DOCSLIB.ORG

Numerical Integration of Odes While Preserving All Polynomial First

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numerical Integration of Odes While Preserving All Polynomial First Numerical integration of ODEs while preserving all polynomial first integrals Benjamin K Tapley∗ Department of Mathematical Sciences, The Norwegian University of Science and Technology, 7491 Trondheim, Norway August 17, 2021 Abstract We present a novel method for solving ordinary differential equations (ODEs) while pre- serving all polynomial first integrals. The method is essentially a symplectic Runge-Kutta method applied to a reformulated version of the ODE under study and is illustrated through a number of examples including Hamiltonian ODEs, a Nambu system and the Toda Lattice. When applied to certain Hamiltonian ODEs, the proposed method yields the averaged vector field method as a special case. 1 Introduction In this paper, we are concerned with the numerical solution of autonomous ODEs in n dimensions x_ = f(x); x 2 Rn; (1) _ that possess the functions Hi(x) for i = 1; :::; m < n satisfying Hi(x) = f(x) · rHi(x) = 0, where d the dot denotes dt . If such functions exist, they are called first integrals of (1). The preservation of first integrals under numerical methods is an important property for the method to inherit when arXiv:2108.06548v1 [math.NA] 14 Aug 2021 performing long-term simulations of dynamical systems [1]. Hamiltonian systems, for instance, are an important class of ODEs that govern many physical phenomena and have subsequently been the subject of many influential studies. Many well designed energy-preserving methods have therefore been proposed for solving such systems, for example, discrete gradient methods [2, 3], Hamiltonian boundary value methods [4] and Runge-Kutta type methods [5{7]. In particular, the energy-preserving averaged vector field method, which was first introduced in [8], has been the building block of many interesting methods for such canonical Hamiltonian ODEs [9, 10]. The study of energy-preserving numerical methods for ODEs often also serves as inspiration for many energy-preserving methods for Hamiltonian PDEs including discrete variational methods [11,12], averaged vector field methods [13, 14], continuous Runge-Kutta type methods [15] and fast lin- early implicit methods based on polarization [16] or discrete gradients [17]. Another successful ∗email: [email protected] 1 energy-preserving method for Hamiltonian PDEs is the scalar auxillary variable approach [18,19]. However, many dynamical systems possess multiple first integrals such as Lie-Poisson systems, lattice equations, Nambu systems and many other integrable systems. As it is desirable for numerical simulations to inherit as many physical properties of the exact solution as possible, numerical methods for preserving multiple first integrals have been considered. Although this is a much more difficult task than preserving a single first integral, many successful methods have been constructed in the past that utilize projection [20,21] and/or discrete gradients [3,22], which is a form of projection [23]. However, as noted in [24], projection is somewhat of a last resort as it destroys other properties of the solution such as the affine covariance and may not give good long time behavior. Other methods for preserving multiple first integrals that are worth mentioning are the line integral methods [25,26]. Here, we present a novel numerical method that preserves all polynomial integrals Hi(x) of an ODE of the form (1). The method is simple and is essentially the midpoint rule, or a higher order symplectic Runge-Kutta method, but applied to a reformulated version of the ODE un- der study in a higher dimensional phase space then projected exactly onto the original phase space. The method is therefore implicit, symmetric, of arbitrary order and roughly the same computational cost of an implicit Runge-Kutta method. Furthermore, when restricted to some canonical Hamiltonian ODEs our method yields the averaged vector field method as a special case. The paper is organized as follows. We begin in section 2.1 by introducing a simplified version of the method for a quartic-Hamiltonian ODE. Here, we illustrate the core concept using an example of the planar quartic oscillator. Once this has been elucidated then extending the method to preserve polynomial integrals (i.e., of arbitrary degree), then multiple polynomial integrals follows straightforwardly. This is presented in sections 2.2 and 2.3, respectively. In section 2.4 we show how to develop higher-order methods and in section 2.5 we briefly discuss the connection between the proposed method and the averaged vector field method. Section 3 presents numerical examples of a Hamiltonian system, a Nambu system and the Toda lattice. Concluding remarks are given in section 4. 