NUMERICAL IMPLEMENTATION FOR SOLVING THE BOUNDARY VALUE PROBLEM FOR IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS WITH PARAMETER
DOI:
https://doi.org/10.26577/JMMCS2023v119i3a2Keywords:
boundary value problem, parametrization method, integro-differential equation with parameter, impulsive effect, numerical algorithmAbstract
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.
References
Halanay. A., Wexler. D., "Qualitative Theory of Impulse Systems", (ditura Academiei Republici Socialiste Romania, Bucuresti, 1968)
Lakshmikantham V., Bainov D.D., Simeonov P.S., "Theory of Impulsive Differential Equations", (World Scientific,
Singapore, (1989)
Samoilenko A.M., Perestyuk N.A., "Impulsive Differential Equations" , (World World Scientific, Singapore (1995)
Nieto J.J., Rodrigues-Lopez R., "New comparison results for impulsive integro-differential equations and applications", Journal of Mathematical Analysis and Applications, 328, (2007): 1343–1368. https://doi.org/10.1016/j.jmaa.2006.06.029
Sousa J., Oliveira D., Oliveira E., "On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation" , Mathematical Methods in the Applied Sciences, 42(4), (2018): 1249-1261. https://doi.org/10.1002/mma.5430
Akhmetov M.U., Zafer A., Sejilova R.D., "The control of boundary value problems for quasilinear impulsive integro- differential equations" , Nonlinear Analysis, 48(2), (2002): 271-286.
He Z., He X., "Monotone iterative technique for impulsive integrodifferential equations with periodic boundary conditions" , Computers and Mathematics with Applications, 48, (2004): 73-84. https://doi.org/10.1016/j.camwa.2004.01.005
Luo Z., Nieto J.J., "New results for the periodic boundary value problem for impulsive integro-differential equations", Nonlinear Analysis: Theory and Methods and Applications, 70, (2009): 2248-260. https://doi.org/10.1016/j.na.2008.03.004
Wang X., Zhang J., "Impulsive anti-periodic boundary value problem for first order integro-differential equations" , Journal of Computational and Applied Mathematics , 234, (2010): 3261-3267. https://doi.org/10.1016/j.cam.2010.04.024
Dzhumabaev D.S. "Solvability of a linear boundary value problem for a fredholm integro-differential equation with impulsive inputs" , Differential Equations, 51(9), (2015): 1180-1196. https://doi.org/10.1134/S0012266115090086
Assanova A.T., Bakirova E.A., Kadirbayeva Zh.M. and Uteshova R.E., "A computational method for solving a problem with parameter for linear systems of integro-differential equations", Comp. Appl. Math., 39, (2020): 248. https://doi.org/10.1007/s40314-020-01298-1
Bakirova E.A., Assanova A.T., Kadirbayeva Zh.M., "A problem with parameter for the integro-differential equations", Mathematical Modelling and Analysis, 26, (2021): 34–54. https://doi.org/10.3846/mma.2021.11977
Dzhumabaev D.S., "Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation" , USSR Comput. Math. Math. Phys., 1, (1989): 34–46. https://doi.org/10.1016/0041-5553(89)90038-4
Dzhumabaev D.S. "On one approach to solve the linear boundary value problems for fredholm integro- differential equations", Journal of Computational and Applied Mathematics, 294(2), (2016): 342-357. https://doi.org/10.1016/j.cam.2015.08.023
Dzhumabaev D.S. "New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems", Journal of Computational and Applied Mathematics, 327, (2018): 79–108. https://doi.org/10.1016/j.cam.2017.06.010
