# NUMERICAL IMPLEMENTATION FOR SOLVING THE BOUNDARY VALUE PROBLEM FOR IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS WITH PARAMETER

## Authors

• E. Bakirova Institute of Mathematics and Mathematical Modeling, Kazakhstan, Almaty and Kazakh National Women’s Teacher Training University, Kazakhstan, Almaty
• N. Iskakova Institute of Mathematics and Mathematical Modeling, Kazakhstan, Almaty and Abai Kazakh National Pedagogical University, Kazakhstan, Almaty
• Zh. Kadirbayeva Institute of Mathematics and Mathematical Modeling, Kazakhstan, Almaty and International Information Technology University, Kazakhstan, Almaty

120 204

## Keywords:

boundary value problem, parametrization method, integro-differential equation with parameter, impulsive effect, numerical algorithm

## Abstract

In this paper, a linear boundary value problem under impulse effects for the system of Fredholm integro-differential equations with a parameter is investigated. The purpose of this research is to provide a method for solving the studied problem numerically. The ideas of the Dzhumabaev parameterisation method, classical numerical methods of solving Cauchy problems and numerical integration techniques were used as a basis for achieving the goal. When applying the method of parameterisation by points of impulse effects, the interval on which the boundary value problem is considered is divided, additional parameters and new unknown functions are introduced. As a consequence, a problem with parameters equivalent to the original problem is obtained. According to the data of the matrices of the integral term of the equation, boundary conditions and impulse conditions, the SLAE with respect to the introduced parameters is compiled. And the unknown functions are found as solutions of the initial-special problem for the system of integro-differential equations. A numerical algorithm for finding a solution to the boundary value problem for impulse integro-differential equations with a parameter is constructed. Numerical methods for solving Cauchy problems for ODE and calculating definite integrals are used for numerical implementation of the constructed algorithm. Numerical calculations are verified on test problem.

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