Review of the Methods of Transition from Partial to Ordinary Differential

Archives of Computational Methods in Engineering https://doi.org/10.1007/s11831-021-09550-5

REVIEW ARTICLE

Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑ to Nano‑structural Dynamics

J. Awrejcewicz1 · V. A. Krysko‑Jr.1 · L. A. Kalutsky2 · M. V. Zhigalov2 · V. A. Krysko2

Received: 23 October 2020 / Accepted: 10 January 2021 © The Author(s) 2021

Abstract This review/research paper deals with the reduction of nonlinear partial diferential equations governing the dynamic behavior of structural mechanical members with emphasis put on theoretical aspects of the applied methods and signal processing. Owing to the rapid development of technology, materials science and in particular micro/nano mechanical systems, there is a need not only to revise approaches to mathematical modeling of structural nonlinear vibrations, but also to choose/pro- pose novel (extended) theoretically based methods and hence, motivating development of numerical algorithms, to get the authentic, reliable, validated and accurate solutions to complex mathematical models derived (nonlinear PDEs). The review introduces the reader to traditional approaches with a broad spectrum of the Fourier-type methods, Galerkin-type methods, Kantorovich–Vlasov methods, variational methods, variational iteration methods, as well as the methods of Vaindiner and Agranovskii–Baglai–Smirnov. While some of them are well known and applied by computational and engineering-oriented community, attention is paid to important (from our point of view) but not widely known and used classical approaches. In addition, the considerations are supported by the most popular and frequently employed algorithms and direct numerical schemes based on the fnite element method (FEM) and fnite diference method (FDM) to validate results obtained. In spite of a general aspect of the review paper, the traditional theoretical methods mentioned so far are quantifed and compared with respect to applications to the novel branch of mechanics, i.e. vibrational behavior of nanostructures, which includes results of our own research presented throughout the paper. Namely, considerable efort has been devoted to investigate dynamic features of the Germain–Lagrange nanoplate (including physical nonlinearity and inhomogeneity of materials). Modifed Germain–Lagrange equations are obtained using Kirchhof’s hypothesis and relations based on the modifed couple stress theory as well as Hamilton’s principle. A comparative analysis is carried out to identify the most efective methods for solving equations of mathematical physics taking as an example the modifed Germain–Lagrange equation for a nanoplate. In numerical experiments with reducing the problem of PDEs to ODEs based on Fourier’s ideas (separation of variables), the Bubnov–Galerkin method of static problems and Faedo–Galerkin method of dynamic problems are employed and quantifed. An exact solution governing the behavior of nanoplates served to quantify the efciency of various reduction methods, including the Bubnov–Galerkin method, Kantorovich–Vlasov method, variational iterations and Vaindiner’s method (the last three methods include theorems regarding their numerical convergence). The numerical solutions have been compared with the solutions obtained by various combinations of the mentioned methods and with solutions obtained by FDM of the second order of accuracy and FEM for triangular and quadrangular fnite elements. The studied methods of reduction to ordinary diferential equations show high accuracy and feasibility to solve numerous problems of mathematical physics and mechanical systems with emphasis put on signal processing.

1 Introduction

In this review, we consider numerous computational approaches derived from mathematical physics and more widely used over the years to fnd nonlinear PDE solutions that govern the dynamic behavior of structural members. * J. Awrejcewicz [email protected] This section will include the description of advantages (sometimes disadvantages) of the classical and extended Extended author information available on the last page of the article

Vol.:(0123456789)1 3 J. Awrejcewicz et al. theoretical variants of methods developed by Fourier and and continuum mechanics are solved with a help of the Fou- Galerkin, variational methods, as well as the variational rier methods. In the former case the system of governing iteration and Kantorovich–Vlasov methods, and others. ODEs is solved using either single (periodicity) or double (quasi-periodicity) Fourier series, while in the latter case 1.1 Fourier Methods (FM) usually the governing PDEs are solved using the Galerkin and Bubnov–Galerkin methods, as well as the separation A number of computational methods aimed at solving a wide of variable method matched with the Fourier series. The variety of problems in mathematical physics and technology Fourier methods and the associated harmonic analysis are were based on the ideas of the great French scientist J. Fou- always involved either explicitly or implicitly in solving a rier including the method of separation of variables or the variety of problems of mathematical physics including the method of eigenfunctions [1]. He formulated the methods Navier-Stokes equations, Litllewood–Paley decompositions, which later gave impetus to the use of the variable separation Besov spaces, etc. procedure when creating new approaches to solving equa- Fourier series representation includes both real and com- tions for functions of several variables. William Thomson plex sinusoids. In the case of signals (noise) often measured (Lord Kelvin) called this work “The Great Mathematical in engineering problems, the Fourier representation has a Poem”. direct physical interpretation. Various waveforms have Fou- The advantage of the Fourier method is that it creates rier series representations, and therefore the Fourier-based a simple way to obtain an explicit solution. This method analysis is inherently involved in any engineering action. allows us to present a series of very wide class of functions, Any periodic/quasi-periodic signal with the given period/ since it requires that one smoothness condition is satisfed periods can be approximated by 1D/2D Fourier series. The inside the area, and others on the boundary. Employment Fourier series can approximate odd and even square waves/ of the apparatus of asymptotic analysis to obtain estimates signals. In addition, regular pulse trains, triangle wave- of the Fourier coefcients improved the convergence and form as well as saw tooth waveforms have their Fourier highlighted the features of the series. There are many other expressions. benefts while using Fourier’s method, which are specifc to As it has already been mentioned, in engineering Fourier each particular problem. For example, it includes the clear methods serve as a paradigm for solving majority of prob- physical meaning of Fourier coefcients in vibrations problems dealing with signal processing, including the Fourier lems and heat dissipation problems. transform, discrete-time Fourier transform, discrete Fou- Nowadays the Fourier-based methods have found vari- rier transform, short-time Fourier transform and diferent ous applications while analyzing linear and nonlinear PDEs. wavelets. Though FFT does not belong strictly to the Fou- Propagation processes governed by PDEs can be solved rier methods, it can be understood also as the Fourier idea using the so-called split-step Fourier method. It requires a matched with a computational technique for evaluating the simple numerical implementation and can cause phenomena discrete Fourier transforms. It should be noticed that the which are not detected by other numerical methods. This short-time Fourier transform makes it also possible to trace approach is widely used in the numerical analysis of non- time evolution of the frequency components, although usu- linear optical fber channels [2]. The algorithm is a combi- ally wavelet transforms are used for this purpose. nation of reducing the step-size for the integration along a From the engineering point of view the Fourier methods chosen variable, and then by utilizing a discrete-time rep- represent the strength of a signal measured by its ampli- resentation of the propagating signals to get back and forth tudes versus its frequencies, i.e. an arbitrary signal can be between time and frequency domains using FFT (Fast Fou- presented as a sum of sinusoids (harmonics) of diferent fre- rier Transform). quencies. This allows us to carry out the spectral analysis Though the direct Fourier method and the split-step Fou- and detect numerous important frequencies, including those rier method are faster than most commercial programs, in occurring in periodic, quasi-periodic and chaotic signals. general they are computationally time-consuming, and often The discrete-time/discrete Fourier transforms, and the short the fnite-diference methods are used instead [3]. Fourier transform use the discrete-time complex sinusoids, When analyzing mechanical systems, machines and whereas the Fourier transform employs signals as weighted mechanisms, the signal processing is based on the time- combinations of continuous-time complex-valued sinusoids. frequency resolution of Fourier methods and wavelets. The In mathematical modeling of mechanical/mechatronic Fourier methods belong to the oldest and have been used to systems mainly diferential equations are studied. ODEs solve static and dynamic problems of structural members, describe lumped mass mechanical systems and refer to dis- including rods, beams, plates and shells, and the solutions placements (positions), velocity and accelerations, whereas can be obtained either in the form of stresses or displace- PDEs describe the behavior (statics and dynamics) of ment functions. Numerous problems of nonlinear discrete

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑… continuously distributed mass systems and enable the pre- linearly independent system. It is expected that i (weights) diction of vibrations, sound, fuid fow, waves, etc. can be defined using the condition that the residual (i) A property of the Fourier series, i.e. diferentiation of r = u − f → min . However, a question arises how to a sum of harmonics of diferent frequencies yields a sum defne coordinate functions and test functions to fnd a suit- of harmonics of the same frequencies corresponds to the able form of the residual to satisfy the requirement that uN behavior met in real world processes, physics and engineer- tends to u, where u is the exact solution. Galerkin proposed ing. It is well known that the engineering approaches based a general method and pointed out that the residual should on measuring and control of the mechanical, electric and be orthogonal to a system of test functions (linearly inde- mechatronic systems involve time delay. This is well ftted pendent) where the orthogonality refers to the integral sense by the Fourier series, since a delay of the sum of harmonics [13]. (Nowadays, it is recognized that Galerkin employed of diferent frequencies results in the sum of harmonics of not widely known, brilliant idea published by Bubnov (an the same frequencies. engineer and marine ofcer) who applied his method to ship The Fourier methods are well suited to study signal-sys- building problems. This is why in many cases the method tem interactions. In the case of linear time-invariant sys- is referred to as the Bubnov–Galerkin method, while some tems, the so-called “eigenfunction” property holds, i.e. an researchers distinguish the Galerkin and the Bubnov–Galer- input described as an arbitrary sum of complex harmonics kin methods). of diferent frequencies yields an output given by the sum of Therefore, after modifcations, the general Galerkin-based complex harmonics of the same frequencies. The FMs allow idea is to fnd weight coefcients i using the orthogonality also for employing flters to separate certain frequency com- relation of the form (ii) uN − f , k = 0 for all k ∈ YM, ∕ ponents of the input signal. In addition, they give efcient where uN is given by (i), and YM ⊂ Y is the set of independ- computational results of the FFT algorithms based on the ent test functions in the space⟨ conjugate⟩ to Y; duality pairing ∕ properties of complex harmonics. The Fourier methods are of Y and Y is defned by ⋅, ⋅ . The latter general formula- extensively described in numerous books [4–6]. Fornberg tion includes various special cases: (a) the Bubnov–Galerkin [7] employed the Fourier method based on FFT to approxi- method when X and Y are ⟨Hilbert⟩ spaces and spaces XN and mate space derivatives for hyperbolic PDEs with emphasis YN coincide (see [14]); (b) the Ritz–Galerkin method when put on its stability and accuracy. (ii) is generated by a quadratic equation and comes from Kreiss and Oliger [8] developed a stability theory for the the Euler equation ([15], chapter 4; [16]); (c) the Petrov- Fourier pseudo-spectral method with the application to the Galerkin method where the problem was applied to ft the linear hyperbolic/parabolic PDEs with variable coefcients. eigenvalues ([17–19]); (d) the Taylor–Galerkin method Majda et al. [9] employed the Fourier method to gen- where emphasis is laid on the fnite element method applied eral linear hyperbolic Cauchy problems with nonsmooth to PDEs [20]; the Faedo–Galerkin method where dynamic initial data. It was proven that the appropriate smoothing properties are concerned [21]. techniques give stability, whereas smoothing combined with Nowadays the Galerkin methods play a fundamental role smoothing of the initial data yielded infnite order accuracy in numerical solutions of PDEs including elliptic, parabolic of the obtained results. and hyperbolic equations [22]. Koslof et al. [10] used the Fourier method to construct The method is widely used by the classical and extended a 2D forward modeling algorithm. The solution included discretization algorithms employed explicitly and/or implic- discretization in both space and time. Temporal deriva- itly in the fnite element methods (FEMs), fnite diference tives were approximated by the second order diferencing, methods (FDMs), and spectral element methods (SEMs). whereas spatial derivatives were estimated via FFT. The Nodal (Lagrangian interpolants/polynomials) and modal algorithm was validated using the analytic solutions for 2D Fourier basis and Legendre polynomials may serve as basis Lamb problem. functions in 1D, and they can be extended to higher space Peyret [11] employed the Fourier method to solve the dimensions by using SEM and a concept of tensor products. problems of spatial periodicity in one/two directions It should be emphasized that the global modes can be for- matched with the Chebyshev polynomials. mulated in a mathematical, physical and/or empirical way. A particular role in fnding solutions to a PDE plays the 1.2 Galerkin Method (GM) modifed FEM known as the Galerkin FEM. In this case the Lagrange and Hermite elements are used as trial functions The Galerkin method can be thought as an extension of the and this approach can be formulated in both global and local method of Ritz [12] who considered the problem u = f ( way. stands for a diferential operator acting in a Banach space). More recently the classical FEM includes extension to ft N The approximating solution is u(x)=UN (x)= i=1 iwi(x), the discontinuous problems based on completely discontinu- where N is the natural number and wi ∈ XN ⊂ X forms a ous basis functions instead of the piece-wise polynomials ∑ 1 3 J. Awrejcewicz et al.

