SOLVABILITY AND NUMERICAL SOLUTIONS OF SYSTEMS OF NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND WITH PIECEWISE CONTINUOUS KERNELS

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1 MSC 45D5 SOLVABILITY AND NUMERICAL SOLUTIONS OF SYSTEMS OF NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND WITH PIECEWISE CONTINUOUS KERNELS I.R. Muftahov, Irkutsk National Technical University, Irkutsk, Russia D.N. Sidorov, Energy Systems Institute of Siberian Branch of Russian Academy of Sciences, Irkutsk National Technical University, Irkutsk State University, Irkutsk, Russia The existence theorem for systems of nonlinear Volterra integral equations with piecewise continuous kernels of the first kind is proved. Such equations model evolving dynamical systems. A numerical method for solving nonlinear Volterra integral equations of the first kind with piecewise continuous kernels is proposed using midpoint quadrature rule. Also numerical method for solution of systems of linear Volterra equations of the first kind is described. The examples demonstrate efficiency of proposed algorithms. The accuracy of proposed numerical methods is ON 1. Keywords: Volterra integral equations, discontinuous kernel, ill-posed problem, evolving dynamical systems, quadrature, Dekker-Brent method. 1. Problem Statement and Existence Theorem In this paper we study systems of nonlinear Volterra integral equations of the first kind with piecewise continuous kernels using our previous results [1, 2, 3, 5], where theory of linear scalar Volterra integral equations of the first kind with piecewise continuous kernels have been addressed. Let us consider the following system of nonlinear Volterra equations of the first kind Kt, s, xs ds = ft, s t T, f =. 1 We make the following assumptions: A vector-function Kt, s, xs is defined for < x <, s t T and has a discontinuities of the first kind at the curves s = α i t, i = 1,..., n 1, i. e. Kt, s, xs = K 1 t, sg 1 s, xs, t, s D 1, K n t, sg n s, xs, t, s D n, D i = {t, s αi 1 t < s < α i t}, i = 1,..., n, a t =, a n t t, ft = f 1 t,..., f m t, xt = x 1 t,..., x m t, m m matrices K i are defined, continuous and have continuous derivatives w.r.t. t in the corresponding domains D i, f j t, j = 1,..., m, and α i t, i = 1,..., n 1 have continuous derivatives, < α 1... α n 1 < 1, f j =, α i =, < α 1 t < α 2 t <... < α n 1 t < t for t, T ]; B matrices K i, i = 1,..., n have continuously differentiable w.r.t. t continuation into compact { s t T }. 2??, том?,? 1

2 I. R. Muftahov, D. N. Sidorov For the theory of systems of linear integral equations with piecewise continuous kernels readers may refer to monograph [5]. The objective of this paper is to generalize this theory on nonlinear case and suggest algorithm for numerical solution for such systems. It is to be noted that Volterra equations with piecewise continuous kernels are employed in energetics and in other fields for evolving dynamical systems modeling. Here readers may refer to bibliography in [2, 4]. Let us first formulate the sufficient conditions for existence and uniqueness of the solution of system 1. Theorem 1. Let conditions A and B be fulfilled for s t T. Let Lipchitz condition G i s, x 1 s G i s, x 2 s x 1 s x 2 s q i x 1 x 2, x 1, x 2 R n, be also n 1 fulfilled and q n + α i Kn, 1 K i, K i+1, 1 + qi < 1. Then τ > such as system 1 has unique local solution in C [,τ]. Moreover, if min τ t T t α n 1t = h > then such local solution can be continuously extended on the whole domain [τ, T ] using combination of the method of steps and successive approximations. Proof of this theorem is similar with the proof of Theorem 3.2 in the monograph [2] using the Lipchitz condition. 2. Numerical Solution In this section we construct an algorithm based on of the midpoint quadrature for numerical solution of nonlinear Volterra integral equations 1. We also included the numerical examples to demonstrate the efficiency of the proposed scheme. For sake of clarity we first consider the scalar case. To construct a numerical solution of the equation 1 on the interval [, T ] we introduce the mesh not necessarily uniform = t < t 1 < t 2 <... < t N = T, h = maxt i t i 1 = ON 1. 2,N Following our paper [1] we search an approximate solution of equation 1 as following piecewise constant function x N t = N x i δ i t, t, T ], δ i t = { 1, for t i = t i 1, t i ];, for t / i 3 with undetermined yet coefficients x i, i = 1, N. In order to find x let us differentiate both sides of equation 1 w.r.t. t f t = αi t α i 1 t K i t, s G i s, xs ds+ t +α itk i t, α i tg i α i t, xα i t α i 1tK i t, α i 1 tg i α i 1 t, xα i 1 t. 2 Вестник ЮУрГУ. Серия Математическое моделирование и программирование

