approximations. Journal of Faculty of Education, Ain Shams University, Special Issue for the 1st National Conference of Mathematical & Physical Sciences and Ap- plications, Cairo, 77-90. [15] Ismail, H.N.A. and Ekmekkawy, A.Y. (2001) Restrictive Pade Approximation for Solving Singularly Perturbed Initial-Boundary Value Problem for First-Order Hyperbolic Partial Differential Equation. 9th International Technology ASAT-9, Cairo, 61-69. [16] Wazwaz, A.-M. (2007) The Variational Iteration Method: A Powerful Scheme for Handling Linear and Nonlinear Diffusion Equations. Computers and Mathematics with Applications, 54, 933-939. http://dx.doi.org/10.1016/j.camwa.2006.12.039

2037 Journal of Applied Mathematics and Physics, 2016, 4, 2038-2046 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Gauss-Legendre Iterative Methods and Their Applications on Nonlinear Systems and BVP-ODEs

Zhongli Liu1*, Guoqing Sun2

1College of Biochemical Engineering, Beijing Union University, Beijing, China 2College of Renai, Tianjin University, Tianjin, China

How to cite this paper: Liu, Z.L. and Sun, Abstract G.Q. (2016) Gauss-Legendre Iterative Me- thods and Their Applications on Nonlinear In this paper, a group of Gauss-Legendre iterative methods with cubic convergence Systems and BVP-ODEs. Journal of Applied for solving nonlinear systems are proposed. We construct the iterative schemes based Mathematics and Physics, 4, 2038-2046. on Gauss-Legendre quadrature formula. The cubic convergence and error equation http://dx.doi.org/10.4236/jamp.2016.411203 are proved theoretically, and demonstrated numerically. Several numerical examples Received: October 24, 2016 for solving the system of nonlinear equations and boundary-value problems of non- Accepted: November 15, 2016 linear ordinary differential equations (ODEs) are provided to illustrate the efficiency Published: November 18, 2016 and performance of the suggested iterative methods.

Copyright © 2016 by authors and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative Iterative Method, Gauss-Legendre Quadrature Formula, Nonlinear Systems, Commons Attribution International License (CC BY 4.0). Third-Order Convergence, Nonlinear ODEs http://creativecommons.org/licenses/by/4.0/

Open Access

1. Introduction

We consider the general form of a system of nonlinear equations as follows:

T Fx()()()()= ( f12 x, f x ,, fn x) = 0, (1)

where FD: ⊆→ Rnn R is a given nonlinear vector function, and each function T fxi () can be thought of as mapping of a vector x= () xx12,,, xn , which is the n- dimensional space Rn into the real number R. Numerical solutions for systems of nonlinear equations have always appealed greatly to people in scientific computation and engineering fields. Some boundary-value problems of nonlinear ordinary differential equations (ODEs) can be transformed to nonlinear systems like (1) by the finite difference method. Constructing an efficiently iterative method to approximate the root ξ ∈ D (such that F ()ξ = 0 ) of Equation (1) is a

DOI: 10.4236/jamp.2016.411203 November 18, 2016 Z. L. Liu, G. Q. Sun

typical and important issue in nonlinear numerical computation. The Newton’s method (see [1] [2]) is one of the widely used methods for solving nonlinear equations by iteration as follows: −1 xkk+1 = x− F′()() x k Fx k, k= 0,1, 2, , (2)

−1 where Fx′() is the inverse of first Fréchet derivative Fx′() which is a Jacobian matrix of the nonlinear vector function Fx() . It converges quadratically when an initial guess value x0 is close to the root ξ of Equation (1). In recent years, several iterative methods have been used to solve nonlinear equations and systems of nonlinear equations. In order to improve the order of convergence, a few two-step variants of Newton’s methods with cubic convergence have been proposed in some literature [3]-[11] and references therein for solving systems of nonlinear equations. S. We era- x koon and T. Fernando [5] using the Newton theorem fx()()()= fx + f′ td t, n ∫xn proposed the Newton-type method with third-order convergence for nonlinear equation and systems. M. Darvishi and A. Barati [6] received a third-order convergence iterative method based on Adomian decomposition method to the systems of nonlinear equations. M. Frontini and E. Sormani [7] presented third-order midpoint-methods using numerical quadrature formula. A. Cordero and J. R. Torregrsa [8] developed third-order convergence Newton-Simpson’s method and Open Newton’s method using the simple Simpson’s rule and an open quadrature formula of high order respectively. These are all classic two-step Newton-type methods to approximate the root of a system of nonlinear equations. In Section 2 of this paper, we propose a group of two-step iterative methods with third-order convergence by Gauss-Legendre quadrature formula [12]:

b ba−n  ab +− ba f() xd x= wf + t , (3) ∫a ∑ ii 2i=1  22 where the necessary symmetrical conditions is

∑ wi = 2 , ∑ti = 0 , ∑ wtii= 0 . (4)

Several numerical examples are provided in Section 3 using Gauss-Legendre iterative method for solving systems of nonlinear equations and boundary-value problems of nonlinear ODEs, and we finally make conclusions in Section 4.

2. The Iterative Methods and Cubic Convergence

Assume FD: ⊂→ Rnn R be a Fréchet differentiable function in a convex DR⊂ n . We have the mean-value theorem of multivariable vectors function F(x) in [1]: 1 FxFx() −() = Fxtxx′ +−() () xxt −d, (5) k∫0 ( k kk)

Using the left rectangular integral rule: 1 F′′ x+− tx() x ()()()xx −dt ≈F x xx −, (6) ∫0 ( k k) k kk

And by Fx() = 0 , we can get Newton’s Method (2).

2039 Z. L. Liu, G. Q. Sun

Now, we apply the Gauss-Legendre quadrature formula (3) to approximate the integral on the right side of Equation (5), that is,

1 1 n x+− x xx Fx′′+− txx( ) ( xx −)dt ≈ wF kk+t( xx −), (7) ∫0 ( k k) k∑ iik 22i=1 2 and using Fx( ) = 0 we have −1 n x + x xx − ′ kk x− xk ≈−2 ∑ wF i+ tin F( x ) . (8) i=1 22

Herein, taking x as the next iterative step of xk , we get the following implicit structure: −1 n xx + x − x ′ kk++11 k k xkk+1 = x−+2 ∑ wF iti F( x n) . (9) i=1 22

We use the k-th iteration yk of the Newton’s method to replace xk+1 on the right side of the expression (9), and we construct the following group of two-step iterative schemes:

−1 y= x − F′( x) Fx( )  kk k k  −1  n xy + yx − . (10) ′ k k kk xkk+1 = x−+2 ∑ wF iti F( x k)  i=1 22

We state and prove the convergence theorem for the schemes (10) as follows: Theorem Let FD: ⊂→ Rnn R be a k-time Fréchet differentiable function in a

convex set D with a root ξ ⊂ D and x0 be close to ξ , then the group of iterative methods (10) is cubically convergent, and the error equation is

n 2 3 2 34 ek+1= C 23 −+ C 2 +∑ wtii C3 e k + O( e k) . (11) 8 i=1

(k ) −1 F (ξ ) Proof. As F (ξ ) = 0 , and noting that xe−=ξ , CF= ′(ξ ) . kkk k ! By Taylor’s expansion, we have

112 34 Fx( kk) = F′ (ξξ)( x −+) F′′ ( ξξ)( x k −) + F′′′ ( ξξ)( xk −) + Ox( k − ξ) 2! 3! (12) =′ ξ  ++24 + F( )  ekkk Ce23 Ce O( e k),

and ′′=ξ  +++23 F( xk) F( )  I23 Ce23k Ce kk O( e ) . (13) Then, suppose that −−11−1 F′′ x=  De F ξ ( kk) ( ) ( ) , (14) where

23 De( k) =+++ I23 Ce23 k Ce kk Oe( ) . (15)

The inverse of De( k ) is given by

2040 Z. L. Liu, G. Q. Sun

−1 2 D( ek) =++ I Ke12 kk Ke , where K1 and K2 will be satisfied the definition of the following inversion. −−11 De( kk) De( ) = De( k) De( k) = I. (16)

That is

22 (I+23 Ce2n + Ce 3 n)( I ++ Ke 12 nn Ke) = I. (17)

From (17), we have K=−=−2, CK 4 C2 3 C 1 22 2 3. (18) Therefore, ′′−1 =−+−22 + ξ −1 F( xkk)  I2 Ce2( 43 C 23 C) e k F( ) . (19)

And ′ −1 =− +2 − 2 +×+23 + + F( xk) F( x k)  I2 Ce2 k( 43 C 23 C) ek( e kkk Ce2 Ce 3 ) (20) 22 34 =−+ekk Ce2(22 C 23 − C) ek + O( e k) .

By the first step of (10), we have

2 23 4 yk=ξξ + Ce2 k +2( C 32 − C) ekk + O( e ) = +∆, (21)

2 23 4 Here, suppose ∆=Ce2k +2( C 32 − C) ekk + O( e ) . So, xy+ 11 11 kk=+ξξxe +∆=+ +∆, (22) 2 22kk 22 yx− ξ 11 1 1 kk= + ∆−(ξ +ee) =− + ∆. (23) 2 22 2kk 2 2 Furthermore, xy+− yx 11 k k+ kkt =+−++∆=+ξξ(11 te) ( t) d, (24) 22i 2ik 2 i k where, 11 d=(11 − te) +( +∆ t) . (25) k22 ik i Then, xy+− yx ′ k k kk Ft+ i 22 ′ 2 =F(ξ )( I +++23 Cd23kk Cd ) (26) ′ 3 2 23 =F(ξ ) I +−++∆+− C22(11 tik) e C( ti) C 3( 1 ti) e k + Oe( k) 4 ′ 2 23 2 23 =F(ξ ) IC +−+++22(11 teCik) ( teik) C 3( 1 − teOei) k +( k) . 4

Let

2041 Z. L. Liu, G. Q. Sun

n xy+− yx ′ k k kk B = ∑ wFii+ t i=1 22 n 3 2 = ′ ξ +−+++2 2 −23 + ∑ wFi ( )  IC22(11 teCik) ( teik) C 3( 1 teOei) k( k) (27) i=1 4 nn n n ′ 22 3 2 2 3 = F (ξ ) ∑∑wICewi+22 ki(1 −+ tCewt i) ki ∑(1 ++ i) CewttOe 3 ki ∑(12 − ii +) +( k) .  ii=11= i= 14 i=1 

By the conditions (4), we have n ′ 223 2 2 3 B = F (ξ ) 22I+++ C22 ek 2 C e k C 3 e k 2 +∑ wtii  + O( ek) . (28) 4 i=1

−1 So the iterative schemes (10) can be written as xkk+1 = x − 2 BFx( k) . Subtracting ξ from both sides of this equation, we can obtain 11 Be= Be − F( x ) , (29) 22k+1 kk that is

n 2 2 233 32 4 (ICeOeeeCeCeCe++2k( k)) k+1 =+++ k 22 k k 3 k2 +∑ wtOeFxii +( k) −( k) 8 i=1 (30) n 2 33 23 4 =−++(C23 C) ek C 32 ∑ wtii e k + O( e k) 8 i=1

Therefore, the error equation is

n 2 3 2 34 ek+1= C 23 −+ C 2 +∑ wtii C3 e k + O( e k) . (31) 8 i=1 This shows that the group of iterative methods (10) is third-order convergent.

As for the iterative methods (10), when nw=1,11 = 2, t = 0 , then it can be transformed to the following form:

−1 y= x − F′( x) Fx( )  kk k k −1  xy+ . (32) ′ kk xkk+1 = x − F Fx( k)  2

When n=2, ww12 = = 1, t 1 =µµ , t 2 = − , µ ∈ R is a parameter, then the methods (10) can be written as follows:

−1 y= x − F′( x) Fx( )  kk k k  −1  n xy + yx − xy + yx − . (33) ′′k k kk k k kk xkk+1 =−−+ x2 ∑ F µµF +Fx( k)  i=1 22 22

Especially, we take the parameter µ = 0.577350 in the iterative method (33) to make applications in the following section.

3. Numerical Examples

The iterative method (33) with µ = 0.577350 is demonstrated for solving systems of nonlinear equations and some two-point boundary-value problems of nonlinear ODEs.

2042 Z. L. Liu, G. Q. Sun

Example 1. Consider a system of nonlinear equations with variables n = 5 : n 2 ∑ xji−( n −1) x = 0, 1 ≤≤ in, (34) j=1, ji ≠ ′ ′ where x0 = (2, 2, 2, 2, 2) is the guess value. We can get the exact solution ξ = (1,1,1,1,1) by the iterative method (33) for Equation (34). The numerical results are shown in Ta- ble 1, in which k refers to the number of iterations. Example 2. Consider solving the following two-point boundary-value problem of nonlinear ODE:

x′′ ( t) +=∈ xt( ) 0, t ( 0,1)  (35) xx(0) = 0, ( 1) = 1

We discretize the nonlinear ODE (35) with the finite difference method. Taking nodes ti = ih , where hN= 1 and N = 10 herein, we can obtain the following system of nine-variables nonlinear equations: 20xhx−2 −= x  1 12  2 −+xi−+112 xhx i − ii − x =0, i =2,3, ,8 , (36)  − + −−2 =  x8921 x hx 9 0 ′ where x0 = (1,1,1,1,1,1,1,1,1) . Using the iterative method (33), we get the numerical solutions ξ of Equation (36), that is, ξ = (0.1283710267934016 ,0.2531591633178528 ,0.3729158073866021 , 0.4865657681707311 ,0.5932403023143547 ,0.6922126270196785 , 0.7828650200725204 ,0.8646694471440053 ,0.9371751138295893)′ .

The numerical results for the system of nonlinear Equations (36) derived from ODE (35) are shown in Table 2. According to results of the above two numerical experiments, the iterative method (33) can achieve third-order convergence for systems of nonlinear equations, and their numerical solutions show also the method is feasible.

Table 1. The numerical solutions and errors of the system of Equation (34) using the method (33).

k 1 2 3 4 5 6

−ξ xk 2 3.1943e−1 4.3758e−3 1.6660e−8 9.2481e−25 1.5819e−73 7.9180e−220

Fx ( k ) 2 1.4602e−0 1.7537e−2 6.6640e−8 3.6992e−24 6.3278e−73 3.1672e−219

Table 2. The numerical solutions and errors of the system of nonlinear Equation (36).

k 1 2 3 4

− ξ xk 2 4.9234e−4 5.63305e−14 8.6475e−17 8.6475e−17

Fx ( k ) 2 1.3365e−4 9.35871e−15 9.7017e−45 1.5685e−134

2043 Z. L. Liu, G. Q. Sun

Example 3. Solving the following two-point nonlinear boundary-value problem of ODE with exact solution:

y′′ xy=1 + ′ 2  ( )  . (37) 1 − y (0) = 1, y ( 1) = (e + e 1 )  2 1 The exact solution for this problem of ODE (37) is yx( ) = (e−xx+ e ) . We now 2 find the numerical solutions using the present method and compare them with the exact solution yx( ) . By the finite difference method, partitioning the interval [0,1] :

x0=<<<<0 xx 12 xnn− 1 < x, yii+1 =+= y hh,1 n. −1 Let y0= y(0) = 1, y11 = ( x) , , ynn−− 1= yx( 1) , and yynn=( x ) = 0.5(e + e) . And y−+2 yy using the numerical differential formula for the second derivative y′′ = i−+11 ii, i h2 yy− and the first derivative y′ =ii+1 ,( in = 1, 2, , − 1) , we take n = 10 herein, and i h hence obtain the following system of nonlinear equations with nine variables:

2 2 yi−+11−+−2 yyhhyy ii +( i + 1 − i) =0, i = 1, 2, ,9 , (38)

(0) T where y = (0, 0, 0, 0, 0, 0, 0, 0, 0) is an initial value. We obtain the approximate numerical solutions of this problem with the method (33) (iteration number k = 5): y* = (1.00412287612219 ,1.01834640006185 ,1.04286614560037 , 1.07798462092560 ,1.12411585747856 ,1.18179111195371 , 1.25166573656183 ,1.33452729205395 ,1.43130499800870)′ .

Comparison of the numerical results of the boundary-value problem of ODE (37)

and the exact solutions yx( i ) are shown in Table 3, in which xi refers to the nodes. The error comparisons of the numerical solutions and exact solutions at different

nodes xi for solving the problem of ODE (37) are shown in Figure 1.

Table 3. Numerical comparison results for the problem of ODE (37).

* * xi Numerical solutions y Exact solutions yx( i ) The errors yx( i ) − y

0.1 1.00412287612219… 1.0050041680558035039… 8.812e−04 0.2 1.01834640006185… 1.0200667556190758933… 1.720e−03 0.3 1.04286614560037… 1.0453385141288604742… 2.472e−03 0.4 1.07798462092560… 1.0810723718384549485… 3.087e−03 0.5 1.12411585747856… 1.1276259652063806981… 3.510e−03 0.6 1.18179111195371… 1.1854652182422675821… 3.674e−03 0.7 1.25166573656183… 1.2551690056309430243… 3.503e−03 0.8 1.33452729205395… 1.3374349463048447184… 2.907e−03 0.9 1.43130499800870… 1.4330863854487745356… 1.781e−03

2044 Z. L. Liu, G. Q. Sun

Figure 1. Error comparisons for solving the problem of ODE (37).

4. Conclusion

In this paper, we construct a group of iterative methods with cubic convergence for the systems of nonlinear equations by using the Gauss-Legendre quadrature formula. Nu- merical results we gave are in consistence with the theoretical analysis, and meanwhile they also demonstrate that the presented scheme is efficient and feasible to solve systems of nonlinear equations and to solve two-point boundary-value problems of nonlinear ordinary differential equations.

Acknowledgements

The work is supported by the Science and Technology Program of Beijing Municipal Commission of Education (No. KM201511417012).

References [1] Ortega, J.M. and Rheinboldt, W.G. (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York. [2] Traub, J.F. (1964) Iterative Methods for the Solution of Equations. Prentice-Hall, Engle- wood Cliffs. [3] Noor, M.A. and Noor, K.I. (2006) Improved Iterative Methods for Solving Nonlinear Equa- tions. Applied Mathematics and Computation, 183, 774-779. https:/doi.org/10.1016/j.amc.2006.05.084 [4] Chun, C. (2006) A New Ityerative Method for Solving Nonlinear Equations. Applied Ma- thematics and Computation, 178, 415-422. https:/doi.org/10.1016/j.amc.2005.11.055 [5] Weerakoon, S. and Fernando, T.G.I. (2000) A Variant of Newton’s Method with Accele- rated Third-Order Convergence. Applied Mathematics Letters, 13, 87-93. https:/doi.org/10.1016/S0893-9659(00)00100-2 [6] Darvishi, M.T. and Barati, A. (2007) A Third-Order Newton-type Method to Solve Systems of Nonlinear Equations. Applied Mathematics and Computation, 187, 630-635. https:/doi.org/10.1016/j.amc.2006.08.080

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[7] Frontini, M. and Sormani, E. (2004) Third-Order Methods from Quadrature Formulae for Solving Systems of Nonlinear Equations. Applied Mathematics and Computation, 149, 771- 782. https:/doi.org/10.1016/S0096-3003(03)00178-4 [8] Cordero, A. and Torregrosa, J.R. (2007) Variants of Newtons Method Using Fifth-Order Quadrature Formulas. Applied Mathematics and Computation, 190, 686-698. https:/doi.org/10.1016/j.amc.2007.01.062 [9] Noor, M.A. and Wasteem, M. (2009) Some Iterative Methods for Solving a System of Non- linear Equations. Computers and Mathematics with Applications, 57, 101-106. https:/doi.org/10.1016/j.camwa.2008.10.067 [10] Hafiz, M.A. and Bahgat, M.S.M. (2012) An Efficient Two-Step Iterative Method for Solving System of Nonlinear Equations. Journal of Mathematics Research, 4, 28-34. [11] Khirallah, M.Q. and Hafiz, M.A. (2012) Novel Three Order Methods for Solving a System of Nonlinear Equations. Bulletin of Mathematical Sciences & Applications, 1, 1-14. https:/doi.org/10.18052/www.scipress.com/BMSA.2.1 [12] Guan, Z. and Lu, J.F. (1998) Numerical Analysis. High Education Press, Beijing.