[23]. The review paper [23] addressed the problems of the corresponding 1D problems. In addition, they reported new discontinuous Galerkin methods for solving high-order time- results regarding warping of the cross-section and deforma- dependent PDEs with emphasis put on the consideration of tion of the prism axis for diferent types of anisotropy. numerical fux, stability, time discretizations, accuracy of Bespalova [44] proposed also to match the inverse-itera- the solution and error estimates. tion method of successive approximations and the advanced As already mentioned, Galerkin-type approaches have KVM to calculate natural frequencies of an elastic parallel- found great application in both mathematical physics epiped with diferent boundary conditions. The dependence [24–28] and mechanical engineering as well as signal proof lower frequencies of a 3D cantilever beam was reported. cessing [29] (due to the page limit of our review, we have Surianinov and Chaban [45] used the KVM to reduce the omitted here numerous articles dealing with these issues). 2D problem of calculating the anisotropic plate based on Importance of the Galerkin methods, its sound applica- the proposed analytical-numerical method. This approach tion and possible further extension can be found in the edito- yielded the solution of a basis diferential equation of bend- rial written by Repin [30]. ing an anisotropic plate with any boundary conditions and without any restrictions given to the external load. 1.3 Kantorovich–Vlasov Methods (KVM) Nwoji et al. [46] applied the KVM to the fexural investigation of simply supported Kirchhof plates subjected to Among variety of the numerical methods devoted to the transverse uniformly distributed load. The Vlasov approach solution of static and dynamic problems of structural mem- was applied to construct coordinate functions in the x direc- bers, a key role is played by the methods introduced by tion, whereas the Kantorovich method was used to consider Kantorovich [31] and Vlasov [32], further known as the the displacement feld. They reported that the obtained solu- Kantorovich–Vlasov method (KVM). It has the following tions were rapidly convergent for both defection and bend- advantages: ing moments. The results obtained were validated by the Levi-Nadai solutions. (i) solution to a nonlinear PDE(s) considered in the Then, Kerr [47] extended the Kantorovich method to get Cartesian coordinates is represented by a single the exact solution in both directions by using the iterative series of infnite terms as a combination of func- procedure. Kapuria and Kumari [48] applied KVM to study tions with separable variables where all components a 3D problem of elasticity solution of laminated composite of the approximation are unknown. For example, in structures in cylindrical bending. Aghdam et al. [49] used the case of uniformly loaded Kirchhof–Love rec- successfully the Kantorovich method in the static analysis of tangular plate defection, w(x, y) is represented as ∞ moderately thick, functionally graded sector plates. Wu et al. w(x, y)= n=1 f (x)gn(y); [50] found the Kantorovich-based solution when studying (ii) the original 2D (3D) problem is reduced to a system ∑ a simply supported beam under linearly distributed loads. of two (three) coupled 1D problems; The multi-term extended KVM was applied to fnd analyti- (iii) the iterative procedure of KVM is efcient for com- cal solution for the bending problem of moderately thick putations, since it converges rapidly (usually, it takes composite annular sector plates with general boundary con- a few iterations with a few terms in the approximat- ditions and loadings [51]. ing function to achieve the validated and stable solu- The free vibration analysis based on KVM of symmetri- tion); cally laminated fully clamped skew plates and of moderately (iv) KVM is useful and shows advantages over other thick, functionally graded plates was carried out in the works traditional methods of reduction of PDEs to ODEs [52, 53]. Vibrations of laminated plates with various bound- while dealing with vibrations of beams, plates and ary conditions based on the extended Kantorovich method shells. were analyzed [54]. Kapuria et al. [55] studied the 3D elasticity solution of In the beginning, the reduction of PDEs dimension occur- laminated plates with a help of multi-term extended Kan- ring in mathematical physics and mechanics to that of the torovich method, whereas Kumari et al. [56] used the 3D counterpart 1D problem was carried out by Grigorenko and extended Kantorovich solution for Levy-type rectangular his disciples [33–36] as well as by Krysko and his co-work- laminated plates with edge efects. ers [37–39]. Then, Bespalova successfully employed this Huang et al. [57] used KVM to estimate local stresses in method to various problems of nonlinear structural mechan- composite laminates upon polynomial stress function. ics [40–42]. Kumari and Behera [58] obtained a closed form 3D solu- More recently, Bespalova and Urusova [43] used the tion for a rectangular laminated plate with Levy type sup- advanced Kantorovich–Vlasov approach to reduce the port based on the extended Kantorovich method. Natural 3D problem of torsion of an anisotropic prism to three frequencies were estimated for composite and sandwich

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑… laminates. In addition, the efect of the span to thickness The rigorous treatment and extension of the classical ratio and in-plane modulus ratio on the natural frequencies variational methods is ofered by Marsden et al. [67] who was also investigated. Ike and Mama [59] carried out the proposed a variational and multisymplectic formulation of fexural analysis of Kirchhof–Love plates using the vari- compressible/incompressible models of continuum mechan- ational Kantorovich method. ics on general Riemannian manifolds. An interplay of compressible/incompressible continuum mechanics, constrained 1.4 Variational Methods (VM) multisymplectic field theories, symmetries, momentum maps and Noether’s theorem was ofered. The variational methods belong to the fundamental tools Direct applications of variational methods for engineers for solving variety of linear/nonlinear problems not only in can be found in the book [68]. It gives friendly written basic mechanics but in mathematical physics due to the rigorous material devoted to variational methods for algebraic equa- mathematical bases. In the classical mechanics it concerns tions, and to diferential equations, the treatment of Hilbert the mechanics of solids and fuids, materials modeling and and functional spaces feasible for engineers, as well as func- properties (creeps, plasticity and smart materials fabrication, tionals and calculus of variations. materials with memories, etc.), the statics and dynamics of Finally, it should be emphasized that the classical vari- structural members dealing with real engineering construc- ational methods still attract interest of the scientists which tions, heat transfer, and others. It allows us to model com- is documented by the original treatment and extension to the plex behavior of both discrete systems and continuum. thematic issues presented in the monographs/books [69–74]. Theoretical achievements, counted from the 18th century, introduced by seminal works of Euler, Lagrange, Bubnov, 1.5 Variational Iteration Method (VIM) Galerkin, Kantorovich, Nowacki, Vlasov, Vaindiner, Reiss- ner and Ritz along with Hamilton’s principle have recently The next method is the variational iteration method (VIM), attracted great interest of the researches, and have brought which relieves a researcher from the need to construct a sys- back the classical approaches based on the variational tem of approximating functions like in the Galerkin method. methods and energy principles. This is motivated by grow- Arbitrarily defned functions (satisfying only the smoothness ing attractiveness of numerical/computational methods in conditions) are taken and then they are refned in the process applied sciences, and in particular in engineering includ- of calculation by VIM based on the solutions of the original ing the fnal element method, fnite diference method, Ritz system of diferential equations. method, weighted-residual methods, and more. This method was frst proposed and applied in 1933 by The variational methods require good mathematical back- Shunk [75] to calculate the bending of cylindrical panels. ground, as it is shown by the authors of the outstanding mon- But the work went unnoticed and the method was rediscov- ographs [60–65], but it seems that the pressure of rapidly ered again in 1964 by Zhukov [76], who used it to calculate developing technology and engineering approaches shifted a thin rectangular plate. Further, VIM was widely used by and somehow shallowed the fundamental achievements of many researchers to solve the problems of shell and plate variational classical theories to engineering and applied sci- theory [37]. The substantiation of this method for the class ences with direct application to the fnite element methods. of equations described by positive defnite operators was However, the classical variational approaches can be given in [38]. extended to mathematical branches of science (geometry, It is worth noting here that in the literature there is a large topology, analysis), mathematical physics, optimization number of diferent names for the method of variational problems and mathematical modeling and analysis of non- iterations. This method is often called by Kerr [77–79] the linear behavior in biology, medicine, and economics. extended Kantorovich method and it was published 38 years There are numerous publications devoted to the theory after Shunk and 5 years after Zhukov. Given this, we can say and application of variational methods. Since the latter have that the method has been rediscovered. been closely matched with the computational methods, it is The variational iteration method or extended Kantorovich almost impossible and perhaps useless to cite all or major- method has been used over the past half century to solve ity of them. Therefore, only few of them will be mentioned. the problems of statics, stability, natural frequencies and Courant [66] pointed out importance of the calculus of dynamics. variations and its equivalence with the boundary value prob- The VIM was extended by Ji-Huan He [80, 81] who used lems of PDEs. He considered variational problems with ref- the old concept of Lagrange multiplier method, with empha- erence to quadratic functionals, rigid and natural boundary sis put on analytical solutions. The approach proposed by He conditions, the Rayleigh–Ritz method, the method of fnite found many followers and his modifcation to the classical diferences, and the method of gradients. VIM was employed in a wide range of issues of mathematical physics dealing with ODEs and PDEs. On the other hand,

1 3 J. Awrejcewicz et al. the convergence of VIM was discussed in many papers, multipliers to identify the optimal values of parameters in including [82–86]. a functional. Solodov and Svaiter [87] proposed a novel projection Heydari et al. [96] combined the spectral method and algorithm to solve variational inequality. First, an appropri- VIM for heat transfer problems with high order of nonlin- ate hyperplane separating the current iterate from the solu- earity. The method was implemented to solve an unsteady tion was employed. It required a single projection onto the nonlinear convective-radiative equation with two small inde- feasible set and application of the Amijo-type line search pendent parameters. along a feasible direction. The next iteration was a projection Wazwaz [97] applied VIM for solving linear/nonlinear of the current iterate onto the intersection of the feasible set ODEs with variable coefcients. He tested a variety of mod- with the halfspace including the solution. els including the hybrid selection model, Thomas-Fermi Matinfar et al. [88] developed a so called modifed vari- equations, as well as the Kidder and Ricatti equations. ational iteration method (MVIM) by merging the classical Zhang et al. [98] used a new scheme while adopting VIM VIM with He’s polynomials and applied it to solve heat to solve the problem of infrared radiative transfer in a scat- transfer problems governed by PDEs with variable coef- tered medium. Both upward and downward intensities were cients. The proposed algorithms matched the variational iter- calculated separately by VIM and it was found that VIM was ations with the homotopy, perturbation methods and exhib- more accurate comparing to the discrete-ordinates method. ited all advantages of the coupled technique. The authors Narayanamoorthy and Mathankumar [99] used VIM showed that MVIM was employed without any discretization to solve problems governed by the linear Volterra fuzzy procedures and strong assumptions and was free from errors. integro-diferential equations. Two test problems were con- Noor and Mohyud-Din [89] developed an analytical ver- sidered and the approximate versus exact solutions were sion of VIM called the variational decomposition method validated. based on the example of the eighth-order boundary value El-Sayed and El-Mongy [100] employed the modifed problem. Analytical solutions were found as the conven- VIM to solve the free vibration problem of a tapered Euler- tional technique confrmed the reliability, validity and high Bernoulli beams mounted on 2DoF mass-spring-damper accuracy of the proposed method. subsystem. Natural frequencies and mode shapes for uni- Ji-Huan He [90] again considered VIM with emphasis form/tapered beam were studied. It was demonstrated how laid on recent results and new interpretations. Procedures highly accurate results were obtained by adjusting only a illustrating the solution were based on nonlinear oscillators few iterations of VIM. governed by ODEs. Then, He and Wu [91] discussed the application of VIM to nonlinear wave equations and non- 1.6 Vaindiner’s Method (VaM) linear fractional diferential equations occurring in various engineering problems. They noticed that VIM may serve as This method, while rigorous and powerful, did not attract, a candidate to solve problems of strongly nonlinear equa- however, special attention particularly among the western- tions which cannot be attacked by the classical approximate oriented researchers. The method can be viewed as an methods. extension and modifcation of the Kantorovich and Bub- Barari et al. [92] combined the homotopy perturbation nov–Galerkin methods. An approximate solution to a prob- method and VIM to solve nonlinear ODEs with an oscillat- lem governed by PDEs was searched in the form of the sum ing solution. They stressed that the method did not require of products of the coordinate and sought functions with either linearization or small perturbation which is needed in applications to the Fourier general polynomials and series the classical approaches. [101, 102]. In the frst product, a given function depends Saadatmandi and Dehghan [93] employed He’s vari- on x, whereas in the second product it depends on y. The ational method to solve the generalized pantograph equa- approach was then further extended and developed by study- tion. They demonstrated that only a few iterations were suf- ing many problems of the convergence of the methods used, fcient to guarantee a highly accurate solution, and it was not the decreasing order of the studied systems, interpolation of afected by computation round of errors. functions with many variables, lattice collocation, boundary Abbasbandy and Shivanian [94] used VIM successfully value problems and singular integral equations of elasticity to solve nonlinear Volterra’s integro-diferential PDEs. They in 3D [101–107]. observed that VIM yielded more accurate results as com- Vaindiner’s method was widely used to study station- pared to the homotopy perturbation method and the Ado- ary geometrically nonlinear problems of the shell theory mian decomposition method. [108]. The system of nonlinear ordinary diferential equa- Salehoor and Jafari [95] employed the VIM extended by tions was reduced to a nonlinear system of algebraic equa- He to fnd a solution to nonlinear gas dynamics equation tions by the fnite diference method. And then, the problem and Stefan equation. The method involved the Lagrange was solved by the general iteration method. A solution for