3 Then f = α i K i, G i, x α i 1K i, G i, x. The desired coefficient x included nonlinearly in the right hand side. We use the Van Wijngaarden- Dekker-Brent method to find this coefficient x see e.g. [6]. Below we shall use the notationf k = ft k, k = 1,..., N, and v ij denotes number of mesh s segment 2 such as value α i t j belongs to, i.e. α i t j vij. Obviously v ij < j for i =, n 1, j = 1, N. Let us now assume the coefficients x, x 1,..., x k 1 of approximate solution be known. αi t k Equation 1 in the knot t = t k is K i t k, sg i s, xs ds = f k, and it can be α i 1 t k presented as I 1 t k + I 2 t k + + I n t k = f k, where I 1 t k = I n t k = v 1,k 1 j=1 vn 1,k α n 1 t k K 1 t k, sg 1 s, xs ds + K n t k, sg n s, xs ds + 1. If v p 1,k v p,k, p = 2,..., n 1 then I p t k = vp 1,k α p 1 t k K p t k, sg p s, xs ds + + αpt k t vp,k 1 2. If v p 1,k = v p,k, p = 2,..., n 1 then I p t k = α1 t k k t v1,k 1 j=v n 1,k +1 v p,k 1 j=v p 1,k +1 K 1 t k, sg 1 s, xs ds, K n t k, sg n s, xs ds. K p t k, sg p s, xs ds. αpt k α p 1 t k K p t k, sg p s, xs ds+ K p t k, sg p s, xs ds. It is to be noted that number of terms in each row of this formula depends on array v ij, which can be determined based on input data: functions α i t, i = 1, n 1 and fixed based on concrete N mesh. Each integral in the last equation can be approximated using the midpoint quadrature, i.e. αpt k t m K p t k, sg p s, xs ds α p t k t m K p t k, s k G p s k, x N s k, where m = v p,k 1, s k = αpt k+t m 2. Moreover, on the intervals where the desired function is determined we select x N t i.e. t t k 1. On the rest of the intervals the unknown value x k appears in the last few terms. We determine the number of such terms using the array v ij analysis. In order to find x k we use the Van Wijngaarden-Dekker-Brent method and proceed with computation of x k+1. The accuracy of proposed method is ε = max i N xt i x N t i, where xt i and x N t i are exact and approximate solution in t i correspondingly. The method has order of O 1 N. 2??, том?,? 3