2046 Journal of Applied Mathematics and Physics, 2016, 4, 2047-2060 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Nano-Contact Problem with Surface Effects on Triangle Distribution Loading

Liyuan Wang*, Wei Han, Shanlin Wang, Lihong Wang, Yinping Xin

Regional Circular Economy key Laboratory of Gansu Higher Institutions; Department of Mathematics, Longqiao College of Lanzhou University of Finance and Economics, Lanzhou, China

How to cite this paper: Wang, L.Y., Han, Abstract W., Wang, S.L., Wang, L.H. and Xin, Y.P. (2016) Nano-Contact Problem with Surface This work presents a theoretical study of contact problem. The Fourier integral Effects on Triangle Distribution Loading. transform method based on the surface elasticity theory is adopted to derive the Journal of Applied Mathematics and Phys- fundamental solution for the contact problem with surface effects, in which both the ics, 4, 2047-2060. http://dx.doi.org/10.4236/jamp.2016.411204 surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are Received: August 29, 2016 compared with those of the classical contact problem. At nano-scale, the contribu- Accepted: November 15, 2016 tions of the surface tension and the surface elasticity to the stress and displacement Published: November 18, 2016 are not equal at the contact surface. The surface tension plays a major role to the Copyright © 2016 by authors and normal stress, whereas the shear stress is mainly affected by the surface elasticity. In Scientific Research Publishing Inc. addition, the hardness of material depends strongly on the surface effects. This study This work is licensed under the Creative is helpful to characterize and measure the mechanical properties of soft materials Commons Attribution International License (CC BY 4.0). through nanoindentation. http://creativecommons.org/licenses/by/4.0/ Open Access Keywords

Surface Tension, Surface Elasticity, Nano-Contact Problem, Fourier Integral Transforms Method, Triangle Distribution Force

1. Introduction

Nowadays, nanometer material and technology have been widely used in industrial and engineering fields. Many new nano-materials have been developed by utilizing the fact that materials begin to exhibit unique mechanical properties at nano-scale, which sig- nificantly differ from those at larger scale. Nano-indention tests have been widely used to measure such mechanical properties of materials. For micro-nano solids with large surface-to-bulk ratio the significance of surfaces is likely to be important. Form the viewpoint of continuum mechanics, this

DOI: 10.4236/jamp.2016.411204 November 18, 2016 L. Y. Wang et al.

difference can be described by such concepts as surface effects [1]. This is especially true for nano-scale materials or structures. In such cases, the surface tension and the surface elasticity play a critical role and thus have been adding its appeal to many researchers. For example, Miller and Shenoy [2] probed the size-dependent elastic properties of nano-plates and beams. Hang et al. [3] explained the size dependent phenomenon by the strain gradient continuum theory. Dingreville et al. [4] investigated the surface free energy and its effect on elastic behavior of the nano-sized particles, wires, and films. Yang [5] studied the size-dependent effective modulus of spherical nanocavities at dilute concentrations. Gao et al. [6] built a simple model to describe the influence of surface stress on the nanoscale adhesive contact. There are a lot of work regarding the surface/interface energy effects on the nanostructures and solids, and we can only include a small part of them here. For more recent developments in this field, the readers can refer to a review article by Wang et al. [7]. To study the mechanical behavior of nano-materials, the most celebrated continuum-based surface/interface model was first established by Gurtin, Murdoch and co- workers [8] [9] [10], which known as the theory of surface elasticity. In the study of nano-scale problems, all material constants appearing in that constitutive model were commonly calibrated with data obtained from either experimental measurements [11] or atomistic simulations [2] [12]. Therefore, the surface effect has been widely adopted to investigate the mechanical phenomena at nano-scale. Cammarata et al. [13] considered the size-dependent deformation in thin film with surface effects. Gao et al. [14] developed a finite-element method to account for the effect of surface elasticity. Wang et al. [15] studied to a half-plane subjected to normal pressures with surface tension. Zhao and Rajapakse [16] proposed a continuum-based model to study the influence of surface stresses on mechanical responses of an elastic half-space compressed by an axisymmetric, rigid, frictionless nano-indentor. Long and Wang [17] studied the effect of the residual surface stress on the two dimensional Hertz contact problem, and later Long et al. [18] generalized their work to the three dimensional case. Wang [19] derived the general analytical solution of nano-contact problem with surface effects by using the complex variable function method. Gao et al. [20] [21] derived the influence of the surface stress on the JKR adhesive contact, which is investigated by employing the non-classical Boussinesq fundamental solutions. In this paper, Fourier integral transform method is used to solve the non-classical boundary value problems with surface effects.

2. Problem Description

Now we consider a material occupying the upper half-plane z > 0 , we refer to a Carte- sian coordinate system (o-xyz), as shown in Figure 1, where the x axis is along the surface and the z axis perpendicular to the surface. It is assumed that the material is subjected to triangle distribution force px( ) and qx( ) over the region xa≤ . While

the normal and shear force form zero ( O1 and O2 ) uniformly increased to maximum

p0 and q0 at the point O. The plane-strain conditions are assumed to ε 2i = 0 , and

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a a

q0 o2 o1 θ 2 θ θ o 1 x

r2 r1

A(x, y)

Figure 1. Schematic of contact problem under triangle distribution loads.

the contact is assumed to be frictionless. The problem statement is to determine the triangle distribution force exerted by the elastic field (e.g., displacement and stresses) with the half-plane for the influence of surface effects.

3. Basic Equations of Surface Elasticity and General Solutions

In surface elasticity theory, the equilibrium and constitutive equations in the bulk of material are the same as those in classical elastic theory, but the presence of surface stresses gives rise to a non-classical boundary condition.

3.1. Basic Equations

In the absence of body force, the equilibrium equations, constitutive law, and geometry relations in the bulk are as follows

σ ij, j = 0 µ (1) σij =2G  εij + εδkk ij 12− µ where G and µ are the shear modulus and Poisson’s ratio of the bulk material,

σ ij and εij are the stress tensor and strain tensor in the bulk material, respectively.

The strain tensor is related to the displacement vector ui by 1 εij=(uu i,, j + j i ) (2) 2 On the surface, the generalized Young-Laplace equation [22], surface constitutive relation and strain-displacement relationship can be expressed as

2049 L. Y. Wang et al.

s σσβαn β+= βα, β 0 s (3) σijnn i j = σκβα βα

s s s s s s ss σβα= τδ βα +2( µ − τε) βα ++( λ τεδ) γγ βα + τu β, α (4)

where ni denotes the normal to the surface, κβα the curvature tensor of the sur- s face, σαβ is the 22× surface stress tensor on the surface, σ ij is the stress tensor s in the elastic material, and εαβ is the 22× surface strain tensor which is equal to the tangential components of the strain tensor of the elastic material on the surface, τ s is the residual surface tension under unstrained conditions, µ s and λ s are surface Lamé constants which can be determined by atom simulations or experiments [2].

3.2. General Solutions

Based on previous work by Wang [15] and Wang [23], Fourier integral transformation method is adopted to solve the stress and displacement components by

1 ∞ −−ξξ σ= Azξ+−2 ξ B ξξ edix z ξ xx ∫−∞ ( ) ( ) ( ) 2π 1 ∞ −−ξξ σ=−+ξ ξξ2 ix z ξ zz ∫ A( ) zB( ) ed 2π −∞ i ∞ −−ξξ σ= ξ1 −− ξzB ξξξ A edix z ξ (5) xz ∫−∞ ( ) ( ) ( ) 2π i ∞ −−ξξ u xz, =2 −vξξ A + z ξ −−21 v B ξ eix z d ξ+C ( ) ∫−∞ ( ) ( ) ( ( )) ( ) 1 22G π 1 ∞ −−ξξ wxz, = ξξA+−12 v + z ξ B ξeix z d ξ+C ( ) ∫−∞ ( ) ( ) ( ) 2 22G π where A and B are generally functions of ξ as yet to be determined by boundary conditions.

4. Elastic Solution under Normal Triangle Distribution Force

As a particular example, let us consider the effect of a normal triangle distribution

force px( ) over the region xa≤ , and the normal force form zero ( O1 and O2 )

uniformly increased to maximum p0 (O), while remainder of the boundary z = 0 being unstressed as shown in Figure 1. p px( ) =0 ( a − x), xa≤ (6) a Due to the surface tension mostly influences the normal stress [20]. Therefore, we keep only the first term in Equation (4). Then, the surface stresses are given by ss σβα= τδ βα (7) On the surface, the boundary conditions (3) can be written by

σ xz ( x) = 0 τ s (8) px( ) +=−σ ( x) zz Rx( )

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Substituting Equation (8) into Equation (5), one obtains BA= ξ (9)

Due to deformation the radius of curvature of the surface is given by 1 ∂2 wx( ,0) = (10) Rx( ) ∂x2

By substituting Equations (9) and (10) into the surface condition Equation (8), A(ξ ) is determined by p (ξ ) A(ξ ) = (11) (1+ s ξξ) 2 where τ s (1− v) s = (12) G

p 2 1− cos(aξ ) p (ξ ) = 0 (13) a π ξ 2 where s is a length parameter depending on the surface property and material elastic constants. It should be pointed out that this parameter indicates the thickness size of the zone where the surface effect is significant, and plays a critical role in the surface elasticity. For metals, s is estimated on the order of nanometers. Therefore A(ξ ) is given by p 2 1− cos(aξ ) A(ξ ) = 0 (14) a π (1+ s ξξ) 4

Substituting Equation (14) into Equation (5), the stresses component and displaces component are obtained as

2q ∞ zξ −1 cos( xξ ) σ= 0 − ξξ−zξ xx ∫ 2 1 cos(a ) e d πas0 1+ ξ ξ

2q ∞ zξ +1 cos( xξ ) σ=−−0 ξξ−zξ zz ∫ 2 1 cos(a ) e d πab0 1+ ξ ξ

2q ∞ z sin ( xξ ) σ=−−0 ξξ−zξ xz ∫  1 cos(a ) e d (15) πas0 1+ ξξ

p ∞ zvξ +−21 = 0 ξξ− −zξ ξ + u( xz, ) ∫ 3 sin( x)  1 cos( a ) e d C1 πGa 0 ξξ(1+ s )

p ∞ zvξ +−21( ) =0 ξ−+ ξξ−zξ wxz( , ) ∫ 3 cos( x)  1 cos( a) e d C2 πGa 0 ξξ(1+ b )

It is seen, when s = 0 , that is, the surface influence is ignored in Equation (15), the stresses of the half-plane are consistent with those in the classical elastic contact results which is the same result [23], respectively.

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p0 2 σxx =( x − a) θ1 ++( x a) θθ2 −2 x + 2 z ln ( rr12 r ) πa 

p0 σzz =( xa −) θ12 ++( xa) θθ −2 x (16) πa zp σ=−0 [ θθ +−2 θ] xz πa 12 where

22 z zz r2=−+=++=+=( xa) zr22, ( xa) zr22, x 2 z 2 ,tanθθθ ,tan = ,tan = 1 2 12−+ xa xa x. On the contact surface ( z = 0 ), the normal stress is given by −1 ∞ 2 p0 1− costs    x  σ zz ( x,0) =−+ 1t  cos  tt  d (17) π ∫0 t 2 aa   

According to the Saint-Venant’s Principle, we assume that the normal displace-

ment is w specified to be zero at a distance r0 on the contact surface, that is,

wr( 0 ,0) = 0, the displacement on the surface is derived as −1 2p( 1−− va) ∞  1 cos( t) sx   r =0 +−0 wx( ,0) ∫ 3 1  cos t cos t d t (18) πG0 t  a   aa

Assuming that the origin has no displacement in the x direction, that is, u (0,0) = 0 one obtains −1 p(1−− 2 va) ∞  1 cos( t) sx   =−+0 ux( ,0) ∫ 3 1  sin t d t (19) πG0 t  aa  

As show in Figure 2, the results indicated that the normal stress σ zz is a smooth distribution when the surface tension is considered by at the loading boundary ( xa= ± ), However, it is just the reverse with classical results that is the surface ten-

sion is ignored (s/a = 0). In addition, the actual normal stress σ zz is smaller than the classical value in the loading zone and is larger outside of the zone. Due to the different surface tension value, the horizontal displacement is dis-

played in Figure 3, where we set ra0 = 5 , and K1 =(12 − vG) π . It is seen that the horizontal displacement is continuous everywhere on the deformed surface. Howev- er, the classical elasticity theory predicted unreasonably that the horizontal displacement is discontinuously at the load boundary xa= ± , as seen from the curve

of s/a = 0. The indent depth is plotted in Figure 4 with K2 =21( − vG) π , which also shows that the slope of the deformed surface is continuous everywhere. It is also found the indent depth decreases with the increase of surface tension.

5. Elastic Solution under Tangential Triangle Distribution Force

Now, let us consider the effect of a tangential triangle distribution force qx( ) over the

region xa≤ , while the shears force form zero ( O1 and O2 ) uniformly increased to

maximum q0 at the point O, while remainder of the boundary z = 0 being unstressed as shown in Figure 1.

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Figure 2. The distribution of contact normal stress σ zz under normal triangle distribution load.

0.0

s/a=0 -0.2 s/a=0.1 s/a=0.5 s/a=2.0 a 0 p

1 -0.4

u(x,0)/K -0.6

-0.8

0 1 2 3 4 5 x/a Figure 3. The distribution of surface displacement u under normal triangle distribution load.

At this moment, the boundary conditions (3) on the contact surface ( z = 0 ) are simplified to

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3.5

3.0

2.5 s/a=0 s/a=0.1 s/a=0.5 a

0 2.0

p s/a=2.0 2

1.5 w(x,0)/K 1.0

0.5

0.0 0 1 2 3 4 5 x/a Figure 4. The distribution of surface indentation w under normal triangle distribution load.

σ zz = 0 s 2 ddτ s u (20) σ xz +=−+qx( )  k 2 dx dx

where qx( ) is the tangential triangle distribution force applied on the materials surface, and k s=2µλ ss + is a surface constant. q qx( ) =0 ( a − x), xa≤ (21) a Substituting Equation (21) into Equation (20), one can be obtained A(ξ ) = 0 q (ξ ) 1 (22) Bi(ξ ) = ξξ1+ b

where kvs (1− ) b = (23) G

1 ∞ ξ q (ξ ) = ∫ qx( )edix x (24) 2π −∞ where b is a length parameter depending on the material surface property. It should be pointed out that this parameter plays a critical role in the surface elasticity. Substituting Equation (24) into the surface condition Equations (12) and (13), the solution of stresses and displaces under pure shear load were obtained

2054 L. Y. Wang et al.

1 ∞ −−ξξ σ= ξ− ξξix z ξ xx ∫ ( zB2) ( ) ed 2π −∞

1 ∞ −−ξξ σ= − ξξ2 ix z ξ zz ∫ zB( )ed 2π −∞

i ∞ −−ξξ σ= ξξ− ξix z ξ xz ∫ (1zB) ( ) ed (25) 2π −∞

i ∞ −−ξξ = ξ−− ξξix z + u( xz, ) ∫ (z 2(1 v )) B( ) e d C1 22G π −∞

1 ∞ −−ξξ = −+ξξix z ξ+ wxz( ,) ∫ (12v z) B( ) e d C2 22G π −∞ Substituting Equation (24) into Equations (22), one obtains q 2 1− cos (aξ ) q (ξ ) = 0 (26) a π ξ 2

Therefore B(ξ ) is given by iq 2 1− cos (aξ ) B (ξ ) = 0 (27) a π (1+ bξξ) 3

Substituting Equation (27) into Equations (25), the stresses component and displaces component are obtained as

2q ∞ zξ − 2 sin ( xξ ) σ= 0 − ξξ−zξ xx ∫ 2 1 cos(a ) e d πab0 1+ ξ ξ

2q ∞ z sin ( xξ ) σ=−−0 ξξ−zξ zz ∫  1 cos(a ) e d πab0 1+ ξξ

2q ∞ 1− zξ cos( xξ ) σ=−−0 ξξ−zξ xz ∫ 2 1 cos(a ) e d (28) πab0 1+ ξ ξ

q ∞ zvξ −−21( ) =−−0 ξξ−zξ ξ + u( xz, ) ∫ 3 cos( x)  1 cos( a ) e d C1 πGa 0 ξξ(1+ b )

q ∞ 12−+vzξ = 0 ξ− ξξ−zξ wxz( , ) ∫ 3 sin( x)  1 cos( a ) e d πGa 0 ξξ(1+ b )

It is seen, when b = 0 , that is, the surface influence is ignored in Equation (28), the stresses and displacements of the half-plane are consistent with those in the classical elastic results [24], respectively. 2q rr r σ=0 12 +2 −θθ +− θ xx 2x ln2 2 az ln 3( 12 2 ) πarr 1 zq σ=−0 ( θθ +−2 θ) (29) zz πa 12

q0 rr12 σxz =( xa −) θ12 ++( xa) θθ −2 x + 2 z ln  πa r 2

On the contact surface z = 0 , the stresses is given by

2055 L. Y. Wang et al.

−1 ∞ 4q0 1− costb    x  σ xx ( x,0) =−+ 1t  sin  tt  d π ∫0 t 2 aa    −1 (30) ∞ 2q0 1− costb    x  σ xz ( x,0) =−+ 1t  cos  tt  d π ∫0 t 2 aa   

Based on the previous assumption, ur( 0 ,0) = 0, the displacement on the surface component is derived as −1 21( − v) qa ∞ 1− costb    x  r =0 +−0 ux( ,0) ∫ 3 1t  cos  t  cos t d t πG0 t a   aa   (31) −1 (12− v) qa0 ∞ 1− costb    x  wx( ,0) =  1+ t  sin  t  d t πG∫0 t3 aa    It is instructive to examine the influence of the surface elasticity on the stresses and displacements of the contact surface and compare them with those in classical contact

problem. Figure 5 and Figure 6 show the distribution of the stresses σ xx and σ xz on the contact surface, where the solution of b/a = 0 is consistent with the classical elastic result.

It can be seen from Figure 7 that the normal stress σ xx transits continuously across the loading boundary xa= ± , which is increasing monotonically with respect to x in the loading region ( xa< 0.75 ), and the inverse is observed outside the loading region ( xa> 0.75 ). It means that the analytical expressions can give approximate results in the region that is far from the loading boundary. It is also found Figure 5 that shear stress changes smoothly across the loading boundary xa= ± , which is different from a

0.0

-0.2

0 -0.4 b/a=0

b/a=0.1

(x,0)/q b/a=0.5 13 σ -0.6 b/a=2

-0.8

-1.0 0 1 2 3 4

x/a

Figure 5. The distribution of the shear stress σ xz under tangential triangle distribution load.

2056 L. Y. Wang et al.

Figure 6. The distribution of surface displacement u under tangential triangle distribution load.