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑… square plates with three types of boundary conditions was with regard to chaotic and bifurcational features (see books obtained including the analysis of a simple support, sliding [112–114] and review paper [115]). Thus, with the appropri- clamping of the contour, a simple support on two opposite ate methodology suitable from the engineering point of view edges and sliding clamping on the other two. More recently, and reliable and accurate approaches to the transition from Vaindiner’s method was used to study the static bending of nonlinear multidimensional problems to a set of nonlinear plates taking into account physical nonlinearity and difer- but one-dimensional problems, progress can be made in the ent moduli of material [109, 110], and also to consider the computational aspects of modern mechanics, aeronautics problems of one-sided corrosion of shells [111]. and civil engineering. A brief review of the literature reveals that there is Worth noting is the interest of engineering community no available comparative analysis of the methods so far in the so called reduced order models (ROMs), where the described and schematically shown in Fig. 1 (description of transition from multidimensional infnite problems governed the Agranovskii–Baglai–Smirnov method (ABSM) is omit- by nonlinear PDEs is reduced to a set of nonlinear ODEs, ted here, because it is presented in Sect. 3.3). Many of the where the problem is highly truncated to the single-mode above methods have not been applied to problems in the approximation. Although this approach allows us to fnd a theory of plates and shells. This is why we have employed relationship between the dynamics of structural members and quantifed the efectiveness of the methods based on the with infnite degrees of freedom and the dynamics of lumped example of the Germain–Lagrange PDE. massed systems with one degree of freedom governed by the second order ODE, in many cases it does not allow us to 1.7 Closing Remarks obtain reliable results. Therefore, recognizing several advantages of this engineering-oriented approach, such as direct It is well recognized that in solving multidimensional prob- ft to the well-known classical 1 DoFs oscillators, includ- lems of mathematical physics, the reduction of dimension ing Dufng, Dufng-Helmholtz and van der Pol archetypal plays a crucial role, and next an attempt to achieve a solu- systems, and capturing the basic dynamic phenomena of tion in a compact way is highly required. In this respect, an structural members under study, we prefer to highlight more original multidimensional problem is substituted by its coun- rigorous mathematically and reliable methods to be used by terpart sequence of one-dimensional problems. This makes engineering community, which is why ROMs are not cov- it possible to obtain solutions without time-consuming ered by our review article. numerical computations, which is important for engineering The rest of the paper is organized in the following way. applications. Since in most cases it is impossible to obtain an Problem statement is given in Sect. 2, whereas solutions exact match between the original multidimensional problem based on the methods described in Sect. 1 are reported in and the sequence of one-dimensional problems, appropriate Sect. 3. Section 4 presents solutions of the problem using functionals are proposed and used to solve the problem in FEM and FDM, and concluding remarks are given in Sect. 6. an approximate way. Nonlinear dynamics of the one-dimensional macro-structural-mechanical issues has been recently deeply revisited

Fig. 1 The scheme of connec- tions of the Fourier method (FM) with the Bubnov–Galerkin method (BGM), Kantorovich– Vlasov method (KVM) and their modifcations

1 3 J. Awrejcewicz et al.

2 Problem Statement components of deformation tensor , components of the deviatory part of the symmetric tensor of the higher order To construct a mathematical model of a Ger- m, and the components of the symmetric part of curvature main–Lagrange nanoplate based on the modifed couple tensor . stress theory, an elastic isotropic rectangular plate is con- Linear deformations and displacements are coupled via sidered. The rectangular coordinate system associated the following relations with the nanoplate is introduced in the following way: 2w 2w 2w ={x, y, z ∕(x, y, z )∈[0, a]×[0, b]× [−h∕2, −h∕2 ]} =−z , =−z , =−2z . xx x2 yy x2 xy xy (2) The nanoplate has dimensions a, b, h along axes x, y, z, z = 0 respectively. Its middle surface at is denoted by The following relation for the non-zeroth component of the B ={x, y ∕(x, y )∈[0, a]×[0, b]} (see Fig. 2). symmetric curvature tensor part holds The displacements along the x, y axes refer to u, v, respec- 2w 2w 1 2w 2w tively. Defection (normal plate movement) along the z axis = , =− , = − . w = w(x, y) xx xy yy xy xy 2 y2 x2 (3) is denoted by . In order to construct the mathematical model, we introduce the following assumptions and hypotheses: (i) Kirchhof hypothesis holds; (ii) rotation iner- For non-zero components of the symmetric part of stress tia of the plate elements are not taken into account; (iii) plate tensor and for the deviatory part of the tensor of the higher m material is homogeneous and isotropic, i.e. it is assumed order , the following relations take place [116]: that for the same stress the same deformations occur in all xx = ( + 2)xx, yy = ( + 2)yy, xy = 2xy, points and elastic properties in each point of the plate are the m = 2l2 , m = 2l2 , m = 2l2 , (4) same in all directions; (iv) all displacement components are xx xx yy yy xy xy considered to be signifcantly smaller than the characteristic E E where: x, y, z, ei = , x, y, z, ei = are size of the considered rectangular plate; (v) the deformations (1+)(1−2 ) 2(1+) xx, yy, xy the Lamé parameters; of middle surface are assumed to be negligible 2 compared to unity (this does not mean that the relationship e = 2 − + 2 + 2 + 3 2 i 3 xx yy yy xx 2 xy describes the between movements and deformations must be linear). √ � � � E Owing to the modifed couple stress theory [116], the intensity of deformation; - Young modulus; - Poisson’s l accumulated energy of deformation U of the plate, taking coefcient; - material length parameter associated with the its size-dependent behavior into account, is governed by the symmetric tensor of the relation gradient. following formula According to the method of variable parameters of elasticity [117], E x, y, z, ei and x, y, z, ei are not con- 1 stant but they depend on the coordinates and intensity of U = xxxx + yyyy + xyxy 2 deformations. (1) Equations (2), (3) and (4) are used to derive deformation energy U of the nanoplate, i.e. we have +mxxxx + myyyy + mxyxy d h a b 2 2 2 , , , , m , m , m , , 1 2w 2w where xx yy xy, xx yy xy, xx yy xy, xx yy xy denote U = ( + 2) −z + ( + 2) −z 2 2 the components of the symmetric part of stress tensor , 2 x y 0 0 − h 2 2 2 2w 1 2w + 2 −z + l2 − xy 2 x2 2 2 1 2w 2w + l2 − + l2 − dzdydx. 2 y2 xy (5) h 2 2 We introduce the following notation: ∫ ( + 2)z dz = 1 − h h h 2 2 2 2 , ∫ z dz = 2 , ∫ dz = 3 . Then, i = i x, y, ei are the − h − h 2 2 functions of coordinates and intensity of deformations. Equation (5) takes the following form Fig. 2 General view of the nanoplate

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

a b 2 2 2 1 2w 2w 2w W = q(x, y, t) w dB, U = + + 2 1 2 2 2 2 � � x y � xy B 0 0 � � � � � � b a (10) 2w(x, y, t) 2 2 2 2 2 2 K = I w dxdy, 1 w w w 2 + l2 + + 2 dxdy, t 3 2 2 0 0 2 � x y xy �⎫ � � � � � � h ⎪ 2 ⎬ (6) I = dz ⎪ where: ∫ . − h where the underlined members include higher order⎭ terms. 2 External work related to the distributed forces takes the Employing Hamilton’s principle, we get following form t1 (K − U + W) dt = 0, W = q(x, y) w(x, y)dB. (11) (7) t0 B and we obtain the following equation of dynamics of the Kinetic plate energy is as follows nanoplate based on the modifed couple stress theory with h 2 variable elastic parameters 1 w 2 K = (x, y, z) dzdB. 2 t (8) B − h 2

The associated variations of U, W, K are as follows

4w 3w 2 2w U = + 2 1 + 1 + 1 x4 x x3 x2 x2 B 4w 3w 2 2w + + 2 1 + 1 1 y4 y y3 y2 y2 2 2w 3w 3w 4w + 2 2 + 2 + 2 + xy xy x xy2 y x2y 2 x2y2 1 4w 3w 2 2w + l2 + 2 3 + 3 2 3 x4 x x3 x2 x2 4w 3w 2 2w + + 2 3 + 3 3 y4 y y3 y2 y2 2 2w 3w 3w 2w +2 3 + 3 + 3 + w dxdy (9) xy xy x xy2 y x2y 3 xy a b b 2w w 2w 3w + − 1 + (w) 1 y2 y y y2 1 y3 0 0 0 b 2 3 2 w w −2 + (w) dx x xy 2 x2y 0 b a 2 a 2 3 w w 1 w w + − + (w) 1 x2 x x x2 1 x3 0 0 0 a a b 2w 3w 2w − 2 2 + (w) dy + 2 (w) , 2 2 2 y xy xy xy 0 0 0

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1 4w 1 4w w x y a2b2 l + l2 + 2 + l2 w̄ = , x = , y = , q = q, � = , 1 2 3 x4 2 2 3 x2y2 h a b Eh4 h2 a (17) 1 4w 1 3w � = . + + l2 + 2 1 + l2 3 b 1 2 3 y4 x 2 x x3 2 1 2 2w 1 3w Then Eq. (14) can be rewritten as follows (bars over the + 1 + l2 3 + 2 1 + l2 3 x2 2 x2 x2 y 2 y y3 dimensionless parameters are omitted) 2 2 2 2 2 2 4 4 4 1 1 2 3 w 2 1 2 3 w 1 w w w + + l + 2 + l D + 2 + 2 = q, y2 2 y2 y2 xy 2 xy xy 2 x4 x2y2 y4 (18) 3 3 2 1 3 w 2 1 3 w 2 + 2 + l2 + 2 + l2 D = 1 + x 2 x xy2 y 2 y x2y where: 12(1− 2) 2(1+ ) , - non-dimensional material 2w length parameter equal to 0 for the classical mechanics case, = q(x, y)+I . t2 and it is in the interval (0; 0.5] for higher order mechanics. (12) Boundary conditions (15), (16) should be attached to The corresponding choice of the boundary conditions is Eq. (18) in the same dimensionless form. The derived PDEs given with regard to the plate edge [0; b]: will be solved both analytically and numerically by several methods: the Kantorovich–Vlasov method, variational itera- b 2w b w tion method, Vaindiner’s method, Bubnov–Galerkin method, = 0 or = 0, 1 y2 y by the use of double trigonometric series, the fnite element 0 0 method and the fnite diference method. In order to keep 2 3 2 3 1 w w 2 w w + 1 − 2 + 2 accuracy of the solution, the Agranovskii–Baglai–Smirnov y y2 y3 x xy x2y (13) method will be used [118, 119]. b 2w Equation (18) can be presented in its counterpart opera- +2 2 = 0 xy 0 tor form b or (w) 0 = 0. L(w)=q, (19) For the case, when elasticity parameters are independent of where L is the operator (in general nonlinear) acting in a both coordinates and time, the modifed couple stress theory, Hilbert space H. Boundary conditions (14) and (15) have the Germain–Lagrange equations of the following form can the following operator representations be used l(w)=f . (20) 4w 4w 4w D + 2 + = q, x4 x2 y2 y4 (14) Remark 1 It should be noted that Eq. (12) with the corre- 3 2 D = Eh + Ehl sponding boundary conditions stands for the generalization where 12(1− 2) 2(1+ ) , and it will be further referred of the modifed Germain–Lagrange equation (MGLE). It to as the modifed Germain–Lagrange equation. allows one to take into account the heterogeneous structure In the future, we will use two types of boundary condi- of the nanoplates. The properties of nanoplates can be het- tions and their combinations: erogeneous prior to deformation and can be changed during 2w loading. This theory is based on the deformation theory of w = 0, = 0 (simple support), B 2 (15) n B plasticity and makes it possible to recalculate elastic moduli and Poisson’s ratios, depending on the deformation diagram and the loading process. w w B = 0, = 0 (clamping), (16) n B Remark 2 Equation (12) is used to consider physical non- where n stands for a normal to the plate boundary. linearity for nanostructures using the iterative scheme of For the numerical study, we use the non-dimensional the method of variable elastic parameters. The method of form of Eq. (14). We introduce the following non-dimen- variable parameters of elasticity [117], used to solve a physi- E (x, y, z, e ) sional variables cally nonlinear problem, employs the fact that i i , vi(x, y, z, ei) are not constant. They depend on the deformed state at a structure point and are defned by the following formulas

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

9K1iGi 1 3K1i − 2Gi Then the counterpart solution in a general form can be writ- Ei = , i = . 3K1i + Gi 2 3K1i + Gi ten as follows N N Here we consider that volumetric deformation 1 sin m x sin n y w = Bmn . K = const 4 2 2 2 (26) 1i . In the deformation theory of plasticity [120], D m=1 n=1 (m ) + (n ) the shear modulus is defned in the following form: For uniformly distributed constant load q = q0, coefcients 1 i(ei) Gi = Bmn take the following form 3 ei 16q0 Bmn = ; m = 1, 3, 5, … ; n = 1, 3, 5, … . (27) where i , ei stand for the plate stress intensity and plate mn 2 deformations, respectively. The relationship i(ei) is determined experimentally for a given nanomaterial. Therefore, we obtain N N 16q sin m x sin n y w = 0 , D 6 2 2 2 2 (28) 3 Analytical and Numerical Methods m=1 n=1 (m + n ) mn for Solving MGLE where 3.1 Exact Solution of MGLE Based on Navier’s a x y = , = , = . Double Trigonometric Series b a b (29)

For Eq. (18), convergence of the solution is studied depend- We consider frst a possibility to get analytical solution to ing on the number of terms of series N in (28), for the fol- Eq. (18) with boundary conditions (15). External transverse lowing fxed values = 0, = 0.3, = 0.5 (see Table 1). load q(x, y) is taken into account in the following form Analysis of the results given in Table 1 shows that for N N = 0 , fve members of the series should be considered, q(x, y)= Bmn sin m x sin n y, (21) whereas for ≠ 0 three members of the series are sufcient m=1 n=1 to get a reliable solution. In what follows, we will consider where this as an exact solution and compare it with the solutions obtained numerically using other methods. 1 1 Bmn = q(x, y) sin m x sin n y dxdy. (22) 0 0 3.2 Bubnov–Galerkin Method