2 Integral-preserving numerical integration 2.1 Quartic-Hamiltonian ODEs n Let Rd[x] be the class of polynomials of degree d in the variables x 2 R . To introduce the method, we first focus our attention on an n-dimensional Hamiltonian ODE with a quartic Hamiltonian H(x) 2 R4[x] for n even. Such an ODE takes the form x_ = JrH(x); (2) 0 −I where J = I 0 , I is the identity matrix in n=2 dimensions and r is the gradient operator with respect to x. We note that the forthcoming method is applicable to ODEs with non-constant symplectic structure and in odd dimension, i.e., when J = J(x). To construct the integral preserving method, we start by introducing the quadratic auxiliary variables yi;j = xixj for j ≤ i; i = 1; :::; n 2 ~ and define what will be referred to as a reduced-degree Hamiltonian H(x; y) 2 R2[x; y] that is quadratic in the variables x and y and satisfies the consistency condition H~ (x; y(x)) = H(x): T m Here, with a slight abuse of notation, we have let y = (y1;1; y1;2; :::; yn;n−1; yn;n) 2 R denote the T m = n(n+1)=2 independent quadratic auxiliary variables and y(x) = (x1x1; x1x2; :::; xnxn−1; xnxn) (i.e., with the argument x written explicitly) denote the y variables as a function of the original phase space variables x. Using the chain rule, the ODE in these new variables now reads ! @ @y(x) T @ x_ =JrH~ (x; y) = J H~ (x; y) + H~ (x; y) := f(x; y) @x @x @y (3) @y(x) y_ = f(x; y) := g(x; y) @x We will refer to f(x; y) as the reduced-degree vector field of f(x) with respect to H~ (x; y), which also satisfies the consistency condition f(x; y(x)) = f(x). Furthermore, we will refer to the collective ODE system (3) as the extended ODE. We now make some observations. Remark 1. The extended ODE (3): (a) is of the following skew gradient form for z = (xT ; yT )T 0 T 1 @y(x) T J − J z_ = S(z)rH~ (z);S(z) = −S(z) = @ @x A : @y(x) @y(x) @y(x) T @x J @x J @x (b) has the reduced-degree Hamiltonian H~ (x; y) as first integral. This follows directly from (a). (c) possesses the m additional quadric integrals Hi;j(x; y) = xixj − yi;j; i = 1; :::; n; j ≤ i: This can be easily seen by differentiating with respect to t _ Hi;j =x _ ixj +x _ jxi − y_i;j = fi(x; y)xj + fj(x; y)xi − (fi(x; y)xj + fj(x; y)xi) = 0: (d) is solved by the solution of (2) with the initial conditions yi;j t=0 = xixj t=0. Due to remark 1d, an integral-preserving method of the ODE (3) will yield an integral-preserving method of (2) as long as the outlined initial conditions yi;j t=0 = xixj t=0 are satisfied. Moreover, ~ the m induced quadric integrals Hi;j(x; y) together with the reduced-degree Hamiltonian H(x; y) are quadratic, therefore any symplectic Runge-Kutta method (i.e., one with vanishing stability matrix [27]) will preserve all such integrals. However, numerically integrating the ODE (3) presents a significant drawback, namely that the phase space dimension has increased from n to n + m = n(n+3)=2. Due to the fact that m scales quadratically with n this quickly becomes computationally prohibitive for large n especially due to the fact all symplectic Runge-Kutta methods are implicit. 3 However, this issue can be circumvented as follows. First we implement the (symplectic) midpoint rule given by x0 − x x0 + x y0 + y = f ; ; (4) h 2 2 y0 − y x0 + x y0 + y = g ; ; (5) h 2 2 where x0 ≈ x(h) denotes the solution of the midpoint rule from initial conditions x = x(0) and similarly for y. Due to the fact that the midpoint rule preserves all quadratic invariants, we have 0 0 preservation of the induced quadric integrals Hi;j(x ; y ) = Hi;j(x; y), which is a linear equation for y0 meaning it's solution can be directly found in terms of x0 and therefore avoiding the need to 0 0 0 solve (5). That is, by substituting yi;j = xixj into (4) we are left with an n dimensional implicit equation for x0 as follows x0 − x x0 + x y(x0) + y(x) = f ; : (6) h 2 2 Doing so circumvents the issue of the increased phase space dimension, thus yielding an efficient method that preserves any quartic Hamiltonian. The method (6) will be referred to as the reduced- degree midpoint method. Remark 2. We observe that the reduced-degree midpoint method: (a) possesses a B-series with respect to the extended ODE vector field. (b) is affinely equivariant with respect to the variables (x; y). (c) is symmetric, that is, it is invariant under the transformation x $ x0 and h ! −h.