0.0

-0.2

-0.4 0 b/a=0 -0.6 b/a=0.1 (x,0)/q xx σ b/a=0.5 -0.8 b/a=2.0

-1.0

-1.2 0 1 2 3 4 x/a

Figure 7. The distribution of the normal stress σ xx under tangential triangle distribution load. singularity predicted by classical elasticity. Due to the different surface elasticity value, the horizontal displacement is displayed in Figure 6, where we set ra0 = 5 , and M1 =21( − vG) π . It is seen that the slope of

2057 L. Y. Wang et al.

0.8

0.6 a 0 q 2

0.4 b/a=0 b/a=0.1 w(x,0)/M b/a=0.5 b/a=2.0 0.2

0.0 0 1 2 3 4 5 x/a Figure 8. The distribution of surface indentation w under tangential triangle distribution load.

the deformed surface for a > 0 is continuous everywhere. It is also found the horizontal displacement decreases with the increase of surface elasticity. The indent depth is plot-

ted in Figure 8 with M2 =(12 − vG) π , which also shows that the normal displacement is continuous everywhere on the deformed surface. In addition, the indent depth decreases continuously with the increase of surface elasticity.

6. Conclusion

In this paper, we consider the two-dimensional contact problem in the light of surface elasticity theory. Fourier integral transform method is adopted solving general analytical solution. For two particular loading cases of triangle distribution forces, the results are analyzed in detail and compared with the classical linear elastic solutions. A series of theoretical and numerical results show that the influences of the surface tension and the surface elasticity on the stresses and displacements are not always equal. It is found that the surface elasticity theory illuminates some interesting characteristics of contact problems at nano-scale, which are distinctly different from the classical solutions of elasticity without surface effects. Therefore, the influence of surface effects should be considered for nano-contact problems.

References [1] Cammarata, R.C. (1994) Surface and Interface Stress Effects in Thin Films. Progress in Sur- face Science, 46, 1-38. http://dx.doi.org/10.1016/0079-6816(94)90005-1 [2] Miller, R.E. and Shenoy, V.B. (2000) Size-Dependent Elastic Properties of Nanosized Structural Elements. Nanotechnology, 11, 139-147.

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http://dx.doi.org/10.1088/0957-4484/11/3/301 [3] Huang, Y., Zhang, F. and Hwang, K.C. (2006) A Model of Size Effects in Nano-Indentation. Journal of the Mechanics & Physics of Solids, 54, 1668-1686. http://dx.doi.org/10.1016/j.jmps.2006.02.002 [4] Dingreville, R., Qu, J. and Cherkaoui, M. (2005) Surface Free Energy and Its Effect on the Elastic Behavior of Nano-Sized Particles, Wires And films. Journal of the Mechanics & Physics of Solids, 53, 1827-1854. http://dx.doi.org/10.1016/j.jmps.2005.02.012 [5] Yang, F. (2004) Size-Dependent Effective Modulus of Elastic Composite Materials: Spheri- cal Nanocavities at Dilute Concentrations. Journal of Applied Physics, 95, 3516-3520. http://dx.doi.org/10.1063/1.1664030 [6] Gao, X., Hao, F. and Huang, Z. (2014) Mechanics of Adhesive Contact at the Nanoscale: The Effect of Surface Stress. International Journal of Solids & Structures, 51, 566-574. http://dx.doi.org/10.1016/j.ijsolstr.2013.10.017 [7] Wang, J.X., Huang, Z.P., et al. (2011) Surface Stress Effect in Mechanics of Nanostructured Materials. Acta Mechanica Solida Sinica, 24, 52-82. http://dx.doi.org/10.1016/s0894-9166(11)60009-8 [8] Gurtin, M.E. and Murdoch, A.I. (1975) A Continuum Theory of Elastic Material Surfaces. Archive for Rational Mechanics & Analysis, 57, 291-323. http://dx.doi.org/10.1007/BF00261375 [9] Gurtin, M.E. and Murdoch, A.I. (1976) Effect of Surface Stress on Wave Propagation in Solids. Journal of Applied Physics, 47, 4414-4421. http://dx.doi.org/10.1063/1.322403 [10] Gurtin, M.E. and Murdoch, A.I. (1978) Surface Stress in Solids. International Journal of Solids & Structures, 14, 431-440. http://dx.doi.org/10.1016/0020-7683(78)90008-2 [11] Jing, G.Y., Duan, H.L., Sun, X.M., et al. (2006) Surface Effects on Elastic Properties of Silver Nanowires: Contact Atomic-Force Microscopy. Physical Review B, 73, 235409. http://dx.doi.org/10.1103/physrevb.73.235409 [12] Shenoy, V.B. (2005) Atomistic Calculations of Elastic Properties of Metallic Fcc Crystal Surfaces. Physical Review B, 71, 4104. http://dx.doi.org/10.1103/PhysRevB.71.094104 [13] Cammarata, R.C., Sieradzki, K. and Spaepen, F. (2000) Simple Model for Interface Stresses with Application to Misfit Dislocation Generation in Epitaxial Thin Films. Journal of Ap- plied Physics, 87, 1227-1234. http://dx.doi.org/10.1063/1.372001 [14] Wei, G., Shouwen, Y. and Ganyun, H. (2014) Finite Element Characterization of the Size- Dependent Mechanical Behaviour in Nanosystems. Nanotechnology, 17, 1118–1122. http://dx.doi.org/10.1088/0957-4484/17/4/045 [15] Wang, G.F. and Feng, X.Q. (2007) Effects of Surface Stresses on Contact Problems at Na- noscale. Journal of Applied Physics, 101, 013510. http://dx.doi.org/10.1063/1.2405127 [16] Zhao, X.J. and Rajapakse, R.K.N.D. (2009) Analytical Solutions for a Surface-Loaded Iso- tropic Elastic Layer with Surface Energy Effects. International Journal of Engineering Science, 47, 1433-1444. http://dx.doi.org/10.1016/j.ijengsci.2008.12.013 [17] Long, J.M., Wang, G.F. and Feng, X.Q. (2012) Two-Dimensional Hertzian Contact Problem with Surface Tension. International Journal of Solids & Structures, 49, 1588-1594. http://dx.doi.org/10.1016/j.ijsolstr.2012.03.017 [18] Long, J.M. and Wang, G.F. (2013) Effects of Surface Tension on Axisymmetric Hertzian Contact Problem. Mechanics of Materials, 56, 65-70. http://dx.doi.org/10.1016/j.mechmat.2012.09.003 [19] Lei, D.X., Wang, L.Y. and Ou, Z.Y. (2012) Elastic Analysis for Nanocontact Problem with

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Surface Stress Effects under Shear Load. Journal of Nanomaterials, 6, 4531-4531. http://dx.doi.org/10.1155/2012/505034 [20] Gao, X., Hao, F., Fang, D., et al. (2013) Boussinesq Problem with the Surface Effect and Its Application to Contact Mechanics at the Nanoscale. International Journal of Solids & Structures, 50, 2620-2630. http://dx.doi.org/10.1016/j.ijsolstr.2013.04.007 [21] Gao, X., Hao, F., Fang, D., et al. (2014) Mechanics of Adhesive Contact at the Nanoscale: The Effect of Surface Stress. International Journal of Solids & Structures, 51, 566-574. http://dx.doi.org/10.1016/j.ijsolstr.2013.10.017 [22] Povstenko, Y.Z. (1993) Theoretical Investigation of Phenomena Caused by Heterogeneous Surface Tension in Solids. Journal of the Mechanics & Physics of Solids, 41, 1499-1514. http://dx.doi.org/10.1016/0022-5096(93)90037-G [23] Wang, L.Y. (2015) Elastic Analysis for Contact Problems with Surface Effects under Normal Load. Mathematics & Mechanics of Solids. http://dx.doi.org/10.1177/1081286514568585 [24] Johnson, K.L. (1985) Contact Mechanics. Cambridge University Press, London. http://dx.doi.org/10.1017/CBO9781139171731

2060 Journal of Applied Mathematics and Physics, 2016, 4, 2061-2068 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals

Heli Gao

Department of Mathematics, Binzhou University, Binzhou, China

How to cite this paper: Gao, H.L. (2016) Abstract Integro-Differential Equations for a Jump- Diffusion Risk Process with Dependence The classical Poisson risk model in ruin theory assumed that the interarrival times between Claim Sizes and Claim Intervals. between two successive claims are mutually independent, and the claim sizes and Journal of Applied Mathematics and Phys- claim intervals are also mutually independent. In this paper, we modify the classical ics, 4, 2061-2068. http://dx.doi.org/10.4236/jamp.2016.411205 Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian mo- Received: October 30, 2016 tion, and assume that the claim sizes and claim intervals are dependent. Using the Accepted: November 19, 2016 properties of conditional expectation, we establish integro-differential equations for Published: November 22, 2016 the Gerber-Shiu function and the ultimate ruin probability. Copyright © 2016 by author and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative Commons Attribution International Jump-Diffusion Risk Process, Diffusion, Geometric Brownian Motion, Gerber-Shiu License (CC BY 4.0). Function http://creativecommons.org/licenses/by/4.0/ Open Access

1. Introduction

Various papers in ruin theory modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. An extension of the classical model is that the risk process perturbed by a diffusion was first introduced by Gerber [1] and has been further studied by many authors during the last few years, e.g. Dufresne and Gerber [2], Gerber and Landry [3], Wang and Wu [4], Tsai and Willmot [5] [6], Chiu and Yin [7], and the references therein. In the risk process that is perturbed by diffusion, the surplus process Ut( ) of an insurance portfolio is given by

DOI: 10.4236/jamp.2016.411205 November 22, 2016 H. L. Gao

Nt( )

U( t) =+− u ct∑ Xi +σ11 B( t), t ≥ 0, (1) i=1

where u ≥ 0 is the initial surplus, c > 0 is the positive constant premium income Nt( )

rate, St( ) = ∑ Xi is the aggregate claims process, in which i=1

Nt( ) =sup{ kT : 12 +++≤ T Tk t}

is the claim number process (denoting the number of claims up to time t), and the in- ∞ terarrival times {T } is a sequence of positive random variables. {Xi,1≥ } is a se- i i=1 i quence of nonnegative independent identically distributed (i.i.d.) random variables with

distribution function Fx( ) =1 − Fx( ) and density function µ = EX( ), {Bt1 ( ),0 t≥ } is a standard Brownian motion that is independent of the aggregate claims process

St( ) , σ1 is a positive constant. ∞ It is explicitly assumed in these papers that the interarrival times {T } and the i i=1 claim sizes {Xii ,1≥ } are mutually independent. However, this assumption is often too restrictive in practice, and there is a need for more general models where the inde- pendence assumptions can be relaxed. Recently, various results have been obtained concerning the asymptotic behavior of the probability of ruin for dependent claims, see [8]-[14], as well as the references therein. Zhao [14] assumed that the distribution of the time between two claim occurrences depends on the previous claim size. Motivated by the results of Zhao [14], the main aim of this paper is to modify the risk model (Eq- uation (1)), and establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability in the new risk model.

2. Improved Risk Model

In this paper, it is assumed that the claim occurrence process to be of the following type:

If a claim X i is larger than a random variable Yi , then the time until the next claim

Ti+1 is exponentially distributed with rate λ1 > 0 , otherwise it is exponentially distri-

buted with rate λ2 > 0 . The quantities Yi are assumed to be i.i.d. random variables with distribution function Gx( ) . Assuming that PX( >≤ Y) PX( Y) µ <+c , λλ12

which is the net profit condition. In the daily operation of insurance company, in addition to the premium income and claim to the operation of spending has a great influence on the outside, and there is also a factor that interest rates should not be neglected. As in [15], this paper assume that the risk model Equation (1) is invested in a stochastic interest process which is assumed ∆t rt+σ 22 B( t) to be a geometric Brownian motion {ee= } , where r and σ2 are positive con-

stants, and {Btt2 ( ),0≥ } is a standard Brownian motion independent of {Ut( ),0 t≥ }. Let Xt( ) denote the surplus of the insurer at time t under this investment assumption. Thus,

2062 H. L. Gao

∆tst −∆ Xt( ) =+e u e d Us( ) , t ≥= 0, X( 0) u . (2) ( ∫0 )

Denote T to be the ruin time (the first time that the surplus becomes negative), i.e., T=inf{ tXt :( ) ≤= 0 X( 0) u} , and T = ∞ if Xt( ) >0, ∀≥ t 0 . This article is interested in the expected discounted penalty (Gerber-Shiu) function: −δT  muEδ ( ) = {e, wXT( −) XTIT( ) ( <∞) X( 0) = u} , (3) where I (⋅) is the indicator function, δ > 0 is the force of interest and wxy( , ) is a nonnegative function of xy>>0, 0 and satisfies w(0, 0) = 1.

Furthermore, let T1 be the time when the first claim occurs, and random variable

T1i being exponentially distributed with rate λi > 0 . Assuming that

PT( = T11 ) = P( X > Y) , PT( = T12 ) = P( X ≤ Y) .

For i = 1, 2 , define = −δT − <∞ = = muEδ (i) ( ) {e wXT( ) , XTIT( ) ( ) X(0,) uTT1i } , (4) such that

mδ ( u) = mδδ(12) ( uPX) ( >+ Y) m( ) ( uPX) ( ≤ Y) , (5) then, mmδ (0) =δδ(12) ( 0) = m( ) ( 01) = .

3. Integro-Differential Equations for muδ (i) ( )

In this section, a system of integro-differential equations with initial value conditions satisfied by the Gerber-Shiu function muδ (i) ( ) is derived. σ 2 Define ar= + 2 , and 2

∆tstt−∆ −∆ s Y=++e u c ed sσ ed Bs( ) . (6) t ( ∫∫0011)

Lemma 3.1 Let AYtt=sup{ ∨ 0} for t > 0 . For ε > 0 define the hitting time st∈[0, ]

Tε =inf{ t > 0, YAtt −=−ε} . Then, for α > 0 , it can be concluded that

−αTε E(e , TXuε <∞( 0) =) = 1 +οε( ) , ε → 0. (7)

Proof Vt:= YA tt − is a reflecting diffusion with generator

2 1 2 22 ddfx( ) fx( ) Lf( x) =(σσ12 +x ) ++(c ax) , 2ddx2 x acting on functions satisfying the reflecting boundary condition f ′(00) = . If ∂f( xt, ) Lf( x, t) + =0,x <> 0, t 0, ∂t

2063 H. L. Gao

∂f( xt, ) and = 0 for t > 0, then, according to Itô’s formula fVt( , ) is a local mar- ∂ t x x=0 tingale. Using the separation variable technique, we find that

−α f( xt,e) = t g( x) z( x)

is a solution, where

o c+ av gx( ) = exp dv ∫x 2 22 , σσ12+ v

zx( ) is a solution of z′′ ( x) += rxzx( ) ( ) 0 .

Here ′′ 2 22 ′ g( x)(σσ12+ x) ++22( c ax) g( x) −α g( x) rx= ( ) 2 22 . gx( )(σσ12+ x)

∂f( xt, ) z′(0) c Using the initial condition = 0 for t > 0 , we get = , conse- ∂ σ 2 x x=0 z (0) 1 quently z (0) c =++1 2 ε οε( ) . z (ε ) σ1

Applying the Optional Stopping Theorem, it follows that

−αTε z(0e) = E( gz( −εε) ( −) , Tε <∞ X( 0) = u) ,

and thus

−αTε z (0) E(e, TXuε <∞( 0) =) = =1 +οε( ) , ε → 0. gz(−−εε) ( )

This ends the proof of Lemma 3.1. Similarly, the following lemma can also be obtained.

Lemma 3.2 Let BYtt=sup{ ∧ 0} for t > 0 . For ε > 0 define the hitting time st∈[0, ]

Tε =inf{ t > 0, YAtt −=ε} . Then, for α > 0 , it can be concluded that

−αTε E(e , TXuε <∞( 0) =) = 1 +οε( ) , ε → 0. (8)

Theorem 3.1 Assuming that muδ (i) ( ) is second order continuously differentiable

functions in u, then muδ (i) ( ) satisfies the following integro-differential equation

1 2 22 ′′ ′ (σ12+ σumucaumu) δ1( ) ++( ) δδ 11( ) −(λδ1 +) mu( ) 2 ( ) ( ) ( ) u +λ PY( ≤ ym) ( u −+ y) PY( > ym) ( u − y) d F( y) , (9) 1 ∫0 δδ(12) ( ) +∞ +λ wuy( ,d −= u) F( y) 0 1 ∫u

2064 H. L. Gao

1 2 22 ′′ ′ (σ12+ σumucaumu) δδδ222( ) ++( ) ( ) −(λδ2 +) mu( ) 2 ( ) ( ) ( ) u +λ PY( ≤ ym) ( u −+ y) PY( > ym) ( u − y) d F( y) , (10) 2 ∫0 δδ(12) ( ) +∞ +λ wuy( ,d −= u) F( y) 0 2 ∫u with the initial value conditions ′′ mmδδ(12) (0) =( ) ( 01) = , mmδδ(12) (0) =( ) ( 00) = .

Proof Let T11 be the time when the first claim occurs which exponentially distributed with rate λ1 > 0 . Consider the risk process {Xt( ) :0 t≥ } defined by Equation (2) in an infinitesimal time interval (0,t) . There are three possible cases in (0,t) as follows.

1) There are no claims in (0,t) with probability 1− λ1t , thus Xt( ) = Yt( ) ;

2) There is exactly one claim in (0,t) with probability λ1t . According to different of the claim amount, there are three possible cases in this case as follows. a) The amount of the claim y< Yt( ) , i.e., ruin does not occur, and thus Xt( ) = Yt( ) − y; b) The amount of the claim y> Yt( ) , i.e., ruin occurs due to the claim; c) The amount of the claim y= Yt( ) , i.e., ruin occurs due to oscillation (observe that the probability that this case occurs is zero). 3) There is more than one claim in (0,t) with probability ο (t) . Thus, considering the three cases above and noting that {Xt( ) :0 t≥ } is a strong Markov process, we have

Y mu( ) =−e1−−δδtt( λλ tEmY) ( ) +e tEPYymYy t ( ≤−) ( ) δδ(11) 11( ) tt∫0 ( δ(1) +>PY ym Y − yd F y ( ) δ (2) ( t )) ( ) (11) +∞ −δ t + eλο1dt wY( tt ,d y −+ Y) F( y) ( t) ∫Yt

−δ By Taylor expansion, we have e1t =−+δοtt( ) , thus Equation (11) becomes λδ+  ( 1 )tE mδ (1) ( Yt ) Y =EmYmu( ) −+( ) e−δ t λ tEPYymYy t ( ≤) ( −) δδ(11) tt( ) 1 ∫0 ( δ(1) (12) +>PY ym Y − yd F y ( ) δ (2) ( t )) ( ) +∞ −δ t +eλο1dt wY( tt ,d y −+ Y) F( y) ( t) ∫Yt

Then, by Itô’s formula we have Em( Y) − m( u) δδ(11) t ( ) ′ 1 2 22 ′′ lim =+(c au) mδδ(12) ( u) ++(σσ12u) m( ) ( u) . (13) t→0 t 2 Therefore, by dividing t on both sides of Equation (12), letting t → 0 , using Equa- tion (13), we obtain Equation (9), and similarly we can obtain Equation (10).

The condition mmδδ(12) (0) =( ) ( 01) = follows from the oscillating nature of the

2065 H. L. Gao

′′ sample paths of {Xt( )} . Now, we prove mmδδ(12) (0) =( ) ( 00) = .