Defection function w(x, y) is taken in the form of the double Further, we will formulate an application scheme of the Bub- trigonometric series nov–Galerkin method using the example of operator equa- N N tion (19) and taking into account boundary conditions (20). {� } ⊂ D(L) H w(x, y)= Amn sin m x sin n y, (23) Let an algebraic base j be chosen in (Hilbert m=1 n=1 space) and approximate solution (19) be sought in the following form and each term of solution (23) satisfes all boundary conditions (15). Equations (21), (23) and (18) allow us to derive the following algebraic equation Table 1 Quantifying reliable solutions of the nanoplate defection N N 2 2 2 versus series terms N for diferent parameter AmnD (m ) + (n ) − Bmn sin m x sin n y = 0. m=1 n=1 Values wN (0.5;0.5), = 0 wN (0.5;0.5), = 0.3wN (0.5;0.5), = 0.5 (24) N Taking into account the linear independence of the terms in 1 0.318 0.231 0.155 Eq. (24), we obtain 2 0.318 0.231 0.155 3 0.310 0.225 0.151 Bmn Amn = . 5 0.311 0.225 0.151 2 2 2 (25) D (m ) + (n ) 10 0.311 0.225 0.151

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N 3. j(x, y) , k(x, y) satisfy rigorously the main boundary w = w0(x, y)+ aj j(x, y), (30) conditions (and arbitrary initial conditions); j=1 4. j(x, y) and k(x, y) should represent the frst N elements of the complete system of functions; where: aj are the unknown numerical coefcients, j(x, y) are 5. j(x, y) and k(x, y) satisfy a completeness property in H. the known analytical functions (approximation), and w0(x, y) is the function satisfying boundary conditions (20). Substitution of (30) into (19) yields the following residual Now, let us illustrate the advantages of BGM. We illustrate the process of fnding solutions of the modifed Ger- R(a0, a1, a2, … aN , x, y)=L(w)−q main–Lagrange equation for boundary conditions (15). N Displacements w can be expanded by using the follow- (31) = L(w0)+ ajL( j)−q. ing series: j=1 N N w(x, y)= A (sin(i x) sin(j y)). In order to determine unknown coefcients aj,j = 1, … , N, ij (35) i=1 j=1 according to the Bubnov–Galerkin method [121, 122], we { }N require that the basis functions j k=1 were also orthogonal After applying the Bubnov–Galerkin method for to residual (31), i.e. so that the following condition q(x, y)=q = const, we fnd w(x, y) 2 2 i j N 64 2q sin sin 2 2 (36) = sin (ix) sin (jy). 4 2 2 2 i,j=1 D(sin (2i) − 2i)(sin (2j) − 2j) j + i Table 2 presents solutions of the defection in the center of (R, )= R dxdy w (0.5;0.5) k k (32) the plate m depending on the number of terms in the series (36) for three values of and fxed parameters q = 7 , = 1. is satisfed. Solutions, which coincide completely with the solution For the Bubnov–Galerkin method the unknown coef- obtained by Navier, are highlighted in Table 2. The given a cients j included in (31) must be determined from the solu- example illustrates the fact that inclusion of in the modifed tion of the following system of linear algebraic equations Germain–Lagrange equation improves the convergence of (R, k)=0, k = 1, 2, … , N (33) the Bubnov–Galerkin method. where R is the residual equation (19), and k are the same 3.3 Agranovskii–Baglai–Smirnov Method (ABSM) analytic functions (weights) that appear in (30). Since the linear problem is considered in (19), Eq. (33) can be written The idea is to construct an iterative procedure for solv- in its counterpart matrix form ing the original equation (19) with the right-hand side N as a residual from the solution at the previous step [118, 119]. We consider the algorithm for applying the aj(L( j)−q, k)=−L(w0, k). (34) j=1 Solving the system of algebraic equations by any method a , aimed at estimating j and substituting the obtained solution Table 2 Quantifcation of reliable solutions for nanoplate defection into (31), we obtain an approximate solution w. versus a series of N terms for diferent ( q = 7 , = 1) Applying the Bubnov–Galerkin method when choosing w (0.5;0.5), = 0 w (0.5;0.5), = 0.3w (0.5;0.5), = 0.5 (x, y) (x, y) Values N N N weights k and approximating functions j , the N following conditions are hypothesized: 1 0.318 0.231 0.155 1. j(x, y)∈H , k(x, y)∈H where H is the Hilbert space; 2 0.318 0.231 0.155 2. ∀ j, k function j(x, y) , k(x, y) are linearly independent 3 0.310 0.225 0.151 and continuous in space ; 5 0.311 0.225 0.151 10 0.311 0.225 0.151

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

2 Agranovskii–Baglai–Smirnov method (ABSM) using an Let G be a bounded space R with boundary . Consider example of the operator equation (19). the frst boundary problem for a quasi-linear elliptic equation A new diferential equation is constructed in the follow- type of order 2m of the following form ing way Lu ≡ (−1) D A (x, D u)=h(x)( ≤ m), (41) L(w1(x, y)) = q(x, y), (37) ≤m where we changed the right-hand side in Eq. (19); now it D u = 0, ≤ m − 1, plays the role of the residual of the solution to the original (42) n equation. This process is repeated times. Finally, the fol- =( + ) where: 1 2 - integer multi-index of diferentiation; lowing series is used as the input solution 1 2 = 1 + 2 ; D = D1 D2 ; Di = ∕ xi ; the same holds for n and . Function A (x, ) where =( 1, 2) is nonlinear w(x, y)= w (x, y). i (38) and depends on all possible indices at ≤ m , x =(x1.x2) . i=1 Finally, let h(x)∈L2(G) . Function u(x) is called the gener◦ - u(x)∈Wm (G) The proof of the convergence of the ABSM method was alized solution of problem (31)◦ and (32 ), if 2 v(x)∈Wm (G) given in references [118, 119]. In the following sections, the and if for any function 2 the following integral method of combinations of the Kantorovich–Vlasov method, identity holds the variational iteration method, and the Vaindiner method (A (x, D u), D v)=(h, v). will be considered. ≤m (43) 3.4 Solving MGLE by Kantorovich–Vlasov Method We introduce a few assumptions, which will be implemented in the next steps of the article. Owing to the Kantorovich–Vlasov method (KVM) [31, 32], a solution to Eq. (19) can be searched in the following form 1. The problem of solving Eq. (41) under condition (42) is N equivalent to solving the operator equation w (x, y)= X (x)Y (y). ◦ ◦ N j j (39) m → m j=1 u = Tu, T ∶ W2 (G) W2 (G), (44) Weight functions Xj(x) satisfy boundary conditions (15) and 2. Area G = E1 × E2 , where E1 and E2 are the bounded sets 1 (16). Functions Yj(y) are the searched functions defned by of spaceR . the following system of projected equations (we substitute 3. Each element of◦ the complete system of functions m (39) into (19) and employ the BGM with respect to co-ordi- i(x) of space W2 (E1 × E2) takes the form nate x): i(x1, x2)=i(x1)i(x2), x1 ∈ E1, x2 ∈ E2, (45) (Lj(w)−q), Xk(x) = 0, k = 1, 2, … , N. (40) and moreover,◦ the system of functions◦ i(x2) is complete Wm (E ) (x ) Wm (E ) Procedure (40) reduces the problem of PDEs to ODEs by in space 2 2 , and i 1 in 2 1 . (In the future, we employing the Bubnov–Galerkin method regarding one of will employ the following lemma). the coordinates. Solving the system of ODEs with the cor- Lemma◦ 1 If sequence vn(x1, x2) of the elements of◦ space responding boundary conditions, we obtain a set of functions m ∗ m W (G) converges in norm to a certain element v ∈ W (G) , Yj(y) , and next substituting them into (39), we obtain the 2 2 i.e. desired solution. 2 m k(v − v∗) ∗ 2 n → → 3.5 Proof of the Convergence of KVM for Some Class vn − v ◦ = dx1dx2 0, n ∞, Wm k1 x k2 x of Problems 2 k=0 (k) 1 2 E1×E2 (46) In this section we will prove that the KVM method for a and for ∀n , function vn can be taken as certain class of problems (including geometrically nonlinear problems of the theory of plates and shells [39]) coincides with the projection methods. A distinctive feature of the method relies on the way of constructing an extremely dense system of subspaces [123].

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N n vn(x1, x2)= j (xx)gi(x2), 2 1 F(x1, x2) D gl(x2)dx2 D (x1)dx1 = 0, j=1 2 1 m � � ≤ E E � 1 ⎡ 2 ⎤ ⎢ ⎥ where gi(x2) are the given elements of the complete system � � N ⎢ ⎥ of functions i(x2) , and N is the fxed number, then func- F(x1, x2)=⎣ A x, D j(x1)gj(x⎦2) − h(x1, x2), ∗ tion v (x1, x2) can also be represented in a similar way, i.e. � j=1 � ◦ � N m ∀(x1)∈W2 (E1), l = 1, … , N. v∗(x , x )= n(x )g (x ). 1 2 j x i 2 (47) (51) j=1 The formulated method with the defned assumptions In order to prove the Lemma, it sufces to consider only satisfes the following convergence theorem. one term in expression (47). Indeed, Theorem 1 Let T be a completely continuous◦ operator on a ∗ 2 → → m ∫ vn − v dx1dx2 0, n ∞, or, according to Fubi- certain non-empty open set �⊂W2 (E1 × E2) , and let Eq. E ×E 1 2 (44) have an isolated solution u0 ∈ with non-zero index, ni’s theorem, we have N N N then there is 0 and 0 such that when ≥ 0 , the system of equations has in the area u − u0 ◦ ≤ 0 at least one n (51) Wm F dx → 0, n → ∞, 2 1 → (48) solution uN and all solutions uN when N ∞ tend to u0 by E1 the norm. n ∗ 2 where 0 ≤ F (x1)=∫ vn − v dx2. E2 Indeed, we see that the system of equations (51) is equiv- Fnk (x ) It follows from (48 ) that there is a sequence 1 alent to the system obtained from projection of Eq. (44) onto such that the set 2 N N nk (x )g (x )−v∗(x , x ) dx → 0, k → ∞, j 1 j 2 1 2 2 (49) MN = v(x1, x2) v(x1, x2)= j(x1)gj(x2) , (52) j=1 E j=1 2 x gj(x2) N ◦ i(x2) for almost all 1. where: - fxed elements from the system ; ∀j (x ) Wm (E ) N Formula (49) shows that a certain sequence of elements j 1 go over the whole space 2 1 , and is the given from a fnite-dimensional subspace of dimension◦ N (which number. m linearly spans N given elements gj(x2)∈W2 (E2) ) converges It is enough to observe that we can substitute ∗ v → x g x x for almost all x1 to the element v (x1, x2) . Since every fnite- ( 1) l( 2) ◦to identity (43), where ( 1) is the arbitrary ∗ Wm E M dimensional subspace is closed, it is obvious that v (x1, x2) function from 2 ( 1) . Obviously, set N is linear and, as has the representation (47). The Lemma is proved. ▪ follows from the Lemma,◦ it is a closed set. Thus, MN is the Wm E E Now let us pay attention to the proof of convergence of subspace of space 2 ( 1 × 2) (in general, of infinite the Kantorovich–Vlasov method. dimension). Taking into account the latter statement and (51), we see that Eq. (44) is projected onto subspace MN Defnition 1 We will say that an approximate solution to orthogonally, and therefore projection operator PN is problem (41) and (42) is sought by the Kantorovich–Vlasov bounded. Further, from the representation◦ of the elements x , x Wm E E method, if the following assumptions hold: of the basic system i( 1 2) of space 2 ( 1 × 2) in the M form (45) it follows that the sequence◦ of subspaces N is Wm (E × E ) (1) the solution to problem (41) and (42) is sought in the extremely◦ dense in 2 1 2 , i.e. m ∀y ∈ W P y − y ◦ → 0, N → ∞ form 2 N m . W2 N The theorem is proved. This proof of the theorem is simi- uN (x1, x2)= j(x1)gj(x2), (50) lar to the proof of the convergence of the Galerkin method j=1 ◦ for nonlinear problems (see [119]). m Comment The obtained results can be easily extended to where: ∀j j(x1)∈W2 (E1), gj(x2) N - known elements ft the nanoplate behavior, and the authors refer to the modi- from the complete system of functions i(x2) , N - fed Germain–Lagrange equation. fxed number; (2) N unknown functions j(x1) are estimated from the following system of equations:

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

3.5.1 Analytical Solution of MGLE by KVM in the First Table 4 Computation of the nanoplate centre defection for various Approximation methods and diference length scale parameter

w(0.5; 0.5) wAg(0.5;0.5) w(0.5; 0.5) The approximate solution w (x, y) can be written in the form (length Analytical solu- Numerical solu- Exact Navier’s w(x, y)=X (x)Y (y). parameter of tion tion 0 1 (53) material) As an initial approximation for boundary conditions (15), KVM KVM + ABSM solution X (x)=sin( x) Q = q we take function 0 and use notation D . 0 0.313 0.308 0.311 Substituting (53) into (18), we obtain 0.3 0.228 0.228 0.225 1 0.5 0.153 0.153 0.151 X (4)Y + 2X(2)Y (2) + 2X Y(4) = Q, 2 0 1 0 1 0 1 (54)