Load more

Recommended publications
  • Global Institute Lahore
    Global Institute Lahore HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION BY GHULAM FARID DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS February, 2015 GLOBAL INSTITUTE LAHORE i HADAMARD k-FRACTINAL INTEGRAL AND ITS APPLICATION BY GHULAM FARID A dissertation submitted to Faculty of Management Sciences In Partial Fullfilment of the Requirements for the Degree of DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS FEBRUARY, 2015 ii IN THE NAME OF ALLAH, THE MOST BENIFICIAL, THE MOST MERCIFUL iii GLOBAL INSTITUTE LAHORE HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION BY GHULAM FARID A dissertation submitted to Faculty of Management Sciences, in partial fullfilment of the requirements for the degree of DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS Dissertation Committee iv DECLARATION This is to certify that this research work has not been submitted for obtaining similar degree from any other university/college. v DEDICATED TO My Beloved Son Muhammad Yahya Khan vi ACKNOWLEDGEMENT Gratitude to almighty Allah for His unending blessings that has reinforced me to complete this humble piece of dissertation. I am also obliged to all the teachers and fellows at Global Institute, Lahore. I am grateful to my supervisor Dr. Ghulam Mustafa Habibullah for his motivation, support and guidance to the right direction. I am highly obliged to Muhammad Kashif Khan, the Chairman Global Institute Lahore and all of the college officials including the Rector, Pro-Rector and all the faculty members of the institute for their guidance, support and favours. The financial support of Higher Education Commission, Pakistan in facilitating me to take up this research work, is most gratefully acknowledged. I am also thankful to my brother Professor Muhammad Zulqarnain Khan and my wife for their continuous support and encouragement.
    [Show full text]
  • Solvability of a Maximum Quadratic Integral Equation of Arbitrary Orders
    Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 11, Number 2, pp. 93–104 (2016) http://campus.mst.edu/adsa Solvability of a Maximum Quadratic Integral Equation of Arbitrary Orders Mohamed Abdalla Darwish King Abdulaziz University Sciences Faculty for Girls Department of Mathematics Jeddah, Saudi Arabia [email protected] Johnny Henderson Baylor University Department of Mathematics Waco, Texas 76798-7328, USA Johnny [email protected] Kishin Sadarangani Universidad de Las Palmas de Gran Canaria Departamento de Matematicas´ Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain [email protected] Abstract We investigate a new quadratic integral equation of arbitrary orders with maxi- mum and prove an existence result for it. We will use a fixed point theorem due to Darbo as well as the monotonicity measure of noncompactness due to Banas´ and Olszowy to prove that our equation has at least one solution in C[0; 1] which is monotonic on [0; 1]. AMS Subject Classifications: 45G10, 47H09, 45M99. Keywords: Fractional, monotonic solutions, monotonicity measure of noncompact- ness, quadratic integral equation, Darbo theorem. Received August 24, 2016; Accepted November 27, 2016 Communicated by Martin Bohner 94 M. A. Darwish, J. Henderson and K. Sadarangani 1 Introduction In several papers, among them [1,11], the authors studied differential and integral equa- tions with maximum. In [6–9] Darwish et al. studied fractional integral equations with supremum. Also, in [4, 5], Caballero et al. studied the Volterra quadratic integral equa- tions with supremum. They showed that these equations have monotonic solutions in the space C[0; 1].