For all ε > 0 , let T=inf{ t > 0, YAtt −=−ε} , τ =TT ∧ 11 . Then, by the strong property of {Xt( )} , it can be concluded that = −δτ τ = mδδ(11) (0) E( e mX( ) ( ( ) X(00) )) =−δτ τ <= E( e mδ (1) ( X( ) IT( T11 ) X(00) )) + −δτ τ > = ≡+ E( e mδ (1) ( X( ) IT( T11 ) X(00) )) I1 I 2

According to Lemma 3.1, it can be concluded that

= −ε −+(αλ1)T <∞ = = −ε + οε Im1 δδ(11) (0) E( e , T X( 00) ) m( ) ( 0) ( ) ,

+∞ +∞ =λ −+(αλ1 )T >=+− = I21∫∫ e PT( sX(00) ) E( mδ (1) ( Ys x) X(00dd) ) sF( x) 00 = οε( ) ′ Thus, mmδδ(11) (00) =( ) ( −+ε) οε( ) , and correspondingly mδ (1) (00−=) . Similar ′ results can be derived for mδ (2) (0 −) .

And for all ε > 0 , let T′ =inf{ t > 0, YAtt −=ε} , τ ′′=TT ∧ 11 , according to Lemma ′′ ′′ 3.2 we obtain mmδδ(12) (0+=) ( ) ( 00 +=) , thus mmδδ(12) (0) =( ) ( 00) = . This ends the proof of Theorem 3.1.

4. Differential Equations for Φi (u)

Let δ = 0 and wxy( ,1) ≡ in Equation (3), correspondingly the expected discounted

penalty function muδ ( ) turns into the ultimate ruin probability Φi (u) . Obviously,

Φ(u) =Φ12( uPX) ( > Y) +Φ( uPX) ( ≤ Y) ,

and

Φ=Φ=Φ=(0) 12( 0) ( 01) .

Suppose that PY( =0) = p, PY( = u) =1 − p q,

and X i is exponentially distributed with rate ββ( > 0) . Then, we get the following theorem.

Theorem 4.1 Assuming that Φi (u) is second order continuously differentiable

functions in u, then Φi (u) satisfies the following integro-differential equation 11  1 σ2+ σ 22 Φ′′′ +−β σ22 + + σ 2 + −βσ 2 Φ′′ ( 12u) 1( u)  2 u( a 2) uc1 1( u) 22  2 (14) ′ +−βauac +( − λ1 − β) Φ 1( uq) + λβ 11 Φ( uq) = λβ 12 Φ( u) 11  1 σ2+ σ 22 Φ′′′ +−β σ22 + + σ 2 + −βσ 2 Φ′′ ( 12u) 2( u)  2 u( a 2) uc1 2( u) 22  2 (15) ′ +−βau +( a − λ2 − c β) Φ 2( u) + p λβ 22 Φ( u) = p λβ 21 Φ( u)

2066 H. L. Gao

with the initial value conditions Φ=Φ=Φ=Φ=(001,000.) ( ) ′′( ) ( ) 12 12 Proof According to Equation (9), it can be concluded that

1 2 22 ′′ ′′ (σσ1+ 2u) Φ++Φ−Φ 1( u) ( c au) 1( u) λ 11( u) 2 u +λp Φ( uy −) +Φ q( uy −) βλed−−ββyu y + e = 0 11∫0  2 1

By taking the derivative with respect to u on both sides of the above formula, and after some careful calculations, we obtain Equation (14). And similarly we can prove that Equation (15) holds. This ends the proof of Theorem 4.1.

5. Conclusion

In this paper, we consider a jump-diffusion risk process compounded by a geometric Brownian motion with dependence between claim sizes and claim intervals. We derive the integro-differential equations for the Gerber-Shiu functions and the ultimate ruin probability by using the martingale measure. Further studies are needed for the numerical solution of Equations (9), (10), (14) and (15). The results derived in this paper can be generalized to similar dependence ruin models.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 11601036), the Natural Science Foundation of Shandong (No. ZR2014GQ005) and the Natural Science Foundation of Binzhou University (No. 2016Y14).

References [1] Gerber, H.U. (1970) An Extension of the Renewal Equation and Its Application in the Col- lective Theory of Risk. Skandinavisk Aktuarietidskrift, 1970, 205-210. [2] Dufresne, F. and Gerber, H.U. (1991) Risk Theory for the Compound Poisson Process That Is Perturbed by Diffusion. Insurance: Mathematics and Economics, 10, 51-59. https://doi.org/10.1016/0167-6687(91)90023-Q [3] Gerber, H.U. and Landry, B. (1998) On the Discounted Penalty at Ruin in a Jump-Diffusion and the Perpetual Put Option. Insurance: Mathematics and Economics, 22, 263-276. https://doi.org/10.1016/s0167-6687(98)00014-6 [4] Wang, G. and Wu, R. (2000) Some Distributions for the Classical Risk Process That Is Per- turbed by Diffusion. Insurance: Mathematics and Economics, 26, 15-24. https://doi.org/10.2307/253675 [5] Tsai, C.C.L. and Willmot, G.E. (2002) A Generalized Defective Renewal Equation for the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 30, 51-66. https://doi.org/10.1016/s0167-6687(01)00096-8 [6] Tsai, C.C.L. and Willmot, G.E. (2002) On the Moments of the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 31, 327-350. https://doi.org/10.1016/s0167-6687(02)00159-2 [7] Chiu, S.N. and Yin, C.C. (2003) The Time of Ruin, the Surplus Prior to Ruin and the Deficit

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at Ruin for the Classical Risk Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 33, 59-66. https://doi.org/10.1016/s0167-6687(03)00143-4 [8] Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006) On a Risk Model with Dependence between Interclaim Arrivals and Claim Sizes. Scandinavian Actuarial Journal, 5, 265-285. https://doi.org/10.1080/03461230600992266 [9] Bargès, M., Cossette, H., Loisel, S. and Marceau, E. (2011) On the Moments of the Aggre- gate Discounted Claims with Dependence Introduced by a FGM Copula. ASTIN Bulletin, 41, 215-238. [10] Li, J. (2012) Asymptotics in a Time-Dependent Renewal Risk Model with Stochastic Return. Journal of Mathematical Analysis and Applications, 387, 1009-1023. https://doi.org/10.1016/j.jmaa.2011.10.012 [11] Yong, W. and Xiang, H. (2012) Differential Equations for Ruin Probability in a Special Risk Model with FGM Copula for the Claim Size and the Inter-Claim Time. Journal of Inequali- ties and Applications, 2012, 156. https://doi.org/10.1186/1029-242X-2012-156 [12] Fu, K.A. and Ng, C.Y.A. (2014) Asymptotics for the Ruin Probability of a Time-Dependent Renewal Risk Model with Geometric Lévy Process Investment Returns and Dominatedly- Varying-Tailed Claims. Insurance Mathematics & Economics, 56, 80-87. https://doi.org/10.1016/j.insmatheco.2014.04.001 [13] Zhang, J. and Xiao, Q. (2015 ) Optimal Investment of a Time-Dependent Renewal Risk Model with Stochastic Return. Journal of Inequalities and Applications, 2015,181. https://doi.org/10.1186/s13660-015-0707-3 [14] Zhao, X.H. and Yin, C.C. (2009) A Jump-Diffusion Risk Model with Dependence between Claim Sizes and Claim Intervals. Acta Mathematica Scientia, 29A, 1657-1664. [15] Cai, J. and Xu, C. (2005) On the Decomposition of the Ruin Probability for a Jump-Diffusion Surplus Process Compounded by a Geometric Brownian Motion. North American Actuarial Journal, 10, 120-132. https://doi.org/10.1080/10920277.2006.10596255

2068 Journal of Applied Mathematics and Physics, 2016, 4, 2069-2078 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

An Alternating Direction Nonmonotone Approximate Newton Algorithm for Inverse Problems

Zhuhan Zhang, Zhensheng Yu, Xinyue Gan

University of Shanghai for Science and Technology, Shanghai, China

How to cite this paper: Zhang, Z.H., Yu, Abstract Z.S. and Gan, X.Y. (2016) An Alternating Direction Nonmonotone Approximate New- In this paper, an alternating direction nonmonotone approximate Newton algorithm ton Algorithm for Inverse Problems. Journal (ADNAN) based on nonmonotone line search is developed for solving inverse prob- of Applied Mathematics and Physics, 4, 2069- lems. It is shown that ADNAN converges to a solution of the inverse problems and 2078. http://dx.doi.org/10.4236/jamp.2016.411206 numerical results provide the effectiveness of the proposed algorithm.

Received: October 28, 2016 Keywords Accepted: November 27, 2016 Published: November 30, 2016 Nonmonotone Line Search, Alternating Direction Method, Bound-Constraints, Newton Method Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International 1. Introduction License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ We consider inverse problems that can be expressed in the form Open Access 1 2 min Ax−+ bφ ( Bx) , (1) 2 ×× where AR∈mn, BR ∈ mn ,:φ R m→( −∞ , ∞) is convex, and bR∈ m . The emphasis of our work is on problems where A and B have a specific structure. It can be applied to many applications, especially in machine learning [1] [2], image reconstruction [3] [4] or model reduction [5]. We assume that the functions in (1) are strictly convex, so both problems has an unique solution x* . In Hong-Chao Zhang’s paper [6], he uses the Alternating Direction Approximate Newton method (ADAN) based on Alternating Direction Method (ADMM) which ori- ginaly in [7] to solve (1). He employs the BB approximation to increase the iterations. In many applications, the optimization problems in ADMM are either easily resolvable, since ADMM iterations can be performed at a low computational cost. Besides, com-

DOI: 10.4236/jamp.2016.411206 November 30, 2016 Z. H. Zhang et al.

bine different Newton-based methods with ADMM have become a trend, see [6] [8] [9], since those methods may achieve the high convergent speed. In alternating direction nonmonotone approximate Newton (ADNAN) algorithm developed in this paper, we adopt the nonmonotone line search to replace the traditional Armijo line search in ADAN, because the nonmonotone schemes can improve the likelihood of finding a global optimum and improve convergence speed in cases where a monotone scheme is forced to creep along the bottom of a narrow curved val- ley in [10]. In the latter context, the first subproblem is to solve the unconstrained minimization problems with Alternating Direction Nonmonotone Approximate Newton algorithm. The purpose is to accelerate the speed of convergence, and then to project or the scale the unconstrained minimizer into the box {wR∈m : lwu ≤≤} , The second subproblem is a bound-constrained optimization problem. The rest of the paper is organized as follows. In Section 2, we give a review of the alternating direction approximate Newton method. In Section 3, we introduce the new algorithm ADNAN. In Section 4, we introduce the gradient-based algorithm of the second subproblem. A preliminary convergence analysis for ADNAN and gradient- based algorithm (GRAD) is given in Section 5. Numerical results presented in Section 6 explain the effectiveness of ADNAN and GRAD.

2. Review of Alternating Direction Approximate Newton Algorithm

In this section, we briefly review the well-known Alternating Direction Approximate Newton (ADAN) method which has been studied in the areas of convex programming and image reconstruction see [4] [6] and references therein. We introduce a new variable w to obtain the split formulation of (1):

1 2 min Axb−+φ ( w) s.t. wBxxRwR = , ∈n , ∈m (2) xw, 2 The augmented Lagrangian function associated with (2) is

1 22β Lxw( ,,λ) = Axb − + φλ( w) +( Bxw −+) Bxw − (3) 22 where β > 0 is the penalty parameter, bR∈ m is a Lagrangian multiplier associated with the constraint w= Bx . In ADMM, each iteration minimizes over x holding w fixed, minimizes over w holding x fixed, and updates an estimate for the multiplier b. More specifically, if λ k is the current approximation to the multiplier, then ADMM [10] [11] applied to the split formulation (3) is given by the iteration: xk+1 = arg min L xw ,kk , λ (4) x ( ) wk++11= arg min Lxkk , w , λ (5) w ( )

k+1 k kk λ=+− λβ(w Bx ) (6)

And (4) can be written as follows:

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xk+1 = arg min fx( ) x

1 2 β − 2 f( x) = Ax −+ b Bx −+ wkkβλ1 (7) 22 ∈ NN× 2 = For any Hermitian matrix QR , we define x Q x, Qx , if Q is a positive de- • finite matrix, then Q is a norm. The proximal version of (4) is xk+1 = arg min fx( ) x

1 2 β − 2 f( x) = Ax − b + Bx − wkk +βλ1 +−x x k (8) 22 Q 2 In ADAN, the choice Q=δ I − AAT will cancel the Ax term in this iteration 2 while Bx is retained. We replace δ by δk , where δk I is a Barzilai-Borwein (BB) [8] [12] approximation to AAT . We can observe the fast convergence of BB approximation in the experiments of Raydan and Svaiter [13]. Moreover, δk is strictly smaller T T than the largest eigenvalue of AA, and Q=δk I − AA is indefinite, so the new convergence analysis is needed. As a result, the updated version for x given in (4) can be expressed as follows:

− kk+1 TT1 x=arg min f( x) =−+ x AAβ BB ∇ fk (9) x ( )

Tk T1kk− k ∇=fk : A( Ax −+ b) β B( Bx − w +βλ) (10)

−1 Here, (.) is the generalized inverse, AATT+ β BB is the Hessian of the objective k k+1 f , and ∇f is the gradient of f at x , gfk = ∇ . The formula for x in (2) is exactly the same formula that we would have gotten if we performed a single iteration of Newton’s method on the equation ∇=fx( ) 0 with starting guess xk . We employ T the BB approximation AA≈ δk I, [14] where

2 δ=arg min Axxkk −−−−−11δ xx kk : δδ ≥ k { ( ) ( ) min }

2 kk−1 Ax( − x ) = δ maxmin , 2 xxkk− −1  and δmin > 0 is a positive lower bound for δk . Hence, the Hessian is approximated by T δβk I+ BB. Since a Fourier transform can be inverted in ON( log N) flops. The in- T T version of δk I+ BB can be accomplished relatively quickly. After replacing AA −1 kk+1 T by δk I , the iteration becomes x= xd + k where dkk=−+∇(δβ I BB) fk T Note that by substituting Q=δk I − AA in (2) and solving for the minimizer, we would get exactly the same formula for the minimizer as that given in (5). When the search direction is determined suitable step size αk along this direction should be kk+1 found to determine the next iterative point such as xx= + αkk d.

The inner product between dk and the objective gradient at xk is ∇=−+δβ22 fkk, d( kk d Bdk) .

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It follows that dk is a descent direction. Since f is a quadratic, the Taylor expan- k+1 sion of fx( ) around xk is as follows:

k+1 k kk ++ 111 kk2 fx( ) = fx( ) +∇ fxk , − x + x − x 2 H where H= AATT + β BB Algorithm 1. Nonmonotone Line search

In this section, we adopt a nonmonotone line search method [9]. The step size αk is chosen in an ordinary Armijo line search which could not admit the more faster speed in unconstrained problems [12]. In contrast, nonmonotone schemes can not only improve the likelihood of finding a global optimum but also improve convergence speed.

Initialization: Choose starting guess x0 , and parameters 01≤≤≤ηηmin max ,

01 0 . Set C00= fx( ) , Q0 = 1, and k = 0 .

Convergence test: If  fx( k ) sufficiently small, then stop.

line search update: set xk+1 =+= x kα kk dd,0 k ,where αk satisfies either the (nonmonotone) Wolfe conditions:

fx( k+αα kk d) ≤+ C k m k ∇ fx( k) d k, (11)

∇fx( k +ασ kkk d) d ≥∇ fx( kk) d (12)

hk or the (nonmonotone) Armijo conditions: αk = αρ , where αk > 0 is the trial step,

and hk is the largest integer such that (11) holds and αµk ≤ .

Cost update: Choose ηk ∈[ ηηmin, max ] , and update

Qk+1=+=ηη kk Q1, C k+1( kkk QC + f( x k++11)) Q k (13)

Observe that Ck+1 is a convex combination of Ck and fx( k+1 ) . Since C00= fx( )

it follows that Ck is a convex combination of the function values fx( 0 ),, f( xk ) . The

choice of ηk controls the degree of nonmonotonicity. If ηk = 0 for each k, then the

line search is the usual monotone Wolfe or Armijo line search. If ηk = 1 for each k, 1 k then CAkk= where Ak= ∑ ffii, = fx( i) , is the average function value. The k +1 i=0

scheme with CAkk= was proposed by Yu-Hong Dai [15]. As ηk approaches 0, the

line search closely approximates the usual monotone line search, and as ηk approaches 1, the scheme becomes more nonmonotone, treating all the previous function

values with equal weight when we compute the average cost value Ck .

Lemma 2.1 If  fx( kk) d≤ 0 for each k, then for the iterates generated by the

nonmonotone line search algorithm, we have fCAkkk≤≤ for each k. Moreover, if

 fx( kk) d< 0 and fx( ) is bounded from below, then there exists αk satisfying either the Wolfe or Armijo conditions of the line search update.

3. Alternating Direction Nonmonotone Approximate Newton Algorithm

In Algorithm 1, we could get the x at each iteration which can be combined with Algo- rithm 2. Then, we use the Algorithm 2 to solve the first subproblem in this paper which

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is an unconstrained minimization problem with ADNAN, then to project or the scale the unconstrained minimizer into the box

1 2 min Ax−+ bφ ( Bx), x ∈ Rn (14) lwu≤≤ 2 the iteration is as follows: xk+1 = arg min L xw ,kk , λ (15) x ( ) wk++11= arg min Lxkk , w , λ (16) lwu≤≤ ( )

k+1 k kk λ=+− λβ(w Bx ) (17)

Later we give the existence and uniqueness result for (1). The solution wk to (5) has the closed-form means

k k kk kλ w= arg min L( x , w , b) = P Bx − lwu≤≤ β with P being the projection map onto the box [lu,.] Algorithm 2. Alternating Direction Nonmonotone Approximate Newton algorithm.

Parameter: 0.5<γ << 1 τρ , > 0,0 < δmin < δ0 , Initialize k = 1. k+1 kk Step 1: If  fx( k ) sufficiently small, then set x= xd, = 0,δδkk = −1 , and branch to Step 4.

2 kk−1 Ax( − x ) −1 =−+∇δβT δδk = maxmin , 2 , dkk( I BB) fk. xxkk− −1 

Step 2: If δαkk>> δ k−−1 α k, δ k max{ δmin , δk 1} then δmin= τδ min .

Step 3: If αk accomplish the Wolfe conditions, then αk := τα max . Step 4: Update x which generated from Algorithm 1. k kk+1 λ Step 5: w= P Bx − . β kk+1 kk++11 Step 6: b=+− bβ ( Bx w ). Step 7: If a stopping criterion is satisfied, terminate the algorithm, Otherwise k = k + 1 and go to Step 1. Lemma 3.1: we show some criteria that are only satisfied a finite number of times, so

δmin converge to positive limits. An upper bound for δmin is the following:

Uniformly in k, we have δmin,kk≤≤ δmax{ δτ , A } where δmin,k is the value of

δmin at the start of iteration k and δδ= min,1 is the starting δmin in ADAN. k Lemma 3.2: If Adkk= Bd = 0 , then x minimizes fx( k ).