4 X (4) = d X0(x) where 0 dx4 . S1y S1y −S1y −S1y Using the Bubnov–Galerkin method, we get Y1(y)=c1e + c2ye + c3e + c4ye + k1, (58) 1 where 1 (4) (2) (2) 2 (4) X0 Y1 + 2X Y1 + X0Y − Q X0dx = 0, 2 0 1 −2S1 −S1 S1 −S1 S1 k1(2e − 2e − 2 − 2S1 + 2e + S1e + S1e ) 0 c1 =− , e−2S1 e2S1 (55) 2 + 2 − 4 −S1 k1(S1 − S1e ) and we obtain the following diferential equation: c2 = , S1 = , 2eS1 − 2e−S1 (4) (2) k (2 − 2S e−S1 − 2e2S1 − 2S + 2eS1 + S e−S1 + S eS1 ) A1Y1 + B1Y1 + C1Y1 = D1, (56) 1 1 1 1 1 c3 =− , 2e−2S1 + 2e2S1 − 4 where k (S − S eS1 ) D c = 1 1 1 , k = 1 . 4 S −S 1 1 1 2e 1 − 2e 1 C1 2 A = 2 X2dx = , B = 2X(2)X dx =−2, (59) 1 0 2 1 0 1 0 0 Finally, substituting (58) into (53) we find the desired 1 1 solution 4 1 (4) 2 C = X X dx = , D = Q X dx = Q . w(x, y)=sin ( x) c eS1y + c yeS1y + c e−S1y + c ye−S1y + k . 1 2 0 0 22 1 0 1 2 3 4 1 0 0 (60) (57) Equation (56) can be solved numerically, for example, by the 2 The following function is used to solve Eq. (56): fnite diference method (FDM) with approximation O(h ) . A very important point plays a number of partitions of variable y. Results of the study on convergence of the solution Table 3 Numerical solution of n Number of parti- wn(0.5;0.5) depending on the number of partitions , are reported in Eq. (25) using FDM tion points n Table 3 for = 0. On the other hand, Table 4 shows the results achieved 5 0.296 using the Kantorovich–Vlasov method (KVM) in the frst 9 0.310 approximation, obtained analytically (54) and numerically 15 0.313 ( wAg ) applying the Agranovskii–Baglai–Smirnov method 19 0.313 (ABSM). The analytical solution of MGLE by the Kan- 25 0.313 torovich–Vlasov method in the frst approximation taking into account the Agranovskii–Baglai–Smirnov approach is difcult to fnd. The numerical solution obtained by match- ing KVM + ABSM is presented in Table 4. Let us consider the solution of Eq. (18) found via the Kantorovich–Vlasov method in the frst approximation for the clamped boundary conditions (16). As an initial approximation, we take function 2 X1(x)=sin ( x) . Similarly, for the boundary condition (15) we obtain the following solution to the original equation:

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Table 5 Nanoplate center c + + k = 0, Number of points n wn(0.5; 0.5) 1 3 1 defection obtained using the −N1 N1 k1 + e (c3 cos(E1)+c4sin(E1)) + e (c1 cos(E1) FDM for diferent number of 5 0.337 partitions n ( = 0) + c sin(E )) = 0, 9 0.308 2 1 (62) 15 0.298 E1c2 + E14 + N1c1 − N1c3 = 0, N1 N1 19 0.296 e (c2E1 cos(E1)−c1E1sin(E1)) + e (c4E1 cos(E1) 25 0.294 − c3E1sin(E1)) = 0. 35 0.293 55 0.292 The equations are solved using the finite difference 75 0.292 method. Table 5 shows the defection values depending on the number of partitions for = 0 . The results reported in Table 5 show that n = 55 partitions are required to achieve Table 6 convergence of the results. However, for boundary condi- Nanoplate center defection using the analytical (KVM) and n = 15 numerical (KVM+ABSM) solutions for diferent tion (15) partitions are sufcient. The increase in the number of partitions is directly related to the edge efect w w (0.5;0.5) (0.5; 0.5) Ag caused by boundary condition (16). (parameter length of Analytical solution Numerical solution Table 6 gives analytical results obtained by the KVM material) method in the frst approximation and numerical results for KVM KVM+ABSM ( wAg ) taking into account the ABSM method for boundary 0 0.292 0.290 condition (16). 0.3 0.212 0.210 3.5.2 Analytical Solution of MGLE by KVM in the Second 0.5 0.142 0.142 Approximation

Now, we consider a similar approach but for the second approximation of the KVM. Defection function is represented by w(x, y)=sin( x)Y1(y)+sin(3 x)Y2(y) . After applying the Bubnov–Galerkin method, the following sys- w(x, y)=sin2( x) eN1y c cos E y + c sin E y + 1 1 2 1 tem of diferential equations is obtained −N1y (61) +e c1 cos E1y + c2 sin E1y + k1 , 1 1 (4) (2) (2) 2 (2) (4) where X1 Y1 + 2X Y1 + X Y − Q X1dx = 0, 2 1 1 1 0 K1,2 = N1 ± E1i, K3,4 =−N1 ± E1i, 1 + 4 N = t cos , 1 (4) (2) (2) 2 (2) (4) 1 2 X Y + 2X Y + X Y − Q X dx = 0, 2 2 2 2 2 2 2 2 √ � � 2 0 � � + 4 −B − B1 − 4A1C1 E = t sin , t = 1 + i, (63) 1 1 � 2 2A1 � 2A1 � √ � � or equivalently 2 � � − B − 4A C (4) (2) 1 1 1 −B1 D1 = arctan ∕ , k = . A1Y1 + B1Y1 + C1Y1 = Ded1, 1 � 1 � 2A1 � 2A1 C1 (4) (2) (64) ⎛⎛ ⎞ � � ⎞ A Y + B Y + C Y = D , ⎜⎜ ⎟ ⎟ 2 2 2 2 2 2 2 ⎜⎜ ⎟ ⎟ ⎜⎜ ⎟ ⎟ A , C , B , D , A , C , B , D Coefcients⎝ ⎝c1, c2, c3, c4 are determined⎠ from⎠ the boundary where coefficients 1 1 1 1 2 2 2 2 are conditions as the solution of the following system of alge- defned in the following way: braic equations

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

1 D1 D2 2 S = , S = 3 , k = , k = , 2 1 2 1 2 A = (X + X )X dx = , C1 C2 1 1 2 1 2 −2S1 −S1 S1 −S1 S1 0 k1(2e − 2e − 2 − 2S1 + 2e + S1e + S1e ) c1 =− , 1 2e−2S1 + 2e2S1 − 4 (2) (2) −S1 B = 2 X + X X dx =−2, k1(S1 − S1e ) 1 1 2 1 c2 = , 2eS1 − 2e−S1 0 −S1 2S1 S1 −S1 S1 1 k1(2 − 2S1e − 2e − 2S1 + 2e + S1e + S1e ) 4 c3 =− , 1 2e−2S1 + 2e2S1 − 4 C = X(4) + X(4) X dx = , 1 2 1 2 1 2 S1 2 k1(S1 − S1e ) 0 c4 = , 2eS1 − 2e−S1 1 k (2e−2S1 − 2e−S2 − 2 − 2S + 2eS2 + S e−S2 + S eS2 ) 2 c =− 2 2 2 2 , D = Q X dx = Q . 5 −2S 2S 1 1 2e 2 + 2e 2 − 4 −S2 0 k2(S2 − S2e ) 1 (65) c6 = , 2eS2 − 2e−S2 2 A = 2 (X + X )X dx = , k (2 − 2S e−S2 − 2e2S2 − 2S + 2eS2 + S e−S2 + S eS2 ) 2 1 2 2 c =− 2 2 2 2 2 , 2 7 −2S 2S 0 2e 2 + 2e 2 − 4 S 1 k (S − S e 2 ) c = 2 2 2 . 8 S S B = 2 X(2) + X(2) X dx =−92, 2e 2 − 2e− 2 2 1 2 2 0 Table 7 shows the results obtained using the Kan- 1 w 4 torovich–Vlasov method in the second approximation 2m 1 (4) (4) C = X + X X dx = , for boundary conditions (15), while w1m is used in the frst 2 2 1 2 2 22 0 approximation. 1 Analysis of the results given in Table 7 leads to the con- 2 D = Q X dx = Q . clusion that the analytical solution of the frst approximation 2 2 according to the Kantorovich–Vlasov method gives results 0 close to those obtained by Navier, the diference is 0.9% = 0 = 0.3 = 0.5 Solving the system of equations (64) we get for , and 1.3% for and . Obtaining an analytical solution using the Agranovskii–Baglai–Smirnov S1y S1y −S1y −S1y w(x, y)=sin ( x) c1e + c2ye + c3e + c4ye + k1 method matched with the Kantorovich–Vlasov method is S2y S2y −S2y −S2y + sin (3 x) c5e + c6ye + c7e + c8ye + k2 , problematic due to the large amount of calculations. Com- parison with the results obtained by the above combinations (66) of methods in comparison with the Navier method gives where

Table 7 Comparison of the nanoplate centre defection using the analytical solutions of the KVM (the frst and second approximation) and the numerical solution of the KVM+ABSM versus the Navier solution for diferent

w1(0.5;0.5) w(0.5;0.5) w2(0.5;0.5) w(0.5; 0.5) (parameter length of Analytical solution of KVM in Numerical solution in the frst Analytical solution of KVM in the Exact material) the frst approximation approximation KVM + ABSM second approximation Navier’s solution

0 0.314 0.308 0.310 0.311 0.3 0.228 0.223 0.227 0.225 0.5 0.153 0.150 0.152 0.151

1 3 J. Awrejcewicz et al. the same result, and the comparison of results obtained by 3.7 Proof of the Convergence of VIM While Solving the Kantorovich–Vlasov method in the second approxima- the Operator Equation tion with the Navier results yields the following errors: 0.3% for = 0 , and 0.7% for = 0.3 and = 0.5 . Though the Let us present a proof of the convergence of the variational Kantorovich–Vlasov method in the second approximation iteration method (VIM) for the operator equation reported slightly improves the solution, it does not coincide with the in [38]. exact one. Apparently, the reason for this lies in the idea of The scheme of VIM can be formally described in the fol- separating variables. lowing way. It is necessary to fnd a solution to the equation

3.6 Variational Iteration Method (VIM) L w(x, y) = q(x, y), x, y ∈ (x, y), (71) where L[w] is the operator defned on the manifold D(L) of This method presents a modification of the Kan- the Hilbert space L2( ) ; q(x, y) stands for a given function torovich–Vlasov method (KVM). Owing to VIM, an n term of two variables x and y; w (x, y) is the searched function of trial solution for a two-dimensional problem is taken to have the same variables; and (x, y) is the space associated with the following form variations of x and y. N If (x, y)=T × P (T - a certain bounded set of variables w (x, y)= X(0)(x)Y (y). N j j (67) x, P bounded set of y), then the solution of Eq. (71) takes j=1 the form (0) Here, Xj (x) is a priori specifed, and in general may not N (1) satisfy the given boundary conditions Yj (y) . We employ wN (x, y)= Xi(x)Yi(y), (72) i=1 the Bubnov–Galerkin method with regard to coordinate x and we get a set of n ordinary diferential equations (ODEs) where functions Xi(x) and Yi(y) are defned by the system y with regard to coordinate . Its solution with an account of of equations the boundary conditions yields the system of functions (1) where Yj (y) is substituted into (67): L w − q X (x)dx = 0, L w − q Y (y)dy = 0, N 1 N 1 N T P w(1) x y X(0) x Y(1) y ⋮⋮ N ( , )= j ( ) j( ). (68) j=1 L w − q X (x)dx = 0, L w − q Y (y)dy = 0, N N N N Therefore, the solutions by the Kantorovich–Vlasov method T P coincides exactly with the frst iteration in the VIM. Further, (73) (1) we use Yj (y) obtained through KVM as the test functions in the following way. A certain system composed of N Xj(x) are unknown to be re-determined after applying the functions with respect to one of the variables, for instance, y 0 0 0 Bubnov–Galerkin method and with regard to we obtain X1 (x), X2 (x), … , XN (x) is given. Then, the frst N equa- a set of n ordinary diferential equations (ODEs) with the tions of system (73) is determined by the system of N x X (1) 1 1 1 respect to . They yield a set of functions j , which allows functions Y1 (x), Y2 (x), … , YN (x) . Next, the obtained us to present the solution in the following form functions are employed to create a new set of functions x, 2 2 2 N i.e. X1 (x), X2 (x), … , XN (x) , which is further used to con- (1) (2) y wN (x, y) = X (x)Y (y). struct a set of new functions with respect to variable , i.e. j j (69) 3 3 3 j=1 Y (x), Y (x), … , Y (x) , and so on. 1 2 N m The so far described procedure can be repeated -times until Defnition 2 The process of calculating a given system of the result converges to a desired degree: functions by another one will be called the VIM step. The N number of steps taken to determine any set of functions cor- w(m) x y X (m−1) x Y(m) y m n N ( , )= j ( ) j ( ), = 1, 2 … . (70) responds to the superscript (number) of functions from the j=1 set under consideration. Breaking the process of fnding functions Xi(x) and Yi(y) This method eventually removes all assumptions on both sets k X x Y y on the -th step (which, for example, corresponds to getting of j( ) and j( ) , but the accuracy of the obtained approxi- (Yk(y), Yk(y),… , Yk (y)) a set of functions 1 N 2 N , allows us to con- mate solution depends on the number of trial solution terms. k k−1 k struct function wN = Xi (x)Yi (y), which is taken as an i=1 approximate solution to∑ Eq. (71) obtained by the VIM.

