A computational method for solving a problem with parameter for linear systems of integro-differential equations
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A computational method for solving a problem with parameter for linear systems of integro-differential equations Anar T. Assanova1,2 Roza E. Uteshova1,3
· Elmira A. Bakirova1,2 · Zhazira M. Kadirbayeva1,3 ·
Received: 27 January 2020 / Revised: 2 July 2020 / Accepted: 8 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical examples. Keywords Problem with parameter · System of integro-differential equations · Solvability criteria · Algorithm · Computational method Mathematics Subject Classification 34B08 · 34H05 · 45J05 · 34K28
Communicated by Antonio José Silva Neto. This research is supported by Ministry of Education and Science of Republic Kazakhstan Grant No. AP 05132455. The project leading to this application has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 873071.
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Anar T. Assanova [email protected]; [email protected] Elmira A. Bakirova [email protected] Zhazira M. Kadirbayeva [email protected] Roza E. Uteshova [email protected]
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Institute of Mathematics and Mathematical Modeling, 125, Pushkin Str., 050010 Almaty, Kazakhstan
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Institute of Information and Computational Technologies, 125, Pushkin Str., 050010 Almaty, Kazakhstan
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International Information Technology University, 34A, Jandossov Str., 050008 Almaty, Kazakhstan 0123456789().: V,-vol
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1 Introduction Control problems, also referred to as boundary value problems with parameters or as parameter identification problems, for ordinary differential and integro-differential equations have been extensively studied by many authors Akhmetov et al. (2002); Alimhan et al. (2015); Dauylbayev and Atakhan (2015); Dauylbaev and Mirzakulova (2017); Kiguradze (1987); Luchka and Nesterenko (2008); Nesterenko (2014); Ronto and Samoilenko (2000). Various methods have been applied to study these problems, such as methods of qualitative theory of differential equations, the calculus of variations and optimization theory, the method of upper and lower solutions, etc. However, there still remain open problems in obtaining effective criteria for the unique solvability of such problems and in developing numerical algorithms to find their optimal solutions. Consider the following problem with a parameter for a system of Fredholm integrodifferential equations with dege
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