    [Show full text]
  • The Geometry of Paths*
    THEGEOMETRY OF PATHS* BY OSWALDVEBLEN AND TRACYYERKES THOMAS 1. Introduction. The first part of this paper is intended as a systematic general account of the geometry of paths and is largely based on the series of notes by Eisenhart and Veblen in volume 8 of the Proceedings of the National Academy of Sciences. The general theory is carried far enough to include an account of a series of tensors defined by means of normal coordinates, and also a series of generalizations of the operation of covariant differentiation. We then turn to a special problem, the investigation of the conditions which must be satisfied by the functions r in order that the differential equations of the paths shall possess homogeneous first integrals.! We first solve a still more special problem for first integrals of the nt\\ degree (§ 15). This includes as a special case the problem solved by Eisenhart and Veblen in the Proceedings of the National Academy of Sciences, vol. 8 (1922), p. 19, of finding the conditions which must be satisfied by the Ps in order that the equations (2.1) shall be the differential equations of the geodesies of a Riemann space. Finally we solve the general problem for the linear and quadratic cases; that is to say, we find algebraic necessary and sufficient conditions on the functions r in order that (2.1) shall possess homogeneous linear and quadratic first integrals. The method used will generalize to homogeneous first integrals of the nth degree. We leave unsolved all the projective problems which corre- spond to the affine problems which we have solved.
    [Show full text]
  • On Solutions of Quadratic Integral Equations
    On solutions of quadratic integral equations Ph.D Thesis Written at The Faculty of Mathematics and Computer Science, Adam Mickiewicz University Pozna´n, Poland, 2013 By MOHAMED METWALI ATIA METWALI Under the guidance of Dr. hab. Mieczys law Cicho´n Department of Differential Equations, Faculty of Mathematics and Computer Science, Adam Mickiewicz University In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Acknowledgments First of all, I am thankful to Allah for all the gifts has given me. I would like to express my gratitude and thanks to my advisor and my professor, Dr. hab. Mieczys law Cicho´n, Department of Differential Equations, Faculty of Mathematics and Computer Science, Adam Mickiewicz University for his help and valuable advice in the preparation of this dissertation. I am thankful to my family (my wife and my lovely sons ”Basem and Eyad”) for their support, encouragement and standing beside me during my stay in Poland. ii Contents 1 Preliminaries 4 1.1 Introduction................................ 4 1.2 Notationandauxiliaryfacts . 4 1.2.1 LebesgueSpaces ......................... 5 1.2.2 Young and N-functions...................... 5 1.2.3 Orliczspaces ........................... 6 1.3 Linearandnonlinearoperators. 7 1.3.1 Thesuperpositionoperators.. 7 1.3.2 Fredholmintegraloperator. 9 1.3.3 Volterraintegraloperator. 10 1.3.4 Urysohnintegraloperator. 11 1.3.5 Themultiplicationoperator. 12 1.4 Monotonefunctions. .. .. .. 14 1.5 Measuresofnoncompactness. 17 1.6 Fixedpointtheorems. .. .. .. 19 2 Monotonic integrable solutions for quadratic integral equations on a half line. 23 2.1 Motivations and historical background. .... 23 2.2 Introduction. ............................... 25 2.3 Mainresult ................................ 26 2.4 Examples ................................