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

0 1 Remark 1 Here and further on, by operator L[w] we mean Consequently, element X (x)Y (y) , which is a result of a certain diferential operator, which is defned on set D(L) the frst step of the variational iteration method, stands as L ( ) 1 of Hilbert space 2 . Thus, at each step, system (73) will the projection of element 0 onto subspace M1 . Therefore, disintegrate into the corresponding system of ordinary dif- the inequality ferential equations, with resolved the solvability problem. w1(x, y) − w ≤ X0(x)Y(y) − w 1 0HL 0HL (77) Remark 2 Function wN (x, y) is called the N-th approximation X0(x)Y(y)∈M1 of the solution for Eq. (71), if the number of terms of the for elements 1 , holds. By a simi- N lar construction, we obtain the inequality for sub- series in representation (70) is equal to . 1 0 1 space M2 ={w(x, y) w(x, y)=X (x)Y (y);X(x)∈ H (T), Y1(y)∈H (P)} Consider the case of the first approximation, i.e. L L which corresponds to the second the solution of Eq. (73) will be sought in the form step of the VIM, i.e. w (x, y) = X(x)Y(y), X(x) Y(y) 1 where functions and are X2(x)Y1(y) − w ≤ X(x)Y1(y) − w 0HL 0HL (78) determined in the above described way from the system of equations 1 1 for arbitrary elements X(x)Y (y) ∈ M2 . From (77) and (79) it X2(x)Y1(y) − w X0(x)Y1(y) − w follows that 0 HT ≤ 0 HL . L X(x) ⋅ Y(y) − q X(x)dx = 0, Arguments similar to those given above are valid for the X functions obtained at the k-step of the VIM. Finally, the (74) L X(x) ⋅ Y(y) − q X(y)dy = 0. mathematical induction method proves the theorem and at the same time inequality (75). ◻ Y Proof of the Theorem (the case of higher approximations) Let operator L in (71) be positive defnite. Let us introduce The the notation: HL(T × P) is the energy space of operator L result of Theorem 1 is extended to the case of higher approx- [123], [∙, ∙] is the scalar product of elements in HL; and w0 is imations. Namely, inequality (75) takes the following form the exact solution to Eq. (71). wm(x, y) − w ≤ wn (x, y) − w , m ≥ n. N 0HL N 0HL

Theorem 2 If L is a positive definite operator with its Further investigation requires proof of the following D L ⊂ H action space ( ) L,then the sequence of elements of lemma. ◻ = wk(x, y) − w is monotonically decreasing, that is, for k 1 0HL arbitrary i and j and if i ≥ j , the following inequality holds Lemma 2 Let each element of the basis system of the space H satisfy the following conditions wi − w wj − w . L 1 0 HT ≤ 1 0 HT (75) i i(x, y) = i(x)i(y), ∀ i(x) ∈ HL(T), i(y) ∈ HL(P). 1 Proof of Theorem 2 Consider subset M1 of space HL , which If the components of some basis function i serve as the ini- 0 takes the form tial approximation of VIM, i.e. X (x) ≡ i(x) , then for any

1 0 0 M1 = w(x, y) w(x, y) = X (x)Y(y), X (x) ∈ HL(T), Y(y) ∈ HL(P) . 1 Obviously, set M1 is the subspace of space HL(T × P) (in the number k of steps taken by VIM, the following inequality holds wk(x, y) − w ≤ c (x) (y) − w where c is the general case the problem has infnite dimension). Accord- 1 0HL i i 0HL, ingly, it is possible to defne the projection of element w0 arbitrary real number. 1 onto space M1 . It is known that to satisfy the condition for 0 ∗ 1 1 0 1 1 element X (x)Y (y) ∈ M1 as the projection of w0 onto M1 it Proof of the Lemma Since X (x)Y (y) ≡ i(x)Y (y) , then by is required that Theorem 2 we have

0 ∗ 0 k 1 X (x)Y (y) − w0, X (x)Y(y) = 0, w (x, y) − w0H ≤ i(x)Y (y) − w0H ≤ c i(x)i(y) − w0H . HL (76) 1 L L L

1 for arbitrary elements X0(x)Y(y) ∈ M1 . Hence, if On the basis of the above Lemma, one criterion for VIM 0 ∗ 1 H X (x)Y (y) ∈ M1 , then Eq. (76) coincides with the frst equa- convergence is formulated.0 First, we identify the space L Wm ( ) tion of system (64). with the space 2 , which is obtained by enclosing the norm

1 3 J. Awrejcewicz et al.

2 Eq. (71) not worse than using the Ritz method in the cor- m kw responding subspace. w Wm = dxdy 2 k1 xk2 y VIM can be extended to cover more variables. For exam- ⎧ k=0 (k) ⎫ � � � � ple, if the desired solution to Eq. (71) is a function of three ⎪ � � ⎪ ‖ ‖ ⎨ � � ⎬ 0 variables x, y, z , then an approximate solution by VIM will � � ∞ of the set ⎪of infnitely diferentiable functions⎪ C ( ) with be searched for in the form Q(x, y, z) = X(x)Y(y)Z(z) , and the ⎩ ⎭ ◻ compact carrier in . VIM procedure can be applied to each variable.

Theorem 30 Let each element of the basis system of the space Remark 4 When implementing VIM, it is not necessary to m W2 (T × P) take the following form construct an initial approximation that satisfes the boundary conditions of the problem. Suppose that operator L[w] i(x, y) = i(x)i(y), (79) 0 defnes a boundary problem. As an initial approximation m where i(x) is the basis0 system in space W2 (T) , simi- we take any function from the domain of defnition of the larly (y) in space Wm (P) and to obtain an arbitrary diferential operator of this problem. Then, in the frst step, i 2 N-th approximation by VIM, the base elements of the system we obtain a system of functions satisfying the boundary con- are taken i(x, y) as the frst approximation. Then, for a ditions in one of the variables, and in the second step - in large value of N VIM gives a unique approximate solution both variables. , N and0 the sequence wN converges by the norm of the space Wm (T × P) to the exact solution w regardless of the Remark 5 These results can be extended to the equations of 2 0 number of steps k, which can be performed for every N-th the nanoplate, which the authors called the modifed Ger- approximation, i.e. main–Lagrange equation.

k w − w 0 → 0, N → ∞. 3.8 Variational Iteration Method for Solving MGLE N 0 m W2 Our goal is to apply the variational iteration method to solve Proof of Theorem 3 If the theorem is proved for the approximations obtained in the frst step of VIM, then based on the the modifed Germain–Lagrange equation with boundary results of the Lemma it will be valid for any chosen k-th conditions (15). step. Therefore, we consider the N-th approximation of the The Kantorovich–Vlasov method in the solution to problem (71), which was obtained in the frst first approximation yielded the solution Y (y)=c eS1y + c yeS1y + c e−S1y + c ye−S1y + k step. Similarly to the statement used in Theorem 2, it can be 1 1 2 3 4 1 to Eq. (56). 1 As mentioned above, further the approximate solution can shown that each wN is the projection of the element w0 into the subspace serve as an initial approximation to obtain the solution with respect to x. The obtained Y1(y) as the specifed functions N and X2(x) as unknown are to be re-determined after applying M1 = w(x, y) w(x, y) = X0(x)Y1(y) , N the Bubnov–Galerkin method. For this, we follow a similar i=1 procedure and, as a result, we fnd the function 0 where Xi (x) consists of N fxed elements from the system 0 i(x) for arbitrary i, Yi(y) takes all values from space m 1 W2 (P) . Thus wN = N w0 , where N is the operator of M1 an orthogonal projection on the subspace N , which is Table 8 Estimation of defection of the nanoplate centre using VIM bounded. Since the elements of the basis system0 i(x, y) have and ABSM versus the exact solutions for diferent m the form (73), it is extremely dense in W2 (T × P) [123]. w(0.5; 0.5) wAg(0.5;0.5) w(0.5; 0.5) The proof can be completed similarly to the proof of Theo- ◻ (parameter length VIM ABSM Exact solution rem 16.2 [123]. of material)

Remark 3 The results of Theorem 1 and the Lemma show 0 0.307 0.308 0.311 that using VIM, one can obtain an approximate solution to 0.3 0.223 0.224 0.225 0.5 0.150 0.150 0.151

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

S2x S2x −S2x −S2x The solution for boundary conditions (16) will be the X2(x)=c5e + c6xe + c7e + c8xe + k2. (80) same as for boundary conditions (15). Following the above Then, using the variational iteration procedure, we fnd func- reasoning, we obtain function Yi, Xi in the following form: tions Xi, Yi that allow us to obtain a solution with a given wi(x, y) accuracy . The solution takes the following form: Nix = e c4i+1 cos Eix + c4i+2 sin Eix Six Six −Six wi(x, y)= c4i+1e + c4i+2xe + c4i+3e −Nix + e c4i+1 cos Eix + c4i+2 sin Eix + ki (83) −Six +c4i+4xe + ki × Niy × e c4i+1 cos Eiy + c4i+2 sin Eiy Siy Siy −Siy (81) × c4i+1e + c4i+2xe + c4i+3e −Niy +e c4i+1 cos Eiy + c4i+2 sin Eiy + ki , −Siy +c4i+4xe + ki , where where c4i+1 + 4i+3 + k1 = 0,

−N1 Bi Di k1 + e (c4i+3 cos(E1)+c4i+4sin(E1)) Si = , ki = , 2C B N1 i i + e (c4i+1 cos(E1)+c4i+2sin(E1)) = 0, −2Si −Si Si −Si Si (84) ki(2e − 2e − 2 − 2Si + 2e + S2e + Sie ) E c + E + N c − N c = 0, c =− , 1 4i+2 14i+4 1 4i+1 1 4i+3 4i+1 −2S 2S 2e 2 + 2e 2 − 4 N1 e (c4i+2E1 cos(E1)−c4i+1E1sin(E1)) k (S − S e−S2 ) i 2 2 N1 c4i+2 = , + e (c4i+4E1 cos(E1)−c4i+3E1sin(E1)) = 0. 2eS2 − 2e−S2 −S2 2S2 S2 −S2 S2 ki(2 − 2S2e − 2e − 2S2 + 2e + S2e + S2e ) w c4i+3 =− , Table 9 shows the values of displacement obtained ana- 2e−2S2 + 2e2S2 − 4 lytically and displacement wAg obtained using the Agranovs- k (S − S eS2 ) c = i 2 2 . kii–Baglai–Smirnov method, for boundary conditions (16). 4i+4 S −S 2e 2 − 2e 2 A few comments on the application of the variational (82) iteration method in the analytical and numerical solution Next, the Agranovskii–Baglai–Smirnov procedure is used using the Agranovskii–Baglai–Smirnov method in the in combination with the variational iteration method in the frst approximation are given below. The numerical results frst term approximation. It should be mentioned that the obtained by the Kantorovich–Vlasov method in the frst and analytical solution of the problem is not an easy task. This second term approximations, by the Kantorovich–Vlasov is why in this case, we use fnite diference method of the method using the variational iteration method and by the second-order accuracy to solve the obtained ODEs. The con- Agranovskii–Baglai–Smirnov method give practically the vergence of the solution is investigated as a function of the same results. The use of the variational iteration method number of partition points along the appropriate coordinate. ofers the following advantage: the system of approximat- It was found that the optimal partitioning in spatial coordi- ing functions does not need to be known, while in the Bub- nates for a hinged support is 15 points, and for clamping it nov–Galerkin method two coordinates should be defned, is 55 points. The same partition is equal for all values of the whereas in the Kantorovich–Vlasov method it is required to dimensionless parameter length of material . know one coordinate. The system of approximating func- Table 8 shows results of the analytical procedure using tions is obtained from the solutions of ODEs. The variational the variational iteration method with respect to the frst term iteration procedure converges even when the set of approxi- approximation for boundary conditions (15), and numeri- mating functions does not satisfy boundary conditions of cal results for wAg using the Agranovskii–Baglai–Smirnov the problem. method versus the exact solution. 3.9 Vaindiner’s Method

Vaindiner’s method (VaM) can be considered as an extension Table 9 Analytical versus numerical (MABS) solutions for diferent of the Kantorovich–Vlasov method. The proof of the con- w w Ag vergence of this method is given in [103] for a linear inho- (parameter length of mate- Analytical solution Numeri- mogeneous second order equation with variable coefcients. rial) cal solution (ABSM) Consider the application of this method to the operator equation (19). This equation can be both linear and non- 0 0.292 0.292 linear and up to the eighth order. In our case, operator equa- 0.3 0.212 0.212 tion (19) is an inhomogeneous fourth-order equation, i.e. 0.5 0.142 0.142