4 I. R. Muftahov, D. N. Sidorov Let us now consider the numerical solution of the following system of linear integral equations Kt, sxs ds = ft, s t T, f =, 4 where m m matrix kernel Kt, s on the compact s t T has 1st kind discontinuities on the curves s = α i t, i = 1,..., n 1. An approximate solution of system 4 we again search as piecewise constant function x N t = x N 1 t,..., x N m t, where x N i t = N j=1 { x N 1, for t i,j δ j = t jt, t, T ], δ j t = j 1, t j ];, for t / j. 5 with coefficients x N i,j, i = 1, m, j = 1, N. Let us introduce the following mesh of knots = t < t 1 < t 2 <... < t N = T, h = maxt i t i 1 = ON 1 6,N for numerical solution of the system 4 on the interval [, T ]. We make the following notation f i,k = f i t k, k = 1,..., N, K i,p t k, s is i, p-th entry of matrix Kt k, s, i = 1, m, p = 1, m. This is the function with 1st of discontinuities on curves α 1 t k < α 2 t k <... < α n 1 t k, i. e. k K i,p t k, sx p s ds = q=1 αqt k α q 1 t k K i,p q t k, sx p s ds. 7 Let us differentiate both sides of equation 4 w.r.t. t in order to find x N αqt f it K q i,p t, s = x p s ds+ t +α itk i,p q q=1 α q 1 t t, α i tx p α i t α i 1tK q i,p t, α i 1 tx p α i 1 t Since all the components x p of the desired vector x can be assumed constant, we gain the following system m f i = K q i,p, α q α q 1 x N p,. 8 q=1 We solve the system of linear algebraic equations 8 and determine x N p, for p = 1, m. Let coefficients x N p,, x N p,1,..., x N p,k 1 be known. We rewrite system 4 in the point t = t k. For ith row we have k t k 1 K i,p t k, sx p s ds = f i t k k 1 t K i,p t k, sx p s ds. 4 Вестник ЮУрГУ. Серия Математическое моделирование и программирование.

5 Taking into account 5 we gain k K i,p t k, s ds x N p,k t k 1 = f i,k k 1 j=1 K i,p t k, s ds x N p,j. Finally taking into account 7 we gain the following system of linear algebraic equations q=1 αqt k α q 1 t k K q i,p t k, s ds x N p,k = f i,k k 1 j=1 q=1 αqt k α q 1 t k K q i,p t k, s ds x N p,j, 9 which we solve and determine coefficients x N p,k for p = 1, m. 3. Numerical Examples 1. /8 1 + t s sin xs ds + /4 t 12 sin xs ds + t/8 t/4 1sin 2 xs + 1 ds = = t + 7t 8 cos t t sin t 8 2t sin t sin t sin 2t, t [, 2]. 4 The exact solution is xt = t. Tab. 1 demonstrates errors ε for various step sizes. Errors for various h example of nonlinear equation h 1/32 1/64 1/128 1/256 1/512 1/124 1/248 ε,345567,178316,9159,4717,23892,1231,637 Table 1 2. /4 + t/2 2 + ts, 1 ts 1 + t + s, 1 ts 1 + t + s, 1 1 1, 1+t x1 s ds + x 2 s x1 s ds = x 2 s /2 t/4 1 t t + s, t + s s, t + s 1+t t t t4 55t x1 s ds+ x 2 s 17t t t4 55t , t where exact solution is xt = 3 /8 t 2, t [, 2]. Tab. 2 demonstrates errors ε i = max x it j x N i t j, i = 1, 2, where x i t j and x N i t j are correspondingly exact j N and computed solutions in the knots t j for various step sizes. 2??, том?,? 5

6 I. R. Muftahov, D. N. Sidorov Table 2 Errors for various h example of systems h 1/32 1/64 1/128 1/256 1/512 1/124 1/248 ε 1,5425,28218,14392,7267,3652,183,916 ε 2,127717,64389,32327,16196,816,455, Conslusion In this brief paper we continue our studies of Volterra integral equations of the first kind with discontinuous kernels [1] [5]. We further develop the numerical methods [1] and suggest methods for solutions of systems of such linear equations and nonlinear equations. The midpoint quadrature rule is employed and the error order is O1/N. The numerical method is evaluated on synthetic data and demonstrated uniform T -convergence [7] of the sequence {x N t i } N to solution xt with rate O1/N. The authors are thankfull to Dr. A. N. Tynda for valuable comments and discussions of the results presented in this article. The second author is partly supported by the International science and technology cooperation program of China and Russia under Grant No. 215DFA758. References 1. Sidorov D.N., Tynda A.N., Muftahov I.R. Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel. Vestnik YuUrGU. Ser. Mat. Model. Progr., 214, vol.7, no. 3, pp Sidorov D.N. Integral Dynamical Models: Singularities, Signals and Control. World Scientific Series on Nonlinear Science, ser. A, vol. 87, Singapore, World Sc. Publ., 215, 26 p. 3. Sidorov D.N. Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels. Vestnik YuUrGU. Ser. Mat. Model. Progr., 212, no. 12, pp Markova E.V., Sidorov D.N. On One Integral Volterra Model of Developing Dynamical Systems. Automation and Remote Control, 214, vol. 75, no 3, pp Sidorov D.N. Solution to Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels. Russian Mathematics, 213, vol. 57, pp Brent R.P. An algorithm with guaranteed convergence for finding a zero of a function. The Computer Journal, 1971, vol. 14, no 4, pp Trenogin V.A. Functional analysis. Moscow, Fizmatlit, Вестник ЮУрГУ. Серия Математическое моделирование и программирование