    [Show full text]
  • Characterisation of the Algebraic Properties of First Integrals of Scalar Ordinary Differential Equations of Maximal Symmetry
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 212, 349]374Ž. 1997 ARTICLE NO. AY975506 Characterisation of the Algebraic Properties of First Integrals of Scalar Ordinary Differential Equations of Maximal Symmetry G. P. Flessas,* K. S. Govinder,² and P. G. L. Leach³ View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Department of Mathematics, Uni¨ersity of the Aegean, Karlo¨assi, 83 200, Samos, Greece Submitted by Thanasis Fokas Received March 10, 1995 We undertake a study of the first integrals of linear nth order scalar ordinary differential equations with maximal symmetry. We establish patterns for the first integrals associated with these equations. It is shown that second and third order equations are the pathological cases in the study of higher order differential equations. The equivalence of contact symmetries for third order equations to non-Cartan symmetries of second order equations is highlighted. Q 1997 Academic Press 1. INTRODUCTION A scalar ordinary differential equation Žn. ExŽ.,y,y9,..., y s0,Ž. 1 *E-mail address: [email protected]. ² Permanent address: Department of Mathematics and Applied Mathematics, University of Natal, Durban, 4041, Republic of South Africa. E-mail address: [email protected]. ³ Permanent address: Department of Mathematics and Applied Mathematics, University of Natal, Durban, 4041, Republic of South Africa. Member of the Centre for Theoretical and Computational Chemistry, University of Natal and associate member of the Centre for Nonlinear Studies, University of the Witwatersrand, Johannesburg, Republic of South Africa. E-mail address: [email protected].
    [Show full text]
  • QUADRATIC INTEGRATION of GAUSSIAN PROCESSES Lim 22Ax
    proceedings of the american mathematical society Volume 81, Number 4, April 1981 QUADRATIC INTEGRATION OF GAUSSIAN PROCESSES T. F. LIN Abstract. Let x(t), 0 < t < r, be a Gaussian process whose covariance function R(s, t) satisfies certain conditions. If G(x) satisfies some mild condition, then the quadratic integral L2-lim'2,kG(x(tk))&x(tk)2 along any sequence of paritions of [0, T] whose mesh goes to zero exists. The differential rule for x(t) is also derived. 0. Introduction. Let x(/), 0 < / < 1, be a Gaussian process with covariance function R(s, /), 0 < s, t < 1. In studying some connections between Gaussian processes and boundary value problems, Baxter showed, under certain conditions, (see [1, Theorem 1, p. 522], or [6, Theorem 20.4, p. 297]) that the quadratic variation of x(/) over [0, 1] is lim 22Ax(/,)2 =['/(/) dt (0.1) where tk = tg = k2~n, 0 < k < 2", Ax(tk) = x(tk + x) - x(tk), 0 < k < 2" - 1, and where/(/), 0 < / < 1, is defined by /(/) = lim {/?(/ + A, /) - R(t, t)}/h - lim {R(t + A, /) - R(t, t)}/h. (0.2) This result was later extended to a wider class of processes (see [5], [2], [4]). The limit in (0.1) is L2-convergent (as well as almost sure convergent). If G(y) is a nice function of y, it is not hard to prove that 2o G(x(tk))Ax(tk)2\ = f{/o'g(x(/))/(/) dt}. (0.3) {2" —1 ) From (0.1) and (0.3), it is reasonable to conjecture that the quadratic integral of G(x(t)) w.r.t.
    [Show full text]
  • Lecture 1 Linear Quadratic Regulator: Discrete-Time Finite Horizon
    EE363 Winter 2008-09 Lecture 1 Linear quadratic regulator: Discrete-time finite horizon LQR cost function • multi-objective interpretation • LQR via least-squares • dynamic programming solution • steady-state LQR control • extensions: time-varying systems, tracking problems • 1–1 LQR problem: background discrete-time system x(t +1) = Ax(t)+ Bu(t), x(0) = x0 problem: choose u(0), u(1),... so that x(0),x(1),... is ‘small’, i.e., we get good regulation or control • u(0), u(1),... is ‘small’, i.e., using small input effort or actuator • authority we’ll define ‘small’ soon • these are usually competing objectives, e.g., a large u can drive x to • zero fast linear quadratic regulator (LQR) theory addresses this question Linear quadratic regulator: Discrete-time finite horizon 1–2 LQR cost function we define quadratic cost function N−1 T T T J(U)= x(τ) Qx(τ)+ u(τ) Ru(τ) + x(N) Qf x(N) τX=0 where U =(u(0),...,u(N 1)) and − Q = QT 0, Q = QT 0, R = RT > 0 ≥ f f ≥ are given state cost, final state cost, and input cost matrices Linear quadratic regulator: Discrete-time finite horizon 1–3 N is called time horizon (we’ll consider N = later) • ∞ first term measures state deviation • second term measures input size or actuator authority • last term measures final state deviation • Q, R set relative weights of state deviation and input usage • R> 0 means any (nonzero) input adds to cost J • LQR problem: find u (0),...,u (N 1) that minimizes J(U) lqr lqr − Linear quadratic regulator: Discrete-time finite horizon 1–4 Comparison to least-norm input c.f.