1 3 J. Awrejcewicz et al. the modifed Germain–Lagrange equation. An n term trial 1 1 solution is assumed in the following form A = 2 X2dx = , 11 1 2 N 0 1 wij(x, y)= X1j(x)Y1j(y)+X2j(x)Y2j(y), i = 1, 2, (85) j=1 B = 2X(2)X dx =− 22, 11 1 1 where weight functions Y1j(y) and X2j(x) , and functions X1j(x) 0 1 and Y2j(y) are unknown. They are determined using the Bub- 1 44 nov–Galerkin procedure, i.e. we have C = X(4)X dx = , 11 2 1 1 2 0 L(wij)−q, Y1k(y) = 0, k = 1, 2, … , N, i = 1, 2, (86) 1 L(w )−q, X (x) = 0, k = 1, 2, … , N, i = 1, 2, 2 ij 2k D = Q X dx = Q , 11 1 1 1 In other words, a system of 2N ordinary diferential equa- 0 X (x) 1 (88) tions is obtained. This allows us to fnd the 1j N and 4 1 2 Y2j(y) functions, and then substituting them in (85) we A = Y dy = , N 12 2 1 2 get a trial solution of the problem. 0 1 3.9.1 Analytical Solution of MGLE by Vaindiner’s Method B = 2Y(2)Y dy =− 22, 12 1 1 To solve (18) with boundary conditions (15) 0 1 using VaM, a trial solution takes the form 4 w(x, y)=sin( x)Y (y)+X (x) sin( y) C = 2 Y(4)Y dy = , 11 12 . Applying the Bub- 12 1 1 2 nov–Galerkin procedure, the following system of diferential 0 equations is obtained 1 2 (4) (2) D12 = Q2 Y1dy = Q2 . A11Y11 + B11Y11 + C11Y11 = D11, (4) (2) (87) 0 A12X12 + B12X12 + C12X12 = D12, Solving the system of equations (87), we get where coefcients A11 , B11 , C11, D11, A12 , B12 , C12, D12 S1y S1y −S1y −S1y are as follows Y11(y)=c11e + c21ye + c31e + c41ye + k11, S1x S1x −S1x −S1x X12(x)=c12e + c22xe + c32e + c42xe + k12, (89) where c11 2k e−2S1 − 2k e−S1 − 2k − 2S k + 2k eS1 + S k e−S1 + S k eS1 =− 11 11 11 1 11 11 1 11 1 11 , 2e−2S1 + 2e2S1 − 4 −S1 S1 −S1 S1k11 − S1k11e S1k11 − S1k11e S1k12 − S1k12e c21 = , c41 = , c22 = , 2eS1 − 2e−S1 2eS1 − 2e−S1 2eS1 − 2e−S1 c31 2k − 2S k e−S1 − 2k e2S1 − 2S k + 2k eS1 + S k e−S1 + S k eS1 =− 11 1 11 11 1 11 11 1 11 1 11 , 2e−2S1 + 2e2S1 − 4 (90) c12 2k e−2S1 − 2k e−S1 − 2k − 2S k + 2k eS1 + S k e−S1 + S k eS1 =− 12 12 12 1 12 12 1 12 1 12 , 2e−2S1 + 2e2S1 − 4 c32 2k − 2S k e−S1 − 2k e2S1 − 2S k + 2k eS1 + S k e−S1 + S k eS1 =− 12 1 12 12 1 12 12 1 12 1 12 , 2e−2S1 + 2e2S1 − 4 S k − S k eS1 D D c = 1 12 1 12 , k = 11 , k = 12 , S = . 42 S −S 11 12 1 2e 1 − 2e 1 C11 C12

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

By substituting functions Y11(y) and X12(x) into the expres- We obtain the functions sion for displacement w(x, y), we obtain the desired solution. X (x)=c eS21x + c xeS21x + c e−S21x + c xe−S21y + k , This iterative procedure can be repeated. 21 51 61 71 81 21 S22y S22y −S22y −S22y If function Y11(y), X12(x) is known, we can fnd function Y22(y)=c52e + c62ye + c72e + c82ye + k22. X12(x), Y22(y) . Then (92) 1 1 1 A = X2dx , B = 2X(2)X dx , 21 2 1 21 1 1 0 0 1 1 C = 2 X(4)X dx , D = Q X dx , 21 1 1 21 1 1 0 0 1 1 A = 2 Y2dy , B = 2Y(2)Y dy , 22 1 22 1 1 0 0 1 1 1 C = Y(4)Y dy , D = Q Y dy, 22 2 1 1 22 2 1 0 0

B21 B22 D21 D22 S21 = , S22 = , k21 = , k22 = , 2C21 2C22 B21 B22 c51 2k e−2S21 − 2k e−S21 − 2k − 2S k + 2k eS1 + S k e−S1 + S k eS21 =− 21 21 21 21 21 21 21 21 21 21 , (91) 2e−2S21 + 2e2S21 − 4 −S21 S22 S21k21 − S1k21e S22k22 − S22k22e c61 = , c81 = , 2eS21 − 2e−S21 2eS22 − 2e−S22 −S22 S22k22 − S22k22e c62 = , 2eS22 − 2e−S22 c71 2k − 2S k e−S1 − 2k e2S21 − 2S k + 2k eS21 + S k e−S21 + S k eS1 =− 21 21 21 21 21 21 21 21 21 21 21 , 2e−2S21 + 2e2S21 − 4 c52 2k e−2S22 − 2k e−S22 − 2k − 2S k + 2k eS22 + S k e−S22 + S k eS22 =− 22 22 22 22 22 22 1 22 1 22 , 2e−2S1 + 2e2S1 − 4 c72 2k − 2S k e−S22 − 2k e2S22 − 2S k + 2k eS22 + S k e−S22 + S k eS22 =− 22 22 22 22 22 22 22 22 22 22 22 , 2e−2S22 + 2e2S22 − 4 S k − S k eS22 D D c = 22 12 22 12 , k = 21 , k = 22 . 82 S −S 21 22 2e 22 − 2e 22 C21 C22

Table 10 Estimation of the nanoplate centre defection using VaM and VaM+ABSM versus the Navier solution for diferent As a result, we have the following displacement function 0.5;0.5 w(x, y)=X (x)Y (y)+X (x)Y (y). w(0.5; 0.5) wAg( ) w(0.5; 0.5) 21 21 22 22 (93) (parameter Analytical solu- Numerical solution Exact length of mate- tion by VaM (VaM+ABSM) Navier’s In order to solve the modifed Germain–Lagrange equa- rial) solution tion, we will use combined Vaindiner’s and the Agranovs- kii–Baglai–Smirnov methods (VaM+ABSM). In this case 0 0.307 0.308 0.311 each ordinary diferential equation is solved by the fnite 0.3 0.223 0.223 0.225 2 diference method with the error O h . The convergence 0.5 0.150 0.150 0.151 of the solution of ordinary diferential equations depending 1 3 J. Awrejcewicz et al.

on the number of partitions along the spatial coordinate is Table 11 Estimation of the nanoplate centre defections using VaM investigated. As a result of a numerical experiment, it is and VaM+ABSM for diferent

found that the required number of partitions in the fnite w(0.5; 0.5) wAg(0.5;0.5) diference method is 9 points for boundary condition (15) (parameter length of Analytical solution Numerical solution by and 55 points for boundary condition (16). material) by VaM VaM+ABSM combina- Table 10 gives displacement wm(0.5;0.5) obtained tion by the analytical Vaindiner’s method and displacement 0 0.289 0.290 w(0.5;0.5) obtained by Vaindiner’s method combined with 0.3 0.210 0.211 the Agranovskii–Baglai–Smirnov method and exact Navier’s 0.5 0.142 0.142 solution. Solution for boundary conditions (16) is obtained like that for boundary conditions (15). In this case, a trial function of where displacement w (x, y) takes the following form 11 + 4 2 2 N11 = t11 cos , w(x, y)=sin ( x)Y11(y)+X12(x)sin ( y). 2 � � � � � 11 + 4 Next, we obtain the system of equations E11 = t11 sin , � � 2 � � � A Y(4) + B Y(2) + C Y = D , + 4 11 11 11 11 11 11 11 N = �t � cos 12 , (4) (2) (94) 12 � 12� A X + B X + C X = D , 2 12 12 12 12 12 12 12 � � � + 4 E = �t �sin 12 , where 12 � 12 � 2 � � � 1 1 � � − B 2 − 4A C 3 2 −B� � 11 11 11 A = 2 X2dx = 2 , B = 2X(2)X dx =− , t = 11 + i, 11 1 11 1 1 11 � 16 2 2A11 � 2A11 � 0 0 1 1 − B 2 − 4A C 4 11 11 11 −B 1 (4) 1 11 C = X X dx = , D = Q X dx = Q , 11 = arctan , 11 2 1 1 2 11 1 1 1 � 2A 2A 4 ⎛⎛ � 11 �⎞ � 11 � ⎞ 0 0 ⎜⎜ ⎟� ⎟ 1 1 D ⎜⎜ ⎟ ⎟ 1 1 3 2 k = 11 , ⎜⎜ ⎟ ⎟ A = Y2dy = , B = 2Y(2)Y dy =− , 11 C ⎝⎝ ⎠ ⎠ 12 2 1 2 16 12 1 1 2 11 0 0 2 −B − B12 − 4A12C12 1 1 t = 12 + i, 24 1 12 � C = 2 Y(4)Y dy = , D = Q Y dy = Q . 2A12 � 2A12 � 12 1 1 2 12 2 1 2 4 0 0 2 − B12 − 4A12C12 −B (95) = arctan 12 , 12 � � 2A12 � 2A12 Solving the system of equations (94), we obtain ⎛⎛ ⎞ � � ⎞ ⎜⎜ ⎟� ⎟ N11y ⎜⎜ ⎟ ⎟ Y11(y)=e c11 cos E11y + c21 sin E11y D12 k12 = . ⎜⎜ ⎟ ⎟ −N11y C ⎝⎝ ⎠ ⎠ + e c31 cos E 11y +c41 sin E 11y + k11, 12 X (x)=eN12x c cos E x + c sin E x (96) 12 12 12 22 12 Substituting the obtained values into the form for the dis- −N12x + e c32 cos E12 x +c42 sin E12 x + k12, placement, we obtain the following solution to the original equation

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

Table 12 n 2 N11y Square plate centre defection for diferent and yielded w (x, y)=sin ( x) e c11 cos E11y + c21 sin E11y by FDM (load equal 7, to boundary conditions (102)) −N11y +e c31 cos E 11y + c41 sinE11y + k11 Number of partition points wn(0.5;0.5) wn(0.5;0.5) wn(0.5;0.5) + eN12x c cos E x + c sin E x for one coordinate 12 12 22 12 = 0 = 0.3 = 0.5 −N12x 2 + e c32 cos E12x + c42 sin E12x + k12 sin ( y), (n = 5) 0.307 0.225 0.150 (97) (n = 11) 0.310 0.225 0.151 where (n = 15) 0.310 0.225 0.151 c11 + 31 + k11 = 0, (n = 21) 0.310 0.225 0.151 k + e−N11 (c cos(E )+c sin(E )) (n = 25) 0.310 0.225 0.151 11 31 11 41 11 N11 Exact Navier’s solution 0.311 0.225 0.151 + e (c11 cos(E11)+c21sin(E11)) = 0, E11c21 + E1141 + N11c11 − N11c31 = 0,

N11 e (c21E11 cos(E11)−c11E11sin(E11))

N11 Table 13 Square plate centre defection for diferent n and yielded + e (c41E11 cos(E11)−c31E11sin(E11)) = 0, by FDM (load equal 21, boundary conditions (103)) (98) c12 + 32 + k12 = 0, 0.5;0.5 0.5;0.5 0.5;0.5 Number of partition wn( ) wn( ) wn( ) −N12 k12 + e (c32 cos(E12)+c42sin(E12)) points for one coordinate N12 = 0 = 0.3 = 0.5 + e (c12 cos(E12)+c22sin(E12)) = 0,

E12c22 + E1242 + N12c12 − N12c32 = 0, (n = 5) 0.412 0.299 0.201 eN12 (c E cos(E )−c E sin(E )) (n = 11) 0.314 0.228 0.153 22 12 12 12 12 12 ( = ) N12 n 15 0.302 0.220 0.148 + e (c42E12 cos(E12)−c32E12sin(E12)) = 0. (n = 21) 0.296 0.215 0.145 (n = 25) 0.294 0.214 0.144 Formula (97) forms a classical solution obtained by Vain- (n = 31) 0.293 0.213 0.143 diner’s method. However, this solution can be refned. Since Y (y), X (x), (n = 35) 0.292 0.212 0.142 we know functions 11 12 we can fnd functions X (x), Y (y) (n = 41) 0.292 0.212 0.142 12 22 . To do this, the iterative procedure can be repeated, and hence

Table 14 Solution to MGLE for Number of Clamping Simple Number of Clamping Simple boundary conditions (15) for w (0.5;0.5) w (0.5;0.5) q = 7 and boundary condition Triangular n Support Triangular n Support q = 21 (16) for Finite wn(0.5;0.5) Finite wn(0.5;0.5) Elements Elements

5 0.291 0.309 5 0.289 0.309 11 0.290 0.310 11 0.289 0.310 21 0.290 0.310 21 0.290 0.310 31 0.290 0.310 31 0.290 0.310 37 0.290 0.310 37 0.290 0.310 Exact Navier’s solution 0.311 Exact Navier’s solution 0.311

1 3 J. Awrejcewicz et al.