4. Algorithm 3: Gradient-Based Algorithm (GRAD)

Next, we consider the second subproblem which is about bound-constrained optimization problem as

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1 2 min Ax−+ bφ ( Bx), w ∈ Rn (18) lxu≤≤ 2 And the iteration is similar with (4) (5) (6) as follows: xk+1 = arg min L xw ,kk ,λ (19) lxu≤≤ ( ) wk++11= arg min Lxkk , w ,λ (20) w ( ) λk+1 =+− λβ k(w kk Bx ) (21) Compute the solution xk from (19), xk= B−−11(λ kk + Pw) Compute the solution wk from (20), ( AATT+ BBw) k =++ Ab Tβλ w kk− 1 Set λk+1 =+− λβ k(w kk Bx )

5. Convergence Analysis

In this section, we show the convergence of proposed algorithms. Obviously, the proofs of the two algorithms are almost the same, and we only prove the convergence of algorithm 2. Lemma 3.1: Let L be the function in (3). The vector ( x**,, w)∈× Rn [ lu] solves (2) if and only if there exists λ* ∈ Rn such that ( xw***,,λ ) solves Lx( **, w ,λ) ≤ Lx( *** , w , λ) ≤ Lxw( ,, λλ*) , ∀( xw ,,) ∈× Rnn[ lu ,] × R. Lemma 3.2: Let L be the function in (3), ( xw***,,λ ) be a saddle-point of L, Then the sequence xwkk, satisfies limxwkk ,= xw** , . ( ) k →∞ ( ) ( )

Theorem 3.1: Let ( xwkk, ) be the sequence of iterates produced by the algorithm 2. * * ** then lim xxk = , lim wwk = and xw, is the optimal point for problem (14) k →∞ k →∞ ( ) Proof From Lemma 3.1, 3.2, we obtain that lim wk+1 − w k =0, lim wxkk −=0 (22)

Since we have a unique minimizer in [lu, ] , so we have lim xxk = * , Then, (22) k →∞ * gives lim wwk = which completes the proof. k →∞

6. Numerical Experiments 6.1. Parameter Settings

In Algorithm 2, the parameters β , the penalty in the augmented Lagrangian (3), are common to these two algorithms, ADAN and ADNAN. Besides β has a vital impact on convergence speed. We choose β = 10−4 based on the results from W. Hager [6].

The choice τ = 1.2 is a compromise between speed and stability, δmin = 0.001 is large enough to ensure invertibility. The search directions were generated by the L-BFGS method developed by No-cedal

in [16] and Liu and Nocedal in [1]. We choose the step size αk to satisfy the Wolfe

conditions with m = 0.09 and σ = 0.9. In addition we employ a fixed value ηk = 0.47 which could get the reasonable results. To obtain a good estimate for the optimal objec-

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tive in (1), we ran them for 100,000 iterations. The optimal objective values for the three data sets were Ψ=0.1275 Ψ=0.2456 Ψ=0.9627 In addition, we timed how long it took ADNAN to reduce the objective error to within 1% of the optimal objective value. The algorithms are coded in MATLAB, version 2011b, and run on a Dell version 4510U with a 2.8 GHz Intel i7 processor. In Algorithm 3, a 256-by-256 gray-scale image was considered, which is compared to the experiment by J. Zhang [8]. The dimensions of the inverse problems are m = n = T 65536 and the constraints are l = −∞ and u = (255, , 255) . The experiments on image deblurring problems show that GRAD algorithm is also effective in terms of quality of the image resolution.

6.2. Experiments Results

This section compares the performance of the ADNAN to ADAN. The main difference between the ADNAN algorithm and the ADAN algorithm is the computation of xk+1 . kk+1 In ADAN xx= + αkk d where αk is the step size. In ADNAN, x generated from Algorithm 1, if the convergence condition in ADAN is satisfied, then the update kk+1 xx= + αkk d could be performed. Here δk is the same choice for them. Hence, there seems to be a significant benefit from using a value for δk smaller than the largest eigenvalue of AAT . The initial guess for x1 , w1 and λ1 was zero for two algorithms. Figures 1-3 show the objective values and objective error as a function of CPU time. Moreover, we give the comparison of objective values and objective error versus CPU time/s for different Ψ conditions. It is observed that ADNAN is slightly stable than ADAN although ADNAN and ADAN are competitive. The ADNAN not only could get more smaller objective error but also get more fast convergence speed (see Figure 3). In addition, ADNAN objective value could get more smaller after a few iterations than ADAN. As a whole, the effect of ADNAN is superior to ADAN.

Figure 1. Ψ=0.1275 .

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Figure 2. Ψ=0.2456 .

Figure 3. Ψ=0.9627 .

On the other hand, the experiment results about Algorithm 3 are as follows:

Original image blurry image deblurred image

Figure 4. lu=−=255, 0 .

Original image blurry image deblurred image

Figure 5. lu=−=255, 150 .

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Original image blurry image deblurred image

Figure 6. lu=−=255, 200 .

Original image blurry image deblurred image

Figure 7. lu=−=255, 255 .

7. Conclusions

According to the Figures 1-3, we can conclude that the nonmonotone line search could accelerate the convergence speed, furthermore ADNAN could get the objective values more stable and fast during the iterations when compared to ADAN. On the other hand, the validness of GRAD is verified. Experiments results on image deblurring problems in Figures 4-7 show that difference constraints on x can also get effective deblurred images.

Acknowledgements

This work is supported by Innovation Program of Shanghai Municipal Education Commission (No. 14YZ094).

References [1] Amit, Y. (2007) Uncovering Shared Structures in Multiclass Classification. Machine Learn- ing, Twenty-fourth International Conference, 227, 17-24. https://doi.org/10.1145/1273496.1273499 [2] Argyriou, A., Evgeniou, T. and Pontil, M. (2007) Multi-Task Feature Learning. Advances in Neural Information Processing Systems, 19, 41-48. [3] Chambolle, A. (2004) An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision, 20, 89-97. [4] Chambolle, A. (2011) A First-Order Primal-Dual Algorithm for Convex Problems with Ap- plications to Imaging. Journal of Mathematical Imaging and Vision, 40, 120-145. [5] Zhang, L. and Vandenberghe, L. (2009) Interior-Point Method for Nuclear Norm Ap-

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proximation with Application to System Identification. SIAM Journal on Matrix Analysis and Applications, 31, 1235-1256. [6] Hager, W. and Zhang, H.-C. (2015) An Alternating Direction Approximate Newton Algo- rithm for Ill-Conditioned Inverse Problems with Application to Parallel MRI. Journal of the Operational Research Society of China, 3, 139-162. [7] Gabay, D. and Mercier, B. (1976) A Dual Algorithm for the Solution of Nonlinear Varia- tional Problems via Finite Element Approximations. Computers & mathematics with applications, 2, 17-40. [8] Zhang, J.J. (2013) Solving Regularized Linear Least-Squares Problems by the Alternating Direction Method with Applications to Image Restoration. Electronic Transactions on Numerical Analysis, 40, 356-372. [9] Zhang, H.C. and Hager, W. (2004) A Nonmonotone Line Search Technique and Its Appli- cation to Unconstrained Optimization. Society for Industrial and Applied Mathematics, 14, 1043-1056. [10] Grippo, L., Lampariello, F. and Lucidi, S. (1986) A Nonmonotone Line Search Technique for Newton’s Method. SIAM Journal on Mathematical Analysis, 23, 717-716. [11] Eckstein, J. and Bertsekas, D. (1992) On the Douglas-Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators. Mathematical Programming, 55, 293-318. https://doi.org/10.1007/BF01581204 [12] Barzilai, J. and Borwein, J. (1988) Two Point Step Size Gradient Methods. IMA Journal of Numerical Analysis, 8, 141-148. https://doi.org/10.1093/imanum/8.1.141 [13] Raydan, M. and Svaiter, B.F. (2002) Relaxed Steepest Descent and Cauchy-Barzilai-Borwein Method. Computational Optimization and Applications, 21, 155-167. https://doi.org/10.1023/A:1013708715892 [14] Block, K.T., Uecker, M. and Frahm J. (2007) Undersampled Radial MRI with Multiple Coils: Iterative Image Reconstruction Using a Total Variation Constraint. Magnetic Re- sonance in Medicine, 57, 1086-1098. https://doi.org/10.1002/mrm.21236 [15] Dai, Y.H. (2002) On the Nonmonotone Line Search. Journal of Control Theory and Appli- cation, 112, 315-330. [16] Grippo, L., Lampariello, F. and Lucidi, S. (1989) A Truncated Newton Method with Non- monotone Line Search for Unconstrained Optimization. Journal of Optimization Theory and Applications, 6, 401-419. https://doi.org/10.1007/BF00940345

2078 Journal of Applied Mathematics and Physics, 2016, 4, 2079-2112 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

High-Capacity Quantum Associative Memories

M. Cristina Diamantini1, Carlo A. Trugenberger2

1Nips Laboratory, INFN and Dipartimento di Fisica, University of Perugia, Perugia, Italy 2Swiss Scientific, Cologny, Switzerland

How to cite this paper: Diamantini M.C. Abstract and Trugenberger, C.A. (2016) High- Capacity Quantum Associative Memories. We review our models of quantum associative memories that represent the “quanti- Journal of Applied Mathematics and Phys- zation” of fully coupled neural networks like the Hopfield model. The idea is to re- ics, 4, 2079-2112. place the classical irreversible attractor dynamics driven by an Ising model with pat- http://dx.doi.org/10.4236/jamp.2016.411207 tern-dependent weights by the reversible rotation of an input quantum state onto an Received: September 6, 2016 output quantum state consisting of a linear superposition with probability ampli- Accepted: November 27, 2016 tudes peaked on the stored pattern closest to the input in Hamming distance, result- Published: November 30, 2016 ing in a high probability of measuring a memory pattern very similar to the input.

Copyright © 2016 by authors and The unitary operator implementing this transformation can be formulated as a se- Scientific Research Publishing Inc. quence of one-qubit and two-qubit elementary quantum gates and is thus the expo- This work is licensed under the Creative nential of an ordered quantum Ising model with sequential operations and with pat- Commons Attribution International tern-dependent interactions, exactly as in the classical case. Probabilistic quantum License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ memories, that make use of postselection of the measurement result of control qu- Open Access bits, overcome the famed linear storage limitation of their classical counterparts be-

cause they permit to completely eliminate crosstalk and spurious memories. The number of control qubits plays the role of an inverse fictitious temperature. The accuracy of pattern retrieval can be tuned by lowering the fictitious temperature under a critical value for quantum content association while the complexity of the retrieval algorithm remains polynomial for any number of patterns polynomial in the number of qubits. These models thus solve the capacity shortage problem of classical associative memories, providing a polynomial improvement in capacity. The price to pay is the probabilistic nature of information retrieval.

Keywords Quantum Information, Associative Memory, Quantum Pattern Recognition

1. Introduction

There is a growing consensus that the fundamental mechanism of human intelligence is

DOI: 10.4236/jamp.2016.411207 November 30, 2016 M. C. Diamantini, C. A. Trugenberger

simply pattern recognition, the retrieval of information based on content association, albeit repeated in ever increasing hierarchical structures [1] [2]. Correspondingly, pattern recognition in machine intelligence [3] has made enormous progress in the last decade or so and such systems are now to be found in applications ranging from medi- cal diagnosis to facial and voice recognition in security and digital personal assistants, the latest addition to the family being self-driving cars. On the other side, the last two decades have seen the birth of, and an explosion of research in a new information- theoretic field: quantum information theory and quantum computation [4] [5]. This chapter deals with quantum pattern recognition, with particular emphasis on models that are both accessible to detailed analytical treatment and efficiently implementable within the framework of the quantum circuit model. Pattern recognizers, which go also under the name of associative memories (or more precisely autoassociative memories), are fundamentally different than von Neumann or Turing machines [6], which have grown into the ubiquitous computers that permeate our information society. Computation is not sequential but, rather, based on collective phenomena due to interactions among a large number of, typically redundant, elementary components. Information is not address-oriented, i.e. stored in look-up tables (random access memories, RAMs) but, rather, distributed in often very complex ways over the connections and interactions parameters. In traditional computers information is identified by a label and stored in a database indexed by these labels. Retrieval requires the exact knowledge of the relevant label, without which information is simply not accessible. This is definitely not how our own brain works. When trying to recognize a person from a blurred photo it is totally useless to know that it is the 16878th person you met in your life. Rather, the recognition process is based on our strong power of association with stored memories that resemble the given picture. Association is what we use every time we solve a crossword puzzle and is distinctive of the human brain. The best known examples of pattern recognizers are neural networks [7] [8] and hidden Markov models [9], the Hopfield model [10] (and its generalization to a bidirectional associative memory [11]) being the paradigm, since it can be studied analyti- cally in detail by the techniques of statistical mechanics [7] [8] [12]. The great advantage of these architectures is that they eliminate the extreme rigidity of RAM memories, which require a precise knowledge of the memory address and, thus, do not permit the retrieval of incomplete or corrupted inputs. In associative memories, on the contrary, recall of information is possible also on the basis of partial knowledge of its content, without knowing a precise storage location, which typically does not even exist. This is why they are also called “content-addressable memories”. Unfortunately, classical associative memories suffer from a severe capacity shortage. When storing multiple patterns, these interfere with each other, a phenomenon that goes under the name of crosstalk. Above a critical number of patterns, crosstalk becomes so strong that a phase transition to a completely disordered spin glass phase [13] takes place. In this phase there is no relation whatsoever between the information en-

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coded in the memory and the original patterns. For the Hopfield model, the critical threshold on the number p of patterns that can be stored in a network of n binary neurons is

pnmax  0.138 [7] [8]. While various possible improvements can be envisaged, the maximum number of patterns remains linear in the number of neurons, pmax = On( ) . The power of quantum computation [4] [5] is mostly associated with the speed-up in computing time it can provide with respect to its classical counterpart, the paramount examples being Shor’s factoring algorithm [14] and Grover’s database search algorithm [15]. The efficiency advantage over classical computation is due essentially to the quantum superposition principle and entanglement, which allow for massively parallel information processing. The bulk of the research effort in quantum computation has focused on the “quantization” of the classical sequential computer architecture, which has led to the quantum circuit model [4] [5], in which information processing is realized by the sequential application of a universal set of elementary one- and two-qubit gates to typically highly entangled quantum states of many qubits. The computation is said to be efficient if the desired unitary evolution of the quantum state can be realized by the application of a polynomial number (in terms of the number of involved qubits) of these elementary quantum gates. However, the question immediately arises if quantum mechanics can be applied successfully also to the collective information processing paradigm typical of machine intelligence algorithms and, specifically, if there are advantages in doing so. While this research has trailed the development of the quantum circuit model, it is presently expe- riencing a flurry of increased interest, so much so that last year NASA and Google have teamed up to found the Quantum Artificial Intelligence Laboratory, entirely dedicated to develop and advance machine intelligence quantum algorithms. While speed has been the main focus of quantum computation, it can be shown that quantum mechanics also offers a way out from the impossibility of reconciling the association power of content-addressable memories with the requirement of large storage capacity. Indeed, one of us pointed out already in 2001 [16] [17] [18] [19] that storage capacity of associative memories can also be greatly enhanced by the quantum superposition principle. The key idea is to exploit the fundamental probabilistic nature of quantum mechanics. If one is willing to abandon the classical paradigm of one-off retrieval and sacrifice some speed by repeating the information retrieval step several times, then it is possible to store any desired polynomial number (in terms of the number of qubits) of patterns in a quantum associative memory and still tune the associative retrieval to a prescribed accuracy, a large advantage with respect to the classical linear limitation described above. Quantum entanglement permits to completely eliminate crosstalk and spurious memories in a tuneable probabilistic content association procedure with polynomial complexity for a polynomial number of stored patterns. Such probabilistic quantum associative memories can thus be implemented efficiently. Simi- lar ideas in this direction were developed simultaneously in [20] [21] [22]. In this chapter, we will review our own work on fundamental aspects of quantum

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associative memories and quantum pattern recognition. We will begin by a short survey of the main features of classical fully coupled neural networks like the Hopfield model and its generalizations, with a special emphasis on the capacity limitation and its origin. We will then describe the quantization of the Hopfield model [23]: the idea is to replace the classical irreversible dynamics that attracts input patterns to the closest minima of an energy function, representing the encoded memories, with a reversible unitary quantum evolution that amplifies an input quantum state to an output quantum state representing one of the stored memories at a given computational time t. In the classical model there is a complex phase diagram in terms of the two noise parameters, the temperature T and the disorder p/n with n the number of bits and p the number of stored patterns. It is, specifically the disorder due to an excessive loading factor p/n that prevents the storage of more than a critical number of patterns by causing the transition to a spin glass phase [13], even at zero temperature. Correspondingly, in the quantum version there are quantum phase transitions due to both disorder and quantum fluctuations, the latter being encoded in the effective coupling Jt, with J being the energy parameter of the model and t being the computational time (throughout the review we will use units in which c = 1 and  = 1 ). These are first examples of quantum collective phenomena typical of quantum machine intelligence. It turns out that, barring periodicity effects due to the unitary time evolution, the phase diagram for the quantum Hopfield model is not so different from its classical counterpart. Specifically, for small loading factors the quantum network has indeed associative power, a very interesting feature by itself, but the maximum loading factor is still limited to pn≤ 1 , above which there is a totally disordered spin glass phase, with no association power for any computational time. The transition to this quantum spin glass phase takes place when one tries to store a number of memories that is not anymore linearly independent. We then turn our attention to probabilistic quantum associative memories [16] [17] [18] [19]. The basic idea underlying their architecture is essentially the same as above, with one crucial difference: they exploit, besides a unitary evolution, a second crucial aspect of quantum mechanics, namely wave function collapse upon measurement [4] [5]. A generic (pure) quantum state is a superposition of basis states with complex coefficients. A measurement projects (collapses) the state probabilistically onto one of the basis states, the probability distribution being governed by the squared absolute values of the superposition coefficients. Probabilistic quantum associative memories involve, besides the memory register itself a certain number b of control qubits. The unitary evolution of the input state is again determined by a Hamiltonian that depends only on the stored patterns. Contrary to quantized Hopfield memories, however, this unitary evolution mixes the memory register and the control qubits. After having applied the unitary evolution to the initial input state, the control qubits are measured. Only if one obtains a certain specific result, one proceeds to measure the memory register. This procedure is called probabilistic postselection of the measurement result and guarantees that the memory register is in a superposition of the stored patterns such that the

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measurement probabilities are peaked on those patterns that minimize the Hamming distance to the input. A measurement of the memory register will thus associate input and stored patterns according to this probability distribution. Of course, if we limit ourselves to a maximum number T of repetitions, there is a non-vanishing probability that the memory retrieval will fail entirely, since the correct control qubit state will never be measured. One can say that information retrieval in these quantum memories consists of two steps: recognition (the correct state of the control qubits has been obtained) and identification (the memory register is measured to give an output). Both steps are probabilistic and both the recognition efficiency and the identification accuracy depend on the distribution of the stored patterns: recognition efficiency is best when the number of stored patterns is large and the input is similar to a substantial cluster of them, while identification accuracy is best for isolated patterns which are very different from all other ones, both very intuitive features. Both recognition efficiency and identification accuracy can be tuned to prescribed levels by varying the repetition threshold T and the number b of control qubits. The accuracy of the input-output association depends only on the choice of the number b of control qubits. Indeed, we will show that tb= 1 plays the role of an effective temperature [19]. The lower t, the sharper is the corresponding effective Boltz- mann distribution on the states closest in Hamming distance to the input and the better becomes the identification. By averaging over the distribution of stored patterns with Hamming distance to the input above a threshold d one can eliminate the dependence on the stored pattern distribution and derive the effective statistical mechanics of quantum associative memories by introducing the usual thermodynamic potentials as a function of d and the effective temperature tb= 1 . In particular, the free energy Ft( ) describes the average behaviour of the recall mechanism and provides concrete criteria to tune the accuracy of the quantum associative memory. By increasing b (lowering t), the associative memory undergoes a phase transition from a disordered phase with no correlation between input and output to an ordered phase with perfect input-output association encoded in the minimal Hamming distance d. This extends to quantum information theory the relation with Ising spin systems known in error-correcting codes [24] [25] [26] and in public key cryptography [27]. The recognition efficiency can be tuned mainly by varying the repetition threshold T: the higher T, the larger the number of input qubits that can be corrupted without affecting recognition. The crucial point is that the recognition probability is bounded 2b from below by ( p −1)(π 2) ( pn2b ) . For any number of patterns, thus, a repetition threshold T polynomial in n guarantees recognition with probability O(1) . Due to the factor ( p −1) in the numerator, whose origin is exclusively quantum mechanical, the number of repetitions required for efficient recognition would actually be polynomial even for a number of patterns exponential in n. The overall complexity of probabilistic associative quantum memories is thus bounded by the complexity Op( (23 n+ )) of the unitary evolution operator. Any polynomial number of patterns p= On( x ) can be encoded and retrieved efficiently in polynomial computing time. The absence of spu-

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rious memories leads to a substantial storage gain with respect to classical associative memories, the price to pay being the probabilistic nature of information recall.