    [Show full text]
  • Solvability of a Q-Fractional Integral Equation Arising in the Study of an Epidemic Model Mohamed Jleli and Bessem Samet*
    Jleli and Samet Advances in Difference Equations (2017)2017:21 DOI 10.1186/s13662-017-1076-7 R E S E A R C H Open Access Solvability of a q-fractional integral equation arising in the study of an epidemic model Mohamed Jleli and Bessem Samet* *Correspondence: [email protected] Abstract Department of Mathematics, College of Science, King Saud We study the solvability of a functional equation involving q-fractional integrals. Such University, P.O. Box 2455, Riyadh, an equation arises in the study of the spread of an infectious disease that does not 11451, Saudi Arabia induce permanent immunity. Our method is based on the noncompactness measure argument in a Banach algebra and an extension of Darbo’s fixed point theorem. MSC: 62P10; 26A33; 45G10 Keywords: epidemic model; q-calculus; fractional order; measure of noncompactness 1 Introduction In this paper, we deal with the solvability of the q-fractional integral equation N t gi(t, x(t)) x(t)= f (t)+ (t – qs)(αi–)u s, x(s) d s , t ∈ I, (.) i (α ) i q i= q i where I = [, ], q ∈ (, ), αi >,fi : I → R,andgi, ui : I × R → R (i =,...,N). For N =,q =,andα =,equation(.) arises in the study of the spread of an infectious disease that does not induce permanent immunity (see [–]). Equation (.)canbewrittenas N αi · · ∈ x(t)= fi(t)+gi t, x(t) Iq ui , x( ) (t) , t I, i= αi where Iq is the q-fractional integral of order αi defined by [] t αi (αi–) ∈ Iq h(t)= (t – qs) h(s) dqs, t I.
    [Show full text]
  • On Some Fractional Quadratic Integral Inequalities
    KYUNGPOOK Math. J. 60(2020), 211-222 https://doi.org/10.5666/KMJ.2020.60.1.211 pISSN 1225-6951 eISSN 0454-8124 c Kyungpook Mathematical Journal On Some Fractional Quadratic Integral Inequalities Ahmed M. A. El-Sayed Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt e-mail : [email protected] Hind H. G. Hashem∗ Department of Mathematics, Faculty of Science, Qassim University, Buraidah, Saudi Arabia e-mail : [email protected] Abstract. Integral inequalities provide a very useful and handy tool for the study of qualitative as well as quantitative properties of solutions of differential and integral equa- tions. The main object of this work is to generalize some integral inequalities of quadratic type not only for integer order but also for arbitrary (fractional) order. We also study some inequalities of Pachpatte type. 1. Introduction and Preliminaries In the theory of differential and integral equations, Gronwall's lemma has been widely used in various applications since its first appearance in the article by Bell- man in 1943. In this article the author gave a fundamental lemma, known as Gronwall-Bellman Lemma, which is used to study the stability and asymptotic be- havior of solutions of differential equations. Gronwall's lemma has seen several generalizations to various forms [3, 11, 28]. The literature on these inequalities and their applications is vast; see [1, 2, 4] and the references given therein [19, 20, 23]. In addition, as the theory of calculus on time scales has developed over the last few years, some Gronwall-type integral inequalities on time scales have been established by many authors [17, 21, 24].