N21y Y21(y)=e c11 cos E21y + c21 sin E21y + 4 Numerical Solution of MGLE by FDM −N21y and FEM + e c31 cos E 21y +c41 sin E 21y + k21, X (x)=eN22x c cos E x + c sin E x + (99) 22 52 22 62 22 We will use the well-known methods to obtain numerical + e−N12x c cos E x + c sin E x + k . 72 22 82 22 22 solutions for numerous PDEs of mathematical physics. This is the fnite diference method (FDM) with approxima- As a result, we have the following displacement function 2 tion O h and the fnite element method (FEM). Recently, w(x, y)=X (x)Y (y)+X (x)Y (y). 21 21 22 22 (100) these methods have been widely used in various felds of continuum mechanics. It seems interesting to compare the Table 11 shows the displacement obtained analytically by solutions obtained by the methods of reducing partial dif- Vaindiner’s method for boundary conditions (16), and the ferential equations to ordinary diferential equations with the numerical solution obtained by the combined methods of solutions obtained by the above methods. Next, we compare Vaindiner and Agranovskii–Baglai–Smirnov. the numerical solutions of the modifed Germain–Lagrange The following conclusions can be formulated. The accu- equation obtained by the methods of reducing partial difer- racy of the solution for two types of boundary conditions ential equations to ordinary diferential equations with the obtained by the Vaindiner method is comparable to the accu- solutions attained by the fnite diference method and fnite racy of the solution obtained by the Kantorovich–Vlasov element method for some types of boundary conditions. method and variational iteration method. It is worth noting that the solution by Vaindiner’s method and the Kan- 4.1 Finite Diference Method torovich–Vlasov method depends on a correct choice of the initial approximation. This limitation can be removed Let us apply the second-order fnite diference method to by using the procedure of the variational iteration method equation (18). We get in each equation of the system obtained by the Vaindiner method (100). Like in the case of the variational iteration D wi+2,j + wi−2,j + wi,j+2 + wi,j−2+ method, this will allow us to use any smooth function for the + 2 wi+1,j+1 + wi+1,j−1 + wi−1,j+1 + wi−1,j−1 initial approximation. 4 +20wi,j − 8 wi+1,j + wi−1,j + wi,j+1 + wi,j−1 = qh . (101)

Table 15 Analytical and Type of Methods = 0 = 0.3 = 0.5 numerical solutions obtained by w w w various methods for estimation solution (0.5; 0.5) (0.5; 0.5) (0.5; 0.5) of the nanoplate centre Analytical solutions Navier method 0.311 0.225 0.151 defection (b.c. (15), q = 7 ) for diferent BGM 0.311 0.228 0.153 KVM (in frst term approximation ) 0.314 0.228 0.153 KVM (in second term approximation) 0.312 0.225 0.152 VIM 0.307 0.223 0.150 VaM 0.307 0.223 0.150 Numerical solution FDM 0.310 0.225 0.151 FEM - (Triangular Elements (350 Elements)) 0.310 0.225 0.151 FEM - (Quadrangular Elements (165 Elements)) 0.310 0.225 0.151 KVM in frst term + ABSM 0.308 0.223 0.150 KVM in second term + ABSM 0.310 0.225 0.151 VIM + ABSM 0.308 0.224 0.150 VaM + ABSM 0.308 0.223 0.150 VaM + VIM 0.308 0.223 0.150

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

Table 16 Analytical and Type of solution Methods = 0 = 0.3 = 0.5 numerical solutions obtained w w w by various methods for the (0.5; 0.5) (0.5; 0.5) (0.5; 0.5) estimation of the nanoplate Analytical solutions KVM centre defection (b.c. (16), q = 21 ) for diferent (in frst term approximation ) 0.289 0.210 0.142 KVM (in second term approximation) 0.291 0.212 0.143 VIM 0.292 0.212 0.142 VaM 0.289 0.210 0.142 Numerical solution FDM 0.294 0.214 0.144 FEM - (Triangular Elements (350 Elements)) 0.297 0.217 0.146 FEM - (Quadrangular Elements (165 Elements)) 0.297 0.217 0.146 KVM in frst term + ABSM 0.290 0.210 0.142 KVM in second term + ABSM 0.291 0.211 0.143 VIM + ABSM 0.292 0.212 0.142 VaM + ABSM 0.290 0.211 0.142 VaM + VIM 0.290 0.210 0.142

Table 18 Various methods of solution versus MGLE and the number Table 17 Various methods of solution versus MGLE and the number of equations required to obtain a reliable solution (b.c. (16), = 0.5) of equations required to obtain a reliable solution (b.c. (15), = 0) The number of The number of equations required equations required to to Solution method obtain a reliable Solution method obtain a reliable solution solution n N n ∗ N n N n ∗ N Kantorovich–Vlasov method in frst term 15 1 15 Kantorovich–Vlasov method in frst term 15 1 15 Kantorovich–Vlasov method in second term 30 1 30 Kantorovich–Vlasov method in second term 30 1 30 Variational iteration method in frst term 15 2 45 Variational iteration method in frst term 15 3 45 Vaindiner’s method in frst term 30 1 30 Vaindiner’s method in frst term 30 3 90 ( 2) 2 Finite diference method O h 121 1 121 Finite diference method O(h ) 121 1 121 Finite element method, triangular elements 1498 1 1498 Finite element method, triangular elements 1498 1 1498 Finite element method, quadrangular elements 1242 1 1242 Finite element method, quadrangular elements 1242 1 1242

2 We obtain a system of linear equations n for each grid node, where n is the number of grid nodes, each on the x and y w0,0 = wn,0 = w0,n = wn,n = 0, axes. We solve this system by the Gauss method. In order to w−1,i =−w1,i, wn+1,i (103) determine the values of the extreme grid nodes, we use the =−wn−1,i , wi,−1 =−wi,1 , wi,n+1 =−wi,n−1 . following boundary conditions. For boundary conditions (15), we have Tables 12 and 13 show the results obtained for bound- w0,0 = wn,0 = w0,n = wn,n = 0, ary conditions (102) and (103), using the fnite diference method for a square plate under the action of a constant w−1,i = w1,i, wn+1,i = wn−1,i , wi,−1 = wi,1 , wi,n+1 = wi,n−1 transverse load. (102) Analysis of the results shows that the material length For boundary conditions (16), we have parameter can significantly influence the results obtained. The solution for the material length parameter

1 3 J. Awrejcewicz et al.

Table 19 Various methods of solution versus MGLE and the number Table 14 shows the results of the study on convergence = 0 of equations required to obtain a reliable solution (b.c. (16), ) for the fnite element method when solving the modifed The number of Germain–Lagrange equation for boundary conditions (15) at equations required a constant transverse load q = 7 and for boundary conditions to (16) for fxed q = 21 , and = 0. Solution method obtain a reliable Thus, the best convergence is achieved for 11 triangular solution fnite elements, both for boundary conditions (15) and (16). n N n ∗ N Table 14 presents the analysis of results of the study on the convergence of the solution of the modifed Ger- Kantorovich–Vlasov method in frst term 55 1 55 main–Lagrange equation by the fnite element method. Thus, Kantorovich–Vlasov method in second term 110 1 110 the best convergence is achieved for 11 triangular fnite ele- Variational iteration method in frst term 55 3 165 ments, for both boundary conditions studied. For quadran- Vaindiner’s method in frst term 110 3 330 gular fnite elements for both types of boundary conditions, O(h2) Finite diference method 1225 1 1225 convergence is achieved for 21 fnite elements. To reduce the Finite element method, triangular elements 1498 1 1498 number of elements when solving the problem, it is possible Finite element method, quadrangular elements 1242 1 1242 to use symmetry of the solution and reduce the number of elements by 4 times, but in this case it was not done.

Table 20 Various methods of solution versus MGLE and the number of equations required to obtain a reliable solution (b.c. (16), = 0.5) 5 Comparison of Results Obtained The number of by the Methods of Reducing PDEs to ODEs equations required and by Numerical Methods to Solution method obtain a reliable Previously, the equation governing bending of a nanoplate solution taking into account the modifed couple stress theory and the n N n ∗ N method of variable parameters of elasticity has been derived. Now, it is solved by some of the already mentioned meth- Kantorovich–Vlasov method in frst term 55 1 55 ods: the Kantorovich–Vlasov method with various modifca- Kantorovich–Vlasov method in second term 110 1 110 tions; variational iteration method, Vaindiner’s method, the Variational iteration method in frst term 55 2 110 Agranovskii–Baglai–Smirnov method, Bubnov–Galerkin Vaindiner’s method in frst term 110 1 110 method, the method of double trigonometric series approxi- O(h2) Finite diference method 1225 1 1225 mation, the fnite element method and the fnite diference Finite element method, triangular elements 1498 1 1498 method. Finite element method, quadrangular elements 1242 1 1242 Table 15 presents the values of the maximum displacement calculated by various methods for solving the equation of bending of a nanoplate under the action of a uniformly = 0.3; = 0.5 coincides with Navier’s exact solution. The distributed load ( q = 7 ) for boundary conditions (15). convergence of the solution obtained by the fnite diference The analysis of the numerical results and final data method depends essentially on the type of boundary condi- reported in Table 15 shows that the solutions obtained by tions. Namely, for boundary conditions (103) three times various numerical methods including the fnite element more partition points are required than in the case of bound- method, finite differences method, Kantorovich–Vlasov ary condition (102). method in N terms and the variational iteration method combined with the Agranovskii–Baglai–Smirnov method 4.2 Finite Element Method are very close to the exact solution. The maximum difference between the exact solution In this case we will use linear triangular and quadrangular and solutions obtained by other methods does not exceed elements while employing FEM. According to the algorithm 1.2%, so these methods are competitive. Methods for reduc- for solving the problem, the area is automatically divided ing PDEs to ODEs allow us to obtain an analytical solution into a given number of elements. The solution is obtained with extremely high accuracy. The most important conclu- by the system of linear algebraic equations using the Gauss sion based on the obtained results is as follows. The cost of method. A convergence study is provided depending on the computer time when using these methods is several times number of the used elements. cheaper than when using the numerical methods based on FEM and FDM.

1 3 Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑…

In addition, Table 16 presents the values of the maximum role in the system modeling, identifcation, monitoring and nanoplate displacement calculated by various methods for control. solving the equation of bending of a nanoplate under the The classical methods aimed at solving the problem have action of uniformly distributed load ( q = 21 ) for boundary their origin in mathematical physics. However, the problem conditions (16). has not yet been fully resolved and, as our review has shown, In this case, the comparative analysis of the numerical it requires revision and more attention to the known meth- results based on the data reported in Table 16 for boundary ods of fnite elements and fnite diferences widely used in condition (16) for square nanoplate leads to the conclusion engineering community but not well recognized as competi- that if = 0.3, = 0.5 the results of all methods are quite tive approaches. The methodology is well known: a given close. The largest diferences in the solutions arise when multidimensional problem should be substituted by its coun- = 0 . It should be emphasized that the values of solutions terpart sequence of one-dimensional problems. However, by the mentioned methods devoted to the reduction of PDEs the obtained results strongly depend on the used methods, to ODEs are close to each other and their diference is 0.6%. which answer numerous questions and require checking of In addition, we have compared the number of algebraic the authenticity, reliability, validity, stability of results, etc. equations while employing the Kantorovich–Vlasov method, The following methods have been reviewed and exam- variational iteration method, Vaindiner’s method, FEM and ined due to their state-of-the art levels of performance and FDM. Table 17 shows a comparison of the number of equa- computational advantages: the Fourier methods, the Galer- tions required to solve Eq. (18) at = 0 for boundary condi- kin-type methods, the variational methods, the variational tions (15). The following notation is used: n is the number iterations, the Kantorovich–Vlasov methods, Vaindiner’s of equations, N - is the number of iterations (if necessary), method, the Agranov- skii-Baglai-Smirnov method with n ∗ N is the total number of equations to obtain a solution. respect to the archetypal fnite element and fnite diference Table 17 presents the results of analysis for diferent num- methods. ber of equations for = 0 , and for various methods. The quantifcation and comparison of the tested (above The same is done in Table 18 but for = 0.5. mentioned) methods was carried out on the basis of the mod- We have also analyzed the same problem regarding ifed Germain–Lagrange PDE governing the dynamics of a equation (14) for = 0 for boundary conditions (16) (see nanoplate and with comparison to the exact results obtained Table 19). on the basis of Navier’s method. The same analysis has been extended for the case when Although the paper has a review character, it includes = 0.5 (see Table 20). also our original results. The whole work can be briefy sum- Based on the analysis of Tables 17, 18, 19 and 20, it can marized based on the following major conclusions. be concluded that in terms of computer time the most expensive are the fnite diference and fnite element methods. • A mathematical model of Germain–Lagrange nanoplates, The analysis of the numerical results given in Tables 17, taking into account the physical nonlinearity and inho- 18, 19 and 20 leads to the conclusion that in terms of com- mogeneity of the material, has been derived. puter time the most expensive are the fnite diference and • The comparative analysis of numerous methods, includ- fnite element methods. The methods that reduce the original ing the Kantorovich–Vlasov method, the variational problem for a partial diferential equation to solving an ordi- iteration method, and Vaindiner’s method aiming at the nary diferential equation use a one-dimensional grid. This reduction of PDEs to ODEs was carried out for the frst leads to a decrease in the number of algebraic equations and time. an increase in the speed of obtaining a solution. • For each method described above, theorems have been both formulated and proved and thus their mathematical justifcation has been given. 6 Concluding Remarks • The proven convergence theorems for solutions obtained by the Kantorovich–Vlasov methods and variational This article presents a comprehensive review of the literature iteration method can be applied to solve the modifed on methods of reducing nonlinear partial diferential equa- Germain–Lagrange equations. tions to a set of nonlinear diferential equations with empha- • The solutions obtained by the mentioned methods of sis put on their reliability, validity, accuracy and computa- reducing the equations of mathematical physics to ordi- tional efciency. Since the nonlinear PDEs govern dynamic nary diferential equations have been compared with the and static behavior of numerous systems in mechanics, aero- exact Navier solution obtained by the authors for the nautics and civil engineering, a proper choice of the feasible modifed Germain–Lagrange equation. computational method to study mass distributed mechanical • These methods appeared to be efective. The computation systems based on the signal processing plays an important time for obtaining a solution by these methods is much

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Authors and Afliations

J. Awrejcewicz1 · V. A. Krysko‑Jr.1 · L. A. Kalutsky2 · M. V. Zhigalov2 · V. A. Krysko2

V. A. Krysko‑Jr. 1 Department of Automation, Biomechanics [email protected] and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90‑924 Lodz, Poland L. A. Kalutsky [email protected] 2 Department of Mathematics and Modeling, Saratov State Technical University, 77 Politehnicheskaya St., Saratov, M. V. Zhigalov Russian Federation 410054 [email protected] V. A. Krysko [email protected]

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