2. The Classical Hopfield Model

Historically, the interest in neural networks [7] [8] has been driven by the desire to build machines capable of performing tasks for which the traditional sequential computer architecture is not well suited, like pattern recognition, categorization and generalization. Since these higher cognitive tasks are typical of biological intelligences, the design of these parallel distributed processing systems has been largely inspired by the physiology of the human brain. The Hopfield model is one of the best studied and most successful neural networks. It was designed to model one particular higher cognitive function of the human brain, that of associative pattern retrieval or associative memory.

The Hopfield model consists of an assembly of n binary neurons si , in= 1, , [28], which can take the values ±1 representing their firing (+1) and resting (−1) states. The

neurons are fully connected by symmetric synapses with coupling strengths wwij= ji

( wii = 0 ). Depending on the signs of these synaptic strengths, the couplings will be ex- citatory (>0) or inhibitory (<0). The model is characterised by an energy function 1 E=−∑ wij ss i j, s i =±= 1, i , j 1, , n , (1) 2 ij≠

and its dynamical evolution is defined by the random sequential updating (in time t) of the neurons according to the rule

stii( +=1) sign( ht( )) , (2)

hi ( t) = ∑ wsij j ( t), (3) ij≠

where hi is called the local magnetization. The synaptic coupling strengths are chosen according to the Hebb rule

1 µµ wij = ∑ ξξi j , (4) n µ =1, , p

µ where ξi , µ = 1, , p are p binary patterns to be memorized. An associative memory is defined as a dynamical mechanism that, upon preparing the network in an initial 0 λ state si retrieves the stored pattern ξi that most closely resembles the presented 0 pattern si , where resemblance is determined by minimizing the Hamming distance, i.e. the total number of different bits in the two patterns. As emerges clearly from this definition, all the memory information in a Hopfield neural network is encoded in the synaptic strengths. It can be easily shown that the dynamical evolution (2) of the Hopfield model satisfies exactly the requirement for an associative memory. This is because: • The dynamical evolution (2) minimizes the energy functional (1), i.e. this energy functional never increases when the network state is updated according to the evo-

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lution rule (2). Since the energy functional is bounded by below, this implies that the network dynamics must eventually reach a stationary point corresponding to a, possibly local, minimum of the energy functional. µ • The stored patterns ξi correspond to, possibly local, minima of the energy functional. This implies that the stored patterns are attractors for the network dynamics (2). An initial pattern will evolve till it overlaps with the closest (in Hamming distance) stored pattern, after which it will not change anymore. Actually, the second of these statements must be qualified. Indeed, the detailed behavior of the Hopfield model depends crucially upon the loading factor α = pn, the ratio between the number of stored memories and the number of available bits. This is best analyzed in the thermodynamic limit p →∞, n →∞, in which the different re- gimes can be studied by statistical mechanics techniques [7] [8] [12] and characterized formally by the values of critical parameters.

For αα< 1crit  0.051 , the system is in a ferromagnetic (F) phase in which there are global energy minima corresponding to all stored memories. The former differ from the original input memories only in a few percent of the total number of bits. Mixing between patterns leads to spurious local energy minima. These, however are destabi- lized at sufficiently high temperatures.

For α1crit  0.051<< αα 2crit 0.138 , the system is in a mixed spin glass (SG) [13] and ferromagnetic phase. There are still minima of sizable overlap with the original memories but they are now only metastable states. The true ground state is the spin glass, characterized by an exponentially increasing number of minima due to the mixing of original memories (crosstalk). The spin glass phase is orthogonal to all stored memories. If an input pattern is sufficiently near (in Hamming distance) to one of the original memories it will be trapped by the corresponding metastable state and the retrieval procedure is successful. On the other hand, if the input pattern is not sufficiently close to one of the stored memories, the network is confused and it will end up in a state very far from all original memories.

For αα> 2crit  0.138 , the system is in a pure spin glass (SG) phase [13] in which all retrieval capabilities are lost due to an uncontrolled proliferation of spurious memories. It is this phase transition to a spin glass that limits the storage capacity of the Hopfield model to α =pn < 0.138 . While various improvements are possible, the storage capacity of classical associative memories remains linearly bounded by the number n of classical bits [7] [8].

3. Quantum Neural Networks and the Quantization of the Hopfield Model

In this section, we introduce a quantum information processing paradigm that is different from the standard quantum circuit model [23]. Instead of one- and two-qubit gates that are switched on and off sequentially, we will consider long-range interactions that define a fully-connected quantum neural network of qubits. This is encoded in a Hamiltonian that generates a unitary evolution in which the operator acting on one qu-

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bit depends on the collective quantum state of all the other qubits. Note that some of the most promising technologies for the implementation of quantum information processing, like optical lattices [29] and arrays of quantum dots [30] rely exactly on similar collective phenomena. In mathematical terms, the simplest classical neural network model is a graph with the following properties:

• A state variable si is associated with each node (neuron) i.

• A real-valued weight wij is associated with each link (synapse) (ij) between two nodes i and j. • = A state-space-valued transfer function fh( i ) of the synaptic potential hi ∑ j wsij j determines the dynamics of the network. Directed graphs correspond to feed-forward neural networks [7] [8] while undirected graphs with symmetric weights contain feed-back loops. If the graph is complete one has fully-connected neural networks like the Hopfield model. Two types of dynamical evolution have been considered: sequential or parallel synchronous. In the first case the neurons are updated one at a time according to  si ( t+=1,) f∑ wsik k ( t) (5) k

while in the second case all neurons are updated at the same time. The simplest model

is obtained when neurons become binary variables taking only the values si = ±1 for all i and the transfer function becomes the sign function. This is the original Mc-Cullogh- Pitts [28] neural network model, in which the two states represent quiescent and firing neurons. As we have seen in the previous section, the Hopfield model [10] is a fully-connected McCullogh-Pitts network in which the synaptic weights are symmetric quantities chosen according to the Hebb rule [7] [8] p 1 µµ wwij= ji = ∑ξξi j,w ii = 0. (6) n µ =1

and in which the the dynamics-defining function f is the sign function, f = sign . This dynamics minimises the energy function 1 E=−∑ wij ss i j, s i =±= 1, i , j 1, , n , (7) 2 ij≠

µ where n is the total number of neurons and ξ are the p binary patterns to be memo- µ rized ( ξi = ±1 ) A quantum McCullogh-Pitts network can correspondingly be defined as a graph that satisfies:

• A two-dimensional Hilbert space i is associated with each node (neuron) i, i.e. each neuron becomes a qubit whose basis states can be labeled as 0 and 1 .

• A vector-valued weight wij is associated with each link (synapse) (ij) between two nodes i and j. • = σ σ = σσσxyz The synaptic potential becomes an operator hi ∑ j wij j , where i( iii,,)

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is the vector of Pauli matrices acting on the Hilbert space i . A unitary operator

Uh( i ) determines the dynamics of the network starting from an initial input quantum state on the product Hilbert space of all qubits. In case of feed-forward quantum networks on directed graphs only a subset of qubits is measured after the unitary evolution, in case of fully connected quantum networks with symmetric weights the state of the whole network is relevant. The crucial difference with respect to classical neural networks concerns the interactions between qubits. In the classical model, the dynamics (5) induced by the transfer function is fully deterministic and irreversible, which is not compatible with quantum mechanics. A first generalization that has been considered is that of stochastic neurons, in which the transfer function determines only the probabilities that the classical state variables will take one of the two values: nti ( +=±11) with probabilities f(± hti ( )) , where f must satisfy fh( → −∞) = 0 , fh( → +∞) = 1 and fh( ) + f( −= h) 1. While this modification makes the dynamics probabilistic by introducing thermal noise, the evolution of the network is still irreversible since the actual values of the neurons are prescribed after an update step. In quantum mechanics the evolution must be reversible and only the magnitudes of the changes in the neuron variables can be postulated. Ac- tually, the dynamics must generate a unitary evolution of the network. It is known that two-level unitary gates are universal, i.e. every unitary matrix on an n-dimensional Hilbert space may be written as a product of two-level unitary matrices. However, an arbitrary unitary evolution cannot be implemented as a sequential succession of a discrete set of elementary gates, nor can it be approximated efficiently with a polynomial number of such gates [4] [5]. In general, quantum neural networks as defined above, have to be thought of as defined by Hamiltonians H that code hard-wired qubit interactions and generate a unitary evolution U= exp( iHt) . This corresponds to the parallel synchronous dynamics of classical neural networks. Only in particular cases, one of which will be the subject of the next section, does this unitary evolution admit a representation as a sequential succession of a discrete set of elementary one- and two- bit gates. In this cases the network admits a sequential dynamics as its classical counterpart. We now describe a direct “quantization” of the Hopfield model in this spirit, i.e. by defining a quantum Hamiltonian that generalizes (7). At first sight one would be z tempted to simply replace the classical spins si of (7) with the third Pauli matrix σ i acting on the Hilbert space Hi . This however would accomplish nothing, the model would still be identical to the original classical model, since all terms in the Hamilto- nian would commute between themselves. A truly quantum model must involve at least two of the three Pauli matrices. In [23] we have proposed the following “transverse” Hamiltonian:

yz  = Jw∑ ijσσ i j , (8) ij where σ k , k= xyz,, denote the Pauli matrices and J is a coupling constant with the dimensions of mass (we remind the reader that we use units in which c =1, = 1 ). This

2087 M. C. Diamantini, C. A. Trugenberger

generates a unitary evolution of the network:

ψψ(t) = exp( it ) 0 , (9)

where ψψ0 =(t = 0) . Specifically, we will choose as initial configuration of the network the uniform superposition of all computational basis states [4] [5]

n 1 21− ψ = x . (10) 0 n ∑ 2 x=0

This corresponds to a “blank memory” in the sense that all possible states have the same probability of being recovered upon measurement. In the language of spin systems this is a state in which all spins are aligned in the x direction. Inputs ξ ext can be accomodated by adding an external transverse magnetic field along the y axis, i.e. modifying the Hamiltonian to

yz ext y  = Jw∑∑ijσσ i j + ghi σ i , (11) ij i

ext = ξ ext where hwi ∑ j ij j . This external magnetic field can be thought of as arising from the interaction of the network with an additional “sensory” qubit register prepared in the state ξ ext , the synaptic weights between the two layers being identical to those of the network self-couplings. Let us now specialize to the simplest case of one assigned memory ξ in which

wnij= ξξ i j . In the classical Hopfield model there are two nominal stable states that represent attractors for the dynamics, the pattern ξ itself and its negative −ξ . Cor- respondingly, the quantum dynamics defined by the Hamiltonian (8) and the initial σ x state (10) have a Z2 symmetry generated by ∏ i i , corresponding to the inversion 01↔ of all qubits. As in the classical case we shall analyze the model in the mean field approximation. In this case, the mean field represents the average over quantum fluctuations rather than thermal ones but the principle remains the same. The mean field model becomes exactly solvable and allows to derive self-consistency conditions on the average overlaps with the stored patterns. In the classical case, the mean field approximation is known to become exact for long-range interactions [31]. In the quantum mean-field approximation operators are decomposed in a sum of their k k kk mean values in a given quantum state and fluctuations around it, σσi= i +−( σσ ii) , and quadratic terms in the fluctuations are neglected in the Hamiltonian. Apart from an irrelevant constant, this gives

y z g ext zy mf =J∑σσi hh i ++ i ii h, i J (12) k kk hwi =∑ ijσξ j = i m, j

kk= σξ where mn(1 )∑i ii is the average overlap of the state of the network with the stored pattern. This means that each qubit i interacts with the average magnetic field k (synaptic potential) hi due to all other qubits: naturally, the correct values of these

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k mean magnetic fields hi have to be determined self-consistently. To this end we compute the average pattern overlaps mk using the mean field Ha- miltonian (12) to generate the time evolution of the quantum state. This reduces to a sequence of factorized rotations in the Hilbert spaces of each qubit, giving m y m y = − sin 2Jt m , m (13) mzz+ ( gJM) mz = sin 2Jt m , m

22 =yz ++ z z = ξξext where m( m) ( m( gJM) ) and Mn(1 ) ∑i ii is the average overlap of the external stimulus with the stored memory. Before we present the detailed solution of these equations, let us illustrate the mechanism underlying the quantum associative memory. To this end we note that, for g = 0 , the pattern overlaps m y and mz in the two directions cannot be simultaneously different from zero. As we show below, only mz ≠ 0 for J > 0 (for J < 0 the roles of m y and mz are interchanged). In this case the evolution of the network becomes a sequence of n rotations zz cos(Jt hii) sin( Jt h )  (14) − zz sin(Jt hii) cos( Jt h ) in the two-dimensional Hilbert spaces of each qubit i. The rotation parameter is exactly the same synaptic potential hi which governs the classical dynamics of the Hopfield model. When these rotations are applied on the initial state (10) they amount to a single update step transforming the qubit spinors into zz+ 1 cos(Jt hii) sin( Jt h ) . (15) 2 zz− cos(Jt hii) sin( Jt h )

This is the generalization to quantum probability amplitudes of the probabilistic formulation of classical stochastic neurons. Indeed, the probabilities for the qubit to be in its eigenstates ±1 after a time t, obtained by squaring the probability amplitudes, are given by fh(± z ) , where f( hzz) =(1 + sin( 2Jt h )) 2 has exactly the properties of an activation function (alternative to the Fermi function), at least in the region Jt < π 4 . In this correspondence, the effective coupling constant Jt plays the role of the inverse temperature, as usual in quantum mechanics. We shall now focus on a network without external inputs. In this case the equation for the average pattern overlaps has only the solution m = 0 for 0<

2089 M. C. Diamantini, C. A. Trugenberger

tion is larger than the deviation itself. Indeed, any so small external perturbation (gJM) z present at the bifurcation time tJ= 12 is sufficient for the network evolution to choose one of the two stable solutions, according to the sign of the external perturbation. The point Jt = 12 represents a quantum phase transition [32] from an amnesia (paramagnetic) phase to an ordered (ferromagnetic) phase in which the network has recall capabilities: the average pattern overlap mz is the corresponding order

parameter. In the ferromagnetic phase the original Z2 symmetry of the model is spon- taneously broken.

For Jt = π 4 , the solution becomes m0 = 1 , which means that the network is capa-

ble of perfect recall of the stored memory. For Jt > π 4 , the solution m0 decreases slowly to 0 again. Due to the periodicity of the time evolution, however, new stable so-

lutions m0 = ±1 appear at Jt=(14 + n) π 4 for every integer n. Also, for Jt ≥ 3π 4 , new solutions with m y ≠ 0 and mz = 0 appear. These, however, correspond all to metastable states. Thus, tJ= π 4 is the ideal computation time for the network. The following picture of quantum associative memories emerges from the above construction. States of the network are generic linear superpositions of computational

basis states. The network is prepared in the state ψ 0 and is then let to unitarily evolve for a time t. After this time the state of the network is measured, giving the result of the computation. During the evolution each qubit updates its quantum state by a rotation that depends on the aggregated synaptic potential determined by the state of all other qubits. These synaptic potentials are subject to large quantum fluctutations which are symmetric around the mean value h z = 0 . If the interaction is strong enough, any external disturbance will cause the fluctuations to collapse onto a collective rotation of all the network’s qubits towards the nearest memory. We will now turn to the more interesting case of a finite density α = pn of stored memories in the limit n →∞. In this case, the state of the network can have a finite µ overlap with several stored memories ξ simultaneously. As in the classical case we shall focus on the most interesting case of a single “condensed pattern”, in which the network uniquely recalls one memory without admixtures. Without loss of generality we will chose this memory to be the first, µ = 1, omitting then the memory superscript on the corresponding overlap m. Correspondingly, we will consider external inputs so that only MMµ =1 = ≠ 0 . For simplicity of presentation, we will focus directly on solutions with a non-vanishing pattern overlap along the z-axis, omitting also the direction superscript z. In case of a finite density of stored patterns, one cannot neglect the noise effect due to the infinite number of memories. This changes (13) to 1 g m= ∑sin 2 Jt m+ M +∆i , nJ i  (16) 1 µµ ∆=i∑ξξ iim . µ ≠1

µ µ As in the classical case we will assume that {ξi } and {m ,1µ ≠ } are all independent random variables with mean zero and we will denote by square brackets the con-

2090 M. C. Diamantini, C. A. Trugenberger

figurational average over the distributions of these random variables. As a consequence of this assumption, the mean and variance of the noise term are given by [∆=i ] 0 and ∆=2 α i r , where 1 2 = µ rm∑ ( ) (17) α µ ≠1  is the spin-glass order parameter [13]. According to the central limit theorem one can −1 now replace n ∑i in (16) by an average over a Gaussian noise, 2 dzg−z  m =∫ e2 sin 2Jt m++ Mα rz . (18) 2π J

The second order parameter r has to be evaluated self-consistently by a similar procedure starting from the equation analogous to Equation (16) for µ ≠ 1. In this case one can use mµ  1 for µ ≠ 1 to expand the transcendental function on the right- hand side in powers of this small parameter, which gives

−z2 dzg2  v =∫ e2 sin 2Jt m++ Mα rz , 2π J 2 (19) dzg−z  x =∫ e2 cos2Jt m++ Mα rz , 2π J

2 where v=(12 − Jtx) r . Solving the integrals gives finally the following coupled equations for the two order parameters m and r:

2 m=sin 2 Jt( m + gJM ) e−2(Jt) α r ,

2 g −8(Jt) α r 1−+ cos4Jt m M e 1 J (20) r = 2 . 2 2 g −2(Jt) α r 1−+ 2Jt cos 2 Jt m M e J

In terms of these order parameters one can distinguish three phases of the network. First of all the value of m determines the presence ( m > 0 ) or absence ( m = 0 ) of ferromagnetic order (F). If m = 0 the network can be in a paramagnetic phase (P) if also r = 0 or a quantum spin glass phase (SG) if r > 0 . The phase structure resulting from a numerical solution of the coupled Equation (20) for g = 0 is shown in Figure 1. For α < 0.025 the picture is not very different from the single memory case. For large enough computation times there exists a ferromagnetic phase in which the m = 0 solution is unstable and the network has recall capabilities. The only difference is that the maximum value of the order parameter m is smaller than 1 (recall is not perfect due to noise) and the ideal computation time t at which the maximum is reached depends on α . For 0.025<<α 1.000 instead, ferromagnetic order coexists as a metastable state with a quantum spin glass state. This means that ending up in the memory retrieval solution depends not only on the presence of an external stimulus but also on its magnitude; in other words, the external pattern has to be close enough to the stored memory in order to be retrieved. For 1 < α all retrieval capabilities are lost and the

2091 M. C. Diamantini, C. A. Trugenberger

Figure 1. The phase structure of quantum associative memories with finite density of stored patterns. P, F and SG denote (quantum) paramagnetic, ferromagnetic and spin-glass phases, respectively. F + SG denotes a mixed phase in which the memory retrieval solution is only locally stable.

network will be in a quantum spin glass state for all computation times (after the transition from the quantum paramagnet). α = 1 is thus the maximum memory capacity of this quantum network. Note that α = 1 corresponds to the maximum possible number of linearly independent memories. For memory densities smaller but close to this maximum value, however, the ferromagnetic solution exists only for a small range of effective couplings centered around Jt  9 : for these high values of Jt the quality of pattern retrieval is poor, the value of the order parameter m being of the order 0.15 - 0.2. Much better retrieval qualities are obtained for smaller effective couplings: e.g. for Jt = 1 the order parameter is larger than 0.9 (corresponding to an error rate smaller than 5%) for memory densities up to 0.1. In this case, however the maximum memory density is 0.175, comparable with the classical result of the Hopfield model. Quantum mechanics, here, does not carry any advantage.