7 УДК РАЗРЕШИМОСТЬ И АЛГОРИТМ ЧИСЛЕННОГО РЕШЕНИЯ СИСТЕМЫ НЕЛИНЕЙНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ ВОЛЬТЕРРА I РОДА С КУСОЧНО-НЕПРЕРЫВНЫМИ ЯДРАМИ И. Р. Муфтахов, Д. Н. Сидоров Литература Доказана теорема существования и разработан численный метод решения систем нелинейных интегральных уравнений Вольтерра первого рода с кусочнонепрерывными ядрами, возникающих в моделировании развивающихся динамических систем. В качестве квадратурной формулы используется метод средних прямоугольников, при этом решение ищется в виде кусочно-постоянной функции. Для решения нелинейного уравнения использован комбинированный метод Дэккера и Брэнта. Приведены результаты расчетов для скалярного нелинейного уравнения и для систем линейных уравнений. Точность предложенных численных методов O1/N. Ключевые слова: интегральные уравнения Вольтерра I рода; развивающиеся системы; разрывное ядро; нелинейные системы; численные методы; метод Дэккера и Брэнта; квадратурной формулы. 1. Сидоров, Д.Н. Численное решение интегральных уравнений Вольтерра I рода с кусочнонепрерывными ядрами/ Д.Н. Сидоров, А.Н. Тында, И.Р. Муфтахов // Вестн. ЮУрГУ. Сер. Матем. моделирование и программирование Том С Sidorov, D. Integral Dynamical Models: Singularities, Signals and Control. World Scientific Series on Nonlinear Science Series A: Vol. 87 / D. Sidorov. Singapore: World Sc. Publ., 215, 26 p. 3. Сидоров, Д. Н. О семействах решений интегральных уравнений Вольтерры первого рода с разрывными ядрами / Д.Н. Сидоров // Вестн. ЮУрГУ. Сер. Матем. моделирование и программирование С Markova, E.V. On One Integral Volterra Model of Developing Dynamical Systems / E.V. Markova, D.N. Sidorov // Automation and Remote Control V. 75, 3. P Sidorov, D.N. Solution to Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels / D.N. Sidorov // Russian Mathematics V. 57. P Brent, R.P. An algorithm with guaranteed convergence for finding a zero of a function / R.P. Brent // The Computer Journal. 1971, V. 14,. 4, P Треногин, В.А. Функциональный анализ / В.А. Треногин. М: Физматлит, 22. Муфтахов Ильдар Ринатович, аспирант кафедры Вычислительной техники, Иркутский национальный исследовательский технический университет г. Иркутск, Россия. Сидоров Денис Николаевич, доктор физико-математических наук, профессор кафедры Вычислительной техники, Иркутский национальный исследовательский технический университет, Иркутский государственный университет, Институт систем энергетики им. Л.А. Мелентьева СО РАН г. Иркутск, Российская Федарация, contact.dns@gmail.com. Поступила в редакцию 2??, том?,? 7