    [Show full text]
  • Quadratic Integral Equations in Reflexive Banach Space
    Discussiones Mathematicae 61 Differential Inclusions, Control and Optimization 30 (2010 ) 61{69 QUADRATIC INTEGRAL EQUATIONS IN REFLEXIVE BANACH SPACE Hussein A.H. Salem Faculty of Sciences Taibah University, Yanbu Saudi Arabia and Department of Mathematics Faculty of Sciences Alexandria University, Egypt e-mail: [email protected] Abstract This paper is devoted to proving the existence of weak solutions to some quadratic integral equations of fractional type in a reflexive Banach space relative to the weak topology. A special case will be considered. Keywords and phrases: Pettis integral, fractional calculus, fixed point theorem, quadratic integral equation. 2000 Mathematics Subject Classification: 26A33. 1. Introduction Prompted by the application of the quadratic functional integral equations to nuclear physics, these equations have provoked some interest in the liter- ature [2, 7] and [14]. Specifically, the so-called quadratic integral equations of Chandrasekher type can be very often encountered in many applications (cf. [2] and [8]). Some problems in the queueing theory and biology lead to the quadratic functional integral equation of fractional type (cf. e.g. [10] and [13]) (1) x(t) = h(t) + g(t; x(t))I αf(t; x(t)); t 2 [0; 1]; α > 0; 62 H.A.H. Salem where Iα denotes the standard (Riemann-Liouville) integral operator (cf. e.g. [15, 16] and [21]) and f is a real-valued function. The quadratic func- tional integral equation of type (1) has provoked some interest by many authors (see [6, 9] and [11] for instance). Similar problems are investigated in [4, 5]. In comparison with the existence results in the references, our assumptions ore more natural.
    [Show full text]
  • A Certain Integral-Recurrence Equation with Discrete-Continuous Auto-Convolution
    ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 245–250 A CERTAIN INTEGRAL-RECURRENCE EQUATION WITH DISCRETE-CONTINUOUS AUTO-CONVOLUTION Mircea I. Cîrnu Abstract. Laplace transform and some of the author’s previous results about first order differential-recurrence equations with discrete auto-convolution are used to solve a new type of non-linear quadratic integral equation. This paper continues the author’s work from other articles in which are considered and solved new types of algebraic-differential or integral equations. 1. Introduction In the earlier paper [4], N. M. Flaisher solved by Fourier transform method a second order differential-recurrence equation. The present author used in [2] the Laplace transform to derive Newton’s formulas about the sums of powers of the roots of a polynomial. In this paper, the Laplace transform will be used to solve an integral-recurrence equation on semi-axis, with discrete-continuous auto-convolution of its unknowns. Namely, applying the Laplace transform on considered equation, we obtain for the transforms of unknowns a first order differential-recurrence equation with discrete auto-convolution of the type studied in [3] and [1]. Using for this equation the general theory given in [3], we find the transforms of unknowns in convenient assumptions, the solutions of the initial equation being obtained by inverse Laplace transform. 2. Convolution products Two convolution products have been imposed over the time. The first, in continuous-variable case, is the bilateral convolution of two integrable functions u(x) and v(x) on real axis, given by formula Z ∞ u(x) ? v(x) = u(t)v(x − t) dt , −∞ 2010 Mathematics Subject Classification: primary 45G10; secondary 44A10.
    [Show full text]
  • Ovidiu Furdui Problems in Mathematical Analysis
    Problem Books in Mathematics Ovidiu Furdui Limits, Series, and Fractional Part Integrals Problems in Mathematical Analysis Problem Books in Mathematics Series Editors: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA For further volumes: http://www.springer.com/series/714 Ovidiu Furdui Limits, Series, and Fractional Part Integrals Problems in Mathematical Analysis 123 Ovidiu Furdui Department of Mathematics Technical University of Cluj-Napoca Cluj-Napoca, Romania ISSN 0941-3502 ISBN 978-1-4614-6761-8 ISBN 978-1-4614-6762-5 (eBook) DOI 10.1007/978-1-4614-6762-5 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013931852 Mathematics Subject Classification (2010): 11B73, 11M06, 33B15, 26A06, 26A42, 40A10, 40B05, 65B10 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer.
    [Show full text]