4. Probabilistic Quantum Memories

We have seen in the last section that crosstalk prevents the amplification of patterns stored in the weights of a simple quantum Hamiltonian like (8) when the loading factor

2092 M. C. Diamantini, C. A. Trugenberger

exceeds a linear bound comparable with the classical one. In this section we show that this limit can be overcome by probabilistic quantum memories, which use postselection of the measurement results of certain control qubits [16] [17] [18] [19]. The price to pay is that such probabilistic memories require repetitions of the retrieval process and that there is non-vanishing probability that this fails entirely. When it is successful, however, it allows retrieval of the most appropriate pattern among a polynomial pool instead of a linear one.

4.1. Storing Patterns

Let us start by describing the elementary quantum gates [4] [5] that we will use in the rest of the paper. First of all there are the single-qbit gates represented by the Pauli matrices σ i , i= xyz,, . The first Pauli matrix σ x , in particular, implements the NOT gate. Another single-qbit gate is the Hadamard gate H, with the matrix representation 1 11 H = . (21) 2 11−

Then, we will use extensively the two-qbit XOR (exclusive OR) gate, which performs a NOT on the second qbit if and only if the first one is in state 1 . In matrix notation this gate is represented as XOR= diag( 1,σ x ) , where 1 denotes a two-dimensional identity matrix and σ x acts on the components 01 and 11 of the Hilbert space. The 2XOR, or Toffoli gate is the three qbit generalization of the XOR gate: it performs a NOT on the third qbit if and only if the first two are both in state 1 . In matrix notation it is given by 2XOR= diag( 1,1,σ x ) . In the storage algorithm we shall make use also of the nXOR generalization of these gates, in which there are n control qbits. This gate is also used in the subroutines implementing the oracles underlying Grover’s algorithm [4] [5] and can be realized using unitary maps affecting only few qbits at a time [33], which makes it efficient. All these are standard gates. In addition to them we introduce the two-qbit controlled gates CS ii=0 0 ⊗+ 1 11 ⊗S , i −11  i i (22) S i = , −−11i  i i

ii for ip= 1, , . These have the matrix notation CS= diag( 1, S ) . For all these gates we shall indicate by subscripts the qbits on which they are applied, the control qbits com- ing always first. The construction of quantum memories relies, of course, on the fundamental fact that one can use entanglement to “store” an arbitrary number p of binary patterns pi of length n in a quantum superposition of just n qubits, 1 p mp= ∑ i . (23) p i=1

2093 M. C. Diamantini, C. A. Trugenberger

The idea of the memory architecture consists thus of two steps: • Generate the state m by a unitary evolution M from a simple prepared state, say

01 , ,0n , mM= 01 , ,0n .

• Given an input state ii= 1,, in , generate from m a superposition of the pattern states that is no more uniform but whose amplitudes define a probability distribution peaked on the pattern states with minimal Hamming distance front the input. It is this step that involves both a unitary evolution and a postselection of the measurement result. The quantum memory itself is the unitary operator M that codes the p patterns. It defines implicitly a Hamiltonian through the formal relation Mi= exp( ) , a Hamil- tonian that represents pattern-dependent interactions among the qubits. This is the quantum generalization of the classical Hopfield model. In order to dispel any possible misunderstandings right away, we point out that this is quite different to the communication of classical information via a quantum channel, limited by the Holevo theorem [34], as we discuss in detail below. In order to construct explicitly the quantum memory M we will start from an algorithm that loads sequentially the classical patterns into an auxiliary register, from which they are then copied into the actual memory register. A first version of such an algorithm was introduced in [35]. The simplified version that we present here is due to [16] [17]. We shall use three registers: a first register p of n qbits in which we will subsequently feed the patterns pi to be stored, a utility register u of two qbits prepared in state 01 , and another register m of n qbits to hold the memory. This latter will be initially

prepared in state 01 , ,0n . The full initial quantum state is thus 111 ψ 01= pp, ,nn ;01;0 1 , ,0 . (24)

The idea of the storage algorithm is to separate this state into two terms, one corresponding to the already stored patterns, and another ready to process a new pattern.

These two parts will be distinguished by the state of the second utility qbit u2 : 0 for the stored patterns and 1 for the processing term. For each pattern pi to be stored one has to perform the operations described below: n ii ψψ10= ∏2.XOR i (25) pumjj2 j=1

i This simply copies pattern p into the memory register of the processing term,

identified by u2 = 1 . n ii ψψ21= ∏NOTm XOR i , j pmjj j=1 (26) ψψii= nXOR . 32m11 mun

The first of these operations makes all qbits of the memory register 1 ’s when the contents of the pattern and memory registers are identical, which is exactly the case only for the processing term. Together, these two operations change the first utility qbit

2094 M. C. Diamantini, C. A. Trugenberger

u1 of the processing term to a 1 , leaving it unchanged for the stored patterns term. ψψi= CS pi+−1 i. 43uu12 (27)

This is the central operation of the storing algorithm. It separates out the new pattern to be stored, already with the correct normalization factor. ψψii= nXOR , 54m11 mun 1 ii (28) ψψ65= ∏XORi NOTm . pmjj j jn=

These two operations are the inverse of Equation (26) and restore the utility qbit u1 and the memory register m to their original values. After these operations on has

i i1 ikpi− ii ψ 6 = ∑ pp;00; + pp;01; . (29) p k =1 p

With the last operation, 1 ii ψψ76= ∏2XORi , (30) pumjj2 jn= one restores the third register m of the processing term, the second term in Equation

(29) above, to its initial value 01 , ,0n . At this point one can load a new pattern into register p and go through the same routine as just described. At the end of the whole process, the m-register is exactly in state m , Equation (23). Any quantum state can be generically obtained by a unitary transformation of the initial state 0, ,0 . This is true also for the memory state m . In the following we will explicitly construct the unitary memory operator M which implements the transformation mM= 0, ,0 . To this end we introduce first the single-qbit unitary gates

iiπ π i Ujj= cos pi 1+ sin  p jσ 2 , (31) 22  where σ 2 is the second Pauli matrix. These operators are such that their product over i the n qbits generates pattern p out of 0, ,0 : pPii= 0, ,0 ,

n ii (32) PU≡ ∏ j . j=1

We now introduce, in addition to the memory register proper, the same two utility qbits as before, also initially in the state 0 . The idea is, exactly as in the sequential algorithm, to split the state into two parts, a storage term with u2 = 0 and a pro- i cessing term with u2 = 1 . Therefore we generalize the operators P defined above to n CPii≡ CU , u22∏ uj (33) j=1

i which loads pattern p into the memory register only for the processing term. It is

2095 M. C. Diamantini, C. A. Trugenberger

then easy to check that mM;00= 0, ,0;00 , p −1 (34) M=  CPi NOT CSpi+−1 XOR CPi× NOT . ∏ ( u2) u1 uu 12 u21 u u 2 u2 i=1  From this construction it is easy to see that the memory operator M involves a number pn(2++ 31) of elementary one- and two-qbit gates. It is thus efficient for any number p of patterns polynomial in the number n of qubits. It is interesting to note that another version of this operator has been recently derived in [36], with a bound of O( pn3 6) on its complexity. This is also linear in p, implying again efficiency for a polynomial number of patterns. While the memory construction we have presented here mirrors its classical counterpart, it is important to stress one notable difference. In classical associative memories, patterns are stored as minima of an energy landscape or, alternatively in the parameters of a dynamical evolution law [7] [8]. This is reflected verbatim in the construction of the unitary operator M in (34), which completely codes the patterns in a dynamical law, albeit reversible in the quantum case. In quantum mechanics, however, there is the possibility of shuffling some (but not all, as we will shortly see) information about the patterns from the unitary evolution law M onto a set of quantum states. The ideal, most compressed quantum memory would indeed be the quantum superposition of patterns m in (23) itself. This, however is impossible. If the memory state has to be used for information retrieval it must be measured and this destroys all information about the patterns (save the one obtained in the measurement). The quantum state must therefore be copied prior to use and this is impossible since the linearity of quantum mechanics forbids exact universal cloning of quantum states [37]. Univer- sal cloning of quantum states is possible only in an approximate sense [38] and has two disadvantages: first of all the copies are imperfect, though optimal [39] [40] and se- condly, the quality of the master copy decreases with each additional copy made. Ap- proximate universal cloning is thus excluded for the purposes of information recall since the memory would be quickly washed out. This leaves state-dependent cloning [41] as the only viable option. State-dependent cloners are designed to reproduce only a finite number of states and this is definitely enough for our purposes. Actually the memory M in (34) is equivalent to a state- dependent cloner for the state m in (23). In this case the information about the stored patterns is completely coded in the memory operator, or equivalently, the state- dependent cloner. It is possible, however, to subdivide the pattern information among an operator and a set of quantum states, obviously including m , by using a probabilistic cloning machine [42]. Probabilistic cloners copy quantum states exactly but the copying process is not guaranteed to succeed and must be repeated until the measurement of an auxiliary register produces a given result associated with copying success. In general, any number of linearly independent states can be copied probabilistically. In the present case for example, it would be sufficient to consider any dummy state d different from m (for more than two states the condition would be linear indepen-

2096 M. C. Diamantini, C. A. Trugenberger

dence) and to construct a probabilistic cloning machine for these two states. This machine would reproduce m with probability pm and d with probability pd ; a flag would tell when the desired state m has been obtained. In order to obtain an exact copy of m one would need then 1 pm trials on average. The master copy would be exactly preserved. The cloning efficiencies of the probabilistic cloner of two states are bounded as follows [42]: 2 pp+≤ . (35) md1+ dm

This bound can be made large by choosing d as nearly orthogonal to m as possible. A simple way to achieve this for a large number of patterns would be, for example, to encode also the state p 1 i+1 dp=∑( −1) i (36) p i=1 together with m when storing information. This can be done easily by using alter- −1 nately the operators S i and (S i ) in the storing algorithm above. For binary patterns which are all different from one would then have dm= 0 , p even, 1 (37) dm= , p odd, p and the bound for the cloning efficiencies would be very close to its maximal value 2 in both cases. The quantum network for the probabilistic cloner of two states has been developed in [43]. It can be constructed exclusively out of the two simple distinguishability tranfer (D) and state separation (S) gates. As expected, these gates embody information about the two states to be cloned. Part of the memory, therefore, still resides in the cloning network. The pattern-dependence of the network cloner can be decreased by choosing a larger set of states in the pool that can be cloned, so that the cloner becomes more and more generic. On one side this decreases also the efficiency of the cloner, so that more repetitions are required, on the other side, since the clonable pool is limited to a set of linearly independent states, one can never eliminate completely the pattern-dependence of the cloning operator. This is why the original claim of an exponential capacity increase of quantum associative memories [16] [17], based on probabilistic cloning of the state m , is excessive. The complexity of the cloner, be it exact as in the memory operator M or probabilistic, remains linear in the number of patterns and the requirement of efficient implementability limits thus p to a polynomial function of the number n of qubits, which is still a large improvement upon classical associative memories.

4.2. Retrieving Patterns

Let us now assume we are given a binary input i that is a corrupted version of one of the patterns stored in the memory. The task of the retrieval algorithm is to “recognize” it,

2097 M. C. Diamantini, C. A. Trugenberger

i.e. output the stored pattern that most resembles this input, where similarity is defined (here) in terms of the Hamming distance, the number of different bits between the two patterns, although other similarity measures [7] could also be incorporated. The retrieval algorithm requires also three registers. The first register i of n qbits contains the input pattern; the second register m, also of n qbits, contains the memory m ; finally there is a control register c with b qbits all initialized in the state 0 . The full initial quantum state is thus: p 1 k ψ 01= ∑ ip; ;0 , ,0b (38) p k =1

where ii= 1,, in denotes the input qbits, the second register, m, contains the memory (23) and all b control qbits are in state 0 . Applying the Hadamard gate to the first control qbit one obtains pp 11kk ψ11= ∑∑ip; ;0 , ,0bb+ ip; ;1 1 , ,0 . (39) 22ppkk=11=

Let us now apply to this state the following combination of quantum gates: n ψψ= NOT XOR , 21∏ mj im jj (40) j=1

As a result of the above operation the memory register qbits are in state 1 if i j k and p j are identical and 0 otherwise: pp 11kk ψ 21= ∑∑id; ;0 , ,0bb+ id; ;1 1 , ,0 , (41) 22ppkk=11=

k k k where d j = 1 if and only if ipjj= and d j = 0 otherwise. Consider now the following Hamiltonian: = ⊗ σ z  (dH )m ( ) , c1 n 1 (42) =σ z + (dH )m ∑, = j 12 m j

where σ z is the third Pauli matrix.  measures the number of 0’s in register m, with

a plus sign if c1 is in state 0 and a minus sign if c1 is in state 1 . Given how we have

prepared the state ψ 2 , this is nothing else than the number of qbits which are different in the input and memory registers i and m. This quantity is called the Hamming distance and represents the (squared) Euclidean distance between two binary patterns. Every term in the superposition (41) is an eigenstate of  with a different eigenva-

lue. Applying thus the unitary operator exp(inπ 2 ) to ψ 2 one obtains π i  2n ψψ32= e, π p i d ip, k 1 2n H ( ) k ψ 31= ∑eid ; ;0 , ,0b (43) 2 p k =1 π p −i d ip, k 1 2n H ( ) k + ∑eid ; ;11 , ,0b , 2 p k =1

2098 M. C. Diamantini, C. A. Trugenberger

k where dH ( ip, ) denotes the Hamming distance bewteen the input i and the stored k pattern p . In the final step we restore the memory gate to the state m by applying the inverse transformation to Equation (40) and we apply the Hadamard gate to the control qbit c1 , thereby obtaining 1 ψψ= H XOR NOT , 43c1 ∏ imjj m j jn= p 1 π kk ψ 41= ∑cosdHb( ip ,) ip ; ;0 , ,0 (44) p k =1 2n p 1 π kk + ∑sindHb( ip ,) ip ;;1,,0.1  p k =1 2n

The idea is now to repeat the above operations sequentially for all b control qbits c1 to cb . This gives p b 1 b−lklkkπ π l ψ fin = ∑∑ cosdHH( ip ,) × sind( ip ,) ∑ ip ;;, J (45) = = 22nnl p kl10 {J } where {J l } denotes the set of all binary numbers of b bits with exactly l bits 1 and (bl− ) bits 0. Note that one could also dispense with a register for the input but, rather, code also the input directly into a unitary operator. Indeed, the auxiliary quantum register for the input is needed only by the operator (40) leading from (39) to (41). The same result (apart from an irrelevant overall sign) can be obtained by applying

n IU= ∏ j , j=1 (46) π π Ujj=sin ii 1 + cos  i jσ 2 , 22  directly on the memory state m . The rest of the algorithm is the same, apart the re- versing of the operator (40) which needs now the operator I −1 . The end effect of the information retrieval algorithm represents thus a rotation of the memory quantum state in the enlarged Hilbert space obtained by adding b control qbits. The overall effect of this rotation is an amplitude concentration on memory states similar to the input, if there is a large number of 0 control qbits in the output state and an amplitude concentration on states different from the input, if there is a large number of 1 control qbits in the output state. As a consequence, the most interesting state for information retrieval purposes is the projection of ψ fin onto the subspace with all control qbits in state 0 . There are two ways of obtaining this projection. The first, and easiest one, is to simply repeat the above algorithm and measure the control register several times, until exactly the desired state for the control register is obtained. If the number of such repetitions exceeds a preset threshold T the input is classified as “non-recognized” and the algorithm is stopped. Otherwise, once cc11,,bb= 0,,0 is obtained, one proceeds

2099 M. C. Diamantini, C. A. Trugenberger

to a measurement of the memory register m, which yields the output pattern of the memory. The second method is to first apply T steps of the amplitude amplification algorithm

[44] rotating ψ fin towards its projection onto the “good” subspace formed by the states with all control qbits in state 0 . To this end it is best to use the version of the retrieving algorithm that does not need an auxiliary register for the input. Let us define as Ri( ) the input-dependent operator which rotates the memory state in the Hilbert

space enlarged by the b control qbits towards the final state ψ fin in Equation (45) (where we now omit the auxiliary register for the input):

ψ fin = Ri( ) m;01 , ,0b . (47)

By adding also the two utility qbits needed for the storing algorithm one can then

obtain ψ fin as a unitary transformation of the initial state with all qbits in state 0 :

ψ fin ;00= RiM( ) 0, ,0;01 , ,0b ;00 . (48)

The amplitude amplification rotation of ψ fin ;00 towards its “good” subspace in which all b control qbits are in state 0 is then obtained [44] by repeated application of the operator

−−11 Q= − R( i) MS0 M R( i) S (49)

on the state ψ fin ;00 . Here S conditionally changes the sign of the amplitude of the

“good” states with the b control qbits in state 0 , while S0 changes the sign of the

amplitude if and only if the state is the zero state 0, ,0;01 , ,0b ;00 . As before, if a measurement of the control registers after the T iterations of the amplitude amplifica-

tion rotation yields 01 , ,0b one proceeds to a measurement of the memory register, otherwise the input is classified as “non-recognized”. The expected number of repetitions needed to measure the desired control register rec state is 1 Pb , with p rec 1 2bkπ PbH= ∑cosd( ip ; ) (50) pnk =1 2

the probability of measuring cc11,,nn= 0,,0. The threshold T governs thus the recognition efficiency of the memory. Note, however, that amplitude amplification rec provides a quadratic boost [44] to the recognition efficiency since only 1 Pb steps

are typically required to rotate ψ fin onto the desired subspace. Accordingly, the threshold T can be lowered to T with respect to the method of projection by measurement. The crucial point is that, due to the quantum nature of the retrieval mechanism, this recognition probability depends on the distribution of all stored patterns. A lower bound on the recognition probability can thus be established as follows. Of all the stored patterns, all but one have Hamming distance from the input smaller or equal than (n −1) . There is only pattern that can have a larger Hamming distance equal to n. So we shall use the upper bound (n −1) for the Hamming distance of all patterns but one, for which we shall use the upper bound n, and this one does not contribute to the recognition probability since the cosine function vanishes. Given that cosine is a de-

2100 M. C. Diamantini, C. A. Trugenberger

creasing function in the interval [0,π 2] , we get the lower bound

rec min p −1 2b π(n −1) PPbb≥= cos . (51) pn2

For n  1 we can now estimate this lower bound as 2b min p −1 π Pb  . (52) pn2

This shows that, independent of the number p of patterns, the threshold T for recognition can be set as a polynomial function of the number n of qubits. Note that this is entirely due to the factor ( p −1) in the numerator of (52), which, in turn, depends on the quantum nature of the memory. In other words, the probabilistic character of the retrieval process does not limit at all the number of possible stored patterns. The typical number of repetitions required would be polynomial even for an exponential number or patterns. The efficient implementability of the quantum memory is limited only by the number of elementary quantum gates in M, which is linear in p. In general, the probability of recognition is determined by comparing (even) powers of cosines and sines of the distances to the stored patterns. It is thus clear that the worst case for recognition is the situation in which there is an isolated pattern, with the remaining patterns forming a tight cluster spanning all the largest distances to the first one. As a consequence, the threshold needed to recognize all patterns diminishes when the number of stored patterns becomes very large, since, in this case, the distribution of patterns becomes necessarily more homogeneous. Indeed, for the maximal number of n rec b stored patterns p = 2 one has Pb = 12 and the recognition efficiency becomes also maximal, as it should be. Once the input pattern i is recognized, the measurement of the memory register k yields the stored pattern p with probability

kb1 2 π k PbH( p) = cosd( ip ,) , (53) Zn2

p rec 2bkπ Z= pPbH = ∑cosd( i , p ) . (54) k =1 2n

Clearly, this probability is peaked around those patterns which have the smallest Hamming distance to the input. The highest probability of retrieval is thus realized for that pattern which is most similar to the input. This is always true, independently of the number of stored patterns. In particular, contrary to classical associative memories, there are no spurious memories: the probability of obtaining as output a non-stored pattern is always zero. This is another manifestation of the fact that there are no restric- tions on the loading factor p/n due to the information retrieval algorithm. In addition to the threshold T, there is a second tunable parameter, namely the number b of control qbits. This new parameter b controls the identification efficiency k of the quantum memory since, increasing b, the probability distribution Ppb ( ) be- k comes more and more peaked on the low dH ( ip, ) states, until

2101 M. C. Diamantini, C. A. Trugenberger

k lim Ppb = δkk , (55) b→∞ ( ) min

where kmin is the index of the pattern (assumed unique for convenience) with the smallest Hamming distance to the input. While the recognition efficiency depends on comparing powers of cosines and sines of the same distances in the distribution, the identification efficiency depends on comparing the (even) powers of cosines of the different distances in the distribution. Spe- cifically, it is best when one of the distances is zero, while all others are as large as possible, such that the probability of retrieval is completely peaked on one pattern. As a consequence, the identification efficiency is best when the recognition efficiency is worst and vice versa. The role of the parameter b becomes familiar upon a closer examination of Equation (53). Indeed, the quantum distribution described by this equation is equivalent to a ca- nonical Boltzmann distribution with (dimensionless) temperature tb= 1 and (dimensionless) energy levels

kkπ E = −2logcosdH ( ip ,) , (56) 2n

with Z playing the role of the partition function. The appearance of an effective thermal distribution suggests studying the average behaviour of quantum associative memories via the corresponding thermodynamic potentials. Before this can be done, however, one must deal with the different distributions of stored patterns characterizing each individual memory. The standard way to do this in similar classical problems is to average over the random distribution of patterns. Typically, one considers quenched averages in which extensive quantities, like the free energy are averaged over the disorder: this is the famed replica trick used to analyze spin glasses [13]. In the present case, however, the disorder cannot lead to spin-glass- like phases since there are no spurious memories: by construction, probabilistic quantum memories can output only one of the stored patterns. The only question is how accurate is the retrieval of the most similar pattern to the input as a function of the fictitious temperature tb= 1 . To address this question we will “quench” only one aspect of the random pattern distribution, namely the minimal Hamming distance d between the input and the stored patterns. The rest of the random pattern distribution will be considered as annealed. In doing so, one obtains an average description of the average memory as a function of the fictitious temperature tb= 1 and the minimal Hamming distance d. To do so we first normalize the pattern representation by adding (modulo 2) to all patterns, input included, the input pattern i. This clearly preserves all Hamming distances and has the effect of choosing the input as the state with all qbits in state 0 . k The Hamming distance dH ( ip, ) becomes thus simply the number of qbits in k pattern p with value 1 . The averaged partition function takes then a particularly simple form:

2102 M. C. Diamantini, C. A. Trugenberger

n pj2b π Zav = ∑∑λ j cos , (57) Nnλ {λ} jd= 2

λ n λ where j describes a probability distribution, ∑ jd= j =1, with the following properties. Let the number of patterns scale as the xth power of the number of qubits, x pn= α x for n  1. Then

λ j =0, jnx >−, 1 (58) λ j ≤=−,,jnx α x x! with all other λ j for jnx<− unconstrained. {λ} is the set of such distributions and Nλ the corresponding normalization factor. Essentially the probability distribution becomes unconstrained in the limit of large n. We now introduce the free energy F( bd, ) by the usual definition

−−bF( b,, d ) bF( b d ) Zav = pe = Zbav ( = 0e) , (59) where we have chosen a normalization such that exp(−bF ) describes the deviation of the partition function from its value for b = 0 (high effective temperature). Since Zp, and consequently also Zpav posses a finite, non-vanishing large-n limit, this normalization ensures that F( bd, ) is intensive, exactly like the energy levels (56), and scales as a constant for large n. This is the only difference with respect to the familiar situation in statistical mechanics. The free energy describes the equilibrium of the system at effective temperature tb= 1 and has the usual expression in terms of the internal energy U and the entropy S: F( td,) = U( td ,) − tStd( ,,) −∂F( td, ) (60) U( td,) = E ,, S( td) = . t ∂t Note that, with the normalization we have chosen in (59), the entropy S is always a negative quantity describing the deviation from its maximal value Smax = 0 at t = ∞ . By inverting Equation (56) with F substituting E one can also define an effective (relative) input/output Hamming distance  at temperature t:

−F( td, ) 2 (td,) = arccose2 . (61) π This corresponds exactly to representing the recognition probability of the average memory as

rec 2b π (Pb ) = cos(bd ,) , (62) av 2 which can also be taken as the primary definition of the effective Hamming distance. The function (bd, ) provides a complete description of the behaviour of the average probabilistic quantum associative memory with a minimal distance Hamming distance d. This can be used to tune its performance. Indeed, suppose that one wants

2103 M. C. Diamantini, C. A. Trugenberger

the memory to recognize and identify inputs with up to n corrupted inputs with an efficiency of ν (01≤≤ν ) . Then one must choose a number b of control qbits sufficiently large that ((bn,1) −≤− ) ( ν ) and a threshold T of repetitions satisfying

2b π T≥ 1 cos(bn , ) , as illustrated in Figure 2. 2 A first hint about the general behaviour of the effective distance function (bd, ) can be obtained by examining closer the energy eigenvalues (56). For small Hamming distance to the input these reduce to

2 2 kk π dHH( ip,,) d( ip) E k , 1. (63)  4 nn

Choosing again the normalization in which i = 00 and introducing a “spin” k k k k si with value si = −12 if qbit i in pattern p has value 0 and si = +12 if qbit k i in pattern p has value 1 , one can express the energy levels for dnH  1 as π22π π 2 k=++kk k E 2 ∑∑ssij si. (64) 16 4nnij, 4 i

Figure 2. Effective input/output distance and entropy (rescaled to [0, 1]) for 1 Mb patterns and dn= 1% .

2104 M. C. Diamantini, C. A. Trugenberger

Apart from a constant, this is the Hamiltonian of an infinite-range antiferromagnetic Ising model in presence of a magnetic field. The antiferromagnetic term favours confi- kk= = gurations k with half the spins up and half down, so that sstot ∑i i 0 , giving k 2 k E = π 16 . The magnetic field, however, tends to align the spins so that sntot = − 2 , giving E k = 0 . Since this is lower than E k = 0 , the ground state configuration is ferromagnetic, with all qbits having value 0 . At very low temperature (high b), where the energy term dominates the free energy, one expects thus an ordered phase of the quantum associative memory with (td, ) = dn. This corresponds to a perfect identification of the presented input. As the temperature is raised (b decreased) however, the thermal energy embodied by the entropy term in the free energy begins to counte- ract the magnetic field. At very high temperatures (low b) the entropy approaches its maximal value St( =∞=) 0 (with the normalization chosen here). If this value is ap- proached faster than 1/t, the free energy will again be dominated by the internal energy. In this case, however, this is not any more determined by the ground state but rather equally distributed on all possible states, giving

−1 1 π Ft( =∞=) Ut( =∞=) d dx 2logcos x d ∫  1− n 2 n (65) 2 dd   =++ 1  2 log2 O , nn   and leading to an effective distance

2 2 2log2 dd =∞=− + (td,.) O (66) 3 π 3 nn

This value corresponds to a disordered phase with no correlation between input and output of the memory. A numerical study of the thermodynamic potentials in (60) and (61) indeed confirms a phase transition from the ordered to the disordered phase as the effective temperature is raised. In Figure 2 we show the effective distance  and the entropy S for 1 Mb ( n =8 × 106 ) patterns and dn= 1% as a function of the inverse temperature b (the entropy is rescaled to the interval [0, 1] for ease of presentation). At high temperature there is indeed a disordered phase with SS=max = 0 and  = 23. At low temperatures, instead, one is in the ordered phase with SS= min and  =dn = 0.01. The effective Hamming distance plays thus the role of the order parameter for this quantum phase transition. −1 The phase transition occurs around bcr  10 . The physical regime of the quantum associative memory (b = positive integer) lies thus just above this transition. For a good accuracy of pattern recognition one should choose a fictitious temperature low enough to be well into the ordered phase. As is clear from Figure 2, this can be achieved already with a number of control qubits bO= (104 ) . Having described at length the information retrieval mechanism for complete, but possibly corrupted patterns, it is easy to incorporate also incomplete ones. To this end

2105 M. C. Diamantini, C. A. Trugenberger

assume that only qn< qbits of the input are known and let us denote these by the in- dices {k1, , kq}. After assigning the remaining qbits randomly, there are two possibil- ities. One can just treat the resulting complete input as a noisy one and proceed as

above or, better, one can limit the operator (dH )m in the Hamiltonian (42) to q σ z +1 = (dH )m ∑ , (67) i=1 2 mki

so that the Hamming distances to the stored patterns are computed on the basis of the known qbits only. After this, the pattern recall process continues exactly as described above. This second possibility has the advantage that it does not introduce random noise in the similarity measure but it has the disadvantage that the operations of the memory have to be adjusted to the inputs. Finally, it is fair to mention that the model of probabilistic quantum associative memory presented here has been criticised [45] on three accounts: • It has been claimed that the same result could have been obtained by storing only one of the p patterns in n classical bits and always using this single pattern as the same output independently of the input, provided the input has a Hamming distance to the unique stored pattern lower than a given threshold, otherwise the input would not be recognized. • It has been claimed that the Holevo theorem bounds the number of patterns that can be stored in a quantum associative memory. • It has been pointed out that the complexity of memory preparation prevents the efficient storing of patterns. This criticism is wrong on the first two accounts and partially justified on the third [46]. It is true that both the quantum memory and the proposed equivalent classical prescription are based on probabilistic recognition and identification processes. In the proposed classical alternative, however the probabilities for both recognition and identification depend on one unique, fixed and random pattern whereas in the quantum memory, exactly due to its quantum character, these probabilities depend on the distribution of all stored patterns. These probabilities are such that an input different from most stored patterns is more difficult to recognize than an input similar to many stored memories and that the identification probability distribution can be peaked with any prescribed accuracy on the stored pattern most similar to the input. In the proposed classical alternative, given that only one single pattern can be stored on the n classical bits, the recognition or lack thereof depends on the distance to a randomly chosen pattern and the identification probability is a delta function peaked on this fixed random pattern. In other words there is no correlation whatsoever between input and output apart from the fact that they have Hamming distance below a certain threshold, a prescription that can hardly qualify as an associative memory: it would indeed be a boring world the one in which every stimulus would produce exactly the same response, if any response at all. Also, the Holevo theorem [34] does not impose any limitation on this type of probabilistic quantum memories. The Holevo theorem applies to the situation

2106 M. C. Diamantini, C. A. Trugenberger

in which Alice codes information about a classical random variable in a quantum state and Bob tries to retrieve the value of this random variable by measurements on the received quantum state. In the present case Alice gives to Bob also corrupted or incomplete classical information about the random variable (the input) and Bob can use also a unitary transformation that encodes both the memories and the input (operator Ri( ) in (47)) in addition to measurements, a completely different situation. Contrary to what the authors of [45] affirm, a memory that “knows the patterns it is supposed to retrieve” not only makes sense but it is actually the very definition of an associative memory: if the memory would not “know” the data it has to retrieve it would just be a random access database, exactly the architecture that one wants to improve by content association, the mechanism whose goal is to recognize and correct corrupted or incomplete inputs. The dynamics of the classical Hopfield model “knows” the patterns it is supposed to retrieve: they are encoded in the neuronal weights. So does any human brain. Finally, the third critique is partially correct. The complexity of the memory operator M is O( pn) and thus the original claim [16] [17] of an exponential capacity gain by quantum associative memories is excessive. This, however, does not invalidate the main claim, a large gain in capacity is made possible by quantum mechanics, albeit only a polynomial one. This correction has been incorporated in the present review.

4.3. Efficiency, Complexity and Memory Tuning

In this last section we would like to address the efficient implementation of probabilistic quantum memories in the quantum circuit model [4] [5] and their accuracy tuning. We have stressed several times that all unitary operators involved in the memory preparation can be realized as a sequence of one- and two-qubit operators. It remains to prove that this is true also for pattern retrieval and that all these operators can be implemented in terms of a small set of universal gates. To this end we would like to point out that, in addition to the standard NOT, H (Hadamard), XOR, 2XOR (Toffoli) and nXOR gates [4] [5] we have introduced only the two-qbit gates CS i in Equation (22) and the unitary operator exp(inπ 2 ) . The latter can, however also be realized by simple gates involving only one or two qbits. To this end we introduce the single-qbit gate

π i 2n U = e0, (68)  01 and the two-qbit controlled gate

CU −−22=0 0 ⊗+ 1 11 ⊗U . (69)

It is then easy to check that exp(inπ 2 ) in Equation (41) can be realized as follows:

π nn i  − e,2n ψψ= CU2 U (70) 22∏∏( )cm m j ij=11i = where c is the control qbit for which one is currently repeating the algorithm. Essen-

2107 M. C. Diamantini, C. A. Trugenberger

tially, this means that one implements first exp(idπ H 2 n) and then one corrects by

implementing exp(−idπ H n) on that part of the quantum state for which the control qbit c is in state 1 . Using this representation for the Hamming distance operator one can count the total number of simple gates that one must apply in order to implement one step of the information retrieval algorithm. This is given by (62n + ) using the auxiliary register for the input and by (42n + ) otherwise. This retrieval step has then to be repeated for each of the b control qbits. Therefore, implementing the projection by repeated measurements, the overall complexity C of information retrieval is bounded by

C≤+ Tb(62 n) CM , (71)

where CM is the complexity of the memory preparation, given by the operator M or a probabilistic cloning machine. In particular, it is given by C= Tb(6 n + 2)( p( 2 n ++ 3) 1,) (72)

for the simplest version of the algorithm, using memory preparation by M and an auxiliary input register. The computation of the overall complexity is easier for the information retrieval algorithm which uses the amplitude amplification technique. In this case the initial memory is prepared only once by a product of the operators M, with complexity pn(2++ 31) and Ri( ) , with complexity bn(42+ ) . Then one applies T times the pn4+ 6 + bn 842 + ++ C + C C operator Q, with complexity ( ) ( ) SS0 , where S and C S0 are the polynomial complexities of the oracles implementing S and S0 . This gives CTpn=(46 ++) bn( 842 ++++) C C + pn(23 ++) bn( 421. ++) SS0 (73) As expected, the memory complexity (be it (72) or (73)) depends on both T and b, the parameters governing the recognition and identification efficiencies. The major limitation comes from the factor p representing the total number of stored patterns. Note however that, contrary to classical associative memories, one can efficiently store and retrieve any polynomial number of patterns due to the absence of spurious memories and crosstalk. Let us finally show how one can tune the accuracy of the quantum memory. Suppose one would like to recognize on average inputs with up to 1% of corrupted or missing bits and identify them with high accuracy. The effective i/o Hamming distance  shown in Figure 2 can then be used to determine the values of the required parameters T and b needed to reach this accuracy for the average memory. For b = 104 e.g., one has  = 0.018 , which gives the average i/o distance (in percent of total qbits) if the minimum possible i/o distance is 0.01. For this value of b the recognition probability is 3.4 × 10−4. With the measurement repetition technique one should thus set the threshold T  3000 . Using amplitude amplification, however, one needs only around T = 54 repetitions. Note that the values of b and T obtained by tuning the memory with the effective i/o Hamming distance become n-independent for large values of n. This is because they are intensive variables unaffected by this “thermodynamic limit”.

2108 M. C. Diamantini, C. A. Trugenberger

For any fixed p polynomial in n, the information retrieval can then be implemented efficiently and the overall complexity is determined by the accuracy requirements via the n-independent parameters T and b.

5. Conclusions

We would like to conclude this review by highlighting the fundamental reason why a probabilistic quantum associative memory works better than its classical counterpart and pointing out about some very intuitive features of the information retrieval process. In classical associative memories, the information about the patterns to recall is typically stored in an energy function. When retrieving information, the input configuration evolves to the corresponding output, driven by the dynamics associated with the memory function. The capacity shortage is due to a phase transition in the statistical ensemble governed by the memory energy function. Spurious memories, i.e. spurious metastable minima not associated with any of the original patterns become important for loading factors p/n above a critical value and wash out completely the memory, a phenomenon that goes by the name of crosstalk. So, in the low p/n phase the memory works perfectly in the sense that it outputs always the stored pattern which is most similar to the input. For p/n above the critical value, instead, there is an abrupt transition to total amnesia caused by spurious memories. Probabilistic quantum associative memories work better than classical ones since they are free from spurious memories. The easiest way to see this is in the formulation

mM= 0. (74)

All the information about the stored patterns is encoded in the unitary operator M. This generates a quantum state in which all components that do not correspond to stored patterns have exactly vanishing amplitudes. An analogy with the classical Hopfield model [7] [8] can be established as follows. Instead of generating the memory state m from the initial zero state m , one can start from a uniform superposition of the computational basis. This is achieved by the operator MW defined by

n 1 21− m= MW j , n ∑ = 2 j 0 (75) n WH≡ ∏ j . j=1

Now, this same result can also be obtained by Grover’s algorithm, or better by its generalization with zero failure rate [47]. Here the state m is obtained by applying to the uniform superposition of the computational basis q times the search operator X defined in

n 1 21− mX= q j, n ∑ 2 j=0 (76)

X≡− WJ0 WJ ,

2109 M. C. Diamantini, C. A. Trugenberger

where J rotates the amplitudes of the states corresponding to the patterns to be stored

by a phase φ which is very close to π (the original Grover value) for large n and J 0 does the same on the zero state. Via the two equations (75) and (76), the memory operator M provides an implicit realization of the phase shift operator J. Being a unitary operator, this can always be written as an exponential of an hermitian Hamiltonian

M , which is the quantum generalization of a classical energy function. By defining

Ji≡−exp( M ) one obtains an energy operator which is diagonal in the computational basis and such that the patterns to be stored have energy eigenvalues E =−−φ  π while all others have energy eigenvalues E = 0 . This formulation is the exact quantum generalization of the Hopfield model; the important point is that the operator M realizes efficiently a dynamics in which the patterns to be stored are always, for any number p of patterns, the exact global minima of a quantum energy landscape, without the appearance of any spurious memories. The price to pay is the probabilistic nature of the information retrieval mechanism. As always in quantum mechanics, the dynamics determines only the evolution of probability distributions and the probabilistic aspect is brought in by the collapse of this probability distributions upon measurement. Therefore, contrary to the classical Hopfield model in the low p/n phase, one does not always have the absolute guarantee that an input is recognized and identified correctly as the stored pattern most similar to the input, even if this state has the highest probability of being measured. But, after all, this is a familiar feature of the most concrete example of associative memory, our own brain, and should thus not be so disturbing. Indeed, it is not only the probabilistic nature of information retrieval that is reminiscent of the behaviour of the human brain but also the properties of the involved probability distributions. These are such that inputs very similar to a cluster of stored patterns will be much easier to recognize than i.

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