Solution of Volterra integral equations of the first kind with discontinuous kernels by using the Adomian decomposition method

Document Type : Research Paper

Authors

1 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran

2 Department of Mathematics, Tabriz Branch Islamic Azad University, Tabriz, Iran

3 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran

Abstract

In this paper, for solving linear and nonlinear Volterra integral equations of the first kind with discontinuous kernels, the Adomian decomposition method, and the modified Adomian decomposition method are presented. We convert  the  main equation into the  Volterra integral equation of the second kind. The numerical examples are given to denote the accuracy of the suggested method.

Keywords

References

  • [1] S. Aghaei Amirkhizi, Y. Mahmoudi, and A. Salimi Shamloo, Legendre polynomials approximation method for solving Volterra integral equations of the first kind with discontinuous kernels, Indian J. Pure Appl. Math., 53 (2022) 492-504.
  • [2] P. M. Anselone, Uniform approximation theory for integral equations with discontinuous kernels, SIAM J. Numer. Anal., 4 (1967), 245–253.
  • [3] I. L. El-kalla, Convergence of Adomian’s method applied to a class of Volterra type integro-differential equations, Int. J. Differ. Equ. Appl., 10 (2005), 225–234.
  • [4] N. Hritonenko and Y. Yatsenko, Mathematical modeling in economics, ecology and the environment, 2nd Edition, Springer 2013.
  • [5] C. Kasumo and E. Moyo, Approximate solutions of nonlinear Volterra integral equations of the first kind, Appl. Math. Sci., 14 (2020), 867-880.
  • [6] A. P. Khromov, Integral operators with discontinuous kernels, Doklady Akademii Nauk, 406 (2006), 317–321.
  • [7] I. R. Muftahov and D. N. Sidorov, and numerical solutions of systems of nonlinear Volterra integral equations of the first kind with piecewise continuous kernels, Bulletin of the South Ural State University. Series-Mathematical Modelling, Programming and Computer Software, 9 (2016), 130-136.
  • [8] I. Muftahov, A. Tynda, and D. Sidorov, Numeric solution of Volterra integral equations of the first kind with discontinuous kernels, J. Comput. Appl. Math., 313 (2017), 119-128.
  • [9] S. Noeiaghdam, D. Sidorov, and V. Sizikov, Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method, Appl. Comput. Math., 19 (2020), 87-105.
  • [10] S. Noeiaghdam, D. Sidorov, A. M. Wazwaz, N. Sidorov, and V. Sizikov, The numerical validation of the Adomian decomposition method for solving Volterra integral equation with discontinuous kernels using the CESTAC method, Mathematics., 9 (2021), 260.
  • [11] D. Sidorov, Volterra integral equations of the first kind with jump discontinuous kernels, INV Follow-up Meeting Isaac Newton Institute for Mathematical Sciences (Feb 10-14, 2014, Cambridge, UK), 2014.
  • [12] D. N. Sidorov, Existence and blow-up of Kantorovich principal continuous solutions of nonlinear integral equations, Diff. Equat., 50 (2014), 1217–1224.
  • [13] D. Sidorov, Integral dynamical models: singularities, signals and control, In: L. O. Chua ed. World Scientific Series on Nonlinear Sciences Series A: Vol. 87, World Scientific Press, Singapore 2015.
  • [14] N. Sidorov, D. Sidorov, and A. Sinitsyn, Toward general theory of differential-operator and kinetic models, World Scientific Series on Nonlinear Science; Series A: Monographs and Treatises, 97; World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, USA 2020.
  • [15] N. A. Sidorov and D. N. Sidorov, On the solvability of a class of Volterra operator equations of the first kind with piecewise continuous kernels, Math. Notes., 96 (2014), 811–826.
  • [16] D. N. Sidorov, On parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel, Differ. Equ., 49 (2013), 210–216.
  • [17] D. Sidorov, I. Muftahov, N. Tomin, D. Karamov, D. Panasetsky, A. Dreglea, F. Liu, and A. Foley, A dynamic analysis of energy storage with renewable and diesel generation using Volterra equations, IEEE Trans. Ind. Inform., 16 (2019), 3451-3459.
  • [18] D. Sidorov, Q. Tao, I. Muftahov, A. Zhukov, D. Karamov, A. Dreglea, and F. Liu, Energy balancing using charge/discharge storages control and load forecasts in a renewable-energy-based grids, Proc. of 38th China Control Conference (July 27-30, 2019, Guangzhou, China), IEEE., (2019), 1-6.
  • [19] D. Sidorov, A. Zhukov, A. Foley, A. Tynda, I. Muftahov, D. Panasetsky, and Y. Li, Volterra models in load leveling problem, E3S Web Conf., 69 (2018), 01015.
  • [20] A. Tynda, S. Noeiaghdam, and D. Sidorov, Polynomial spline collocation method for solving weakly regular Volterra integral equations of the first kind, The Bulletin of Irkutsk State University. Series Mathematics, 39 (2022), 62–79.
Computational Methods for Differential Equations
Volume 12, Issue 1
January 2024
Pages 189-195

Files

History

  • Receive Date: 30 July 2022
  • Revise Date: 22 May 2023
  • Accept Date: 07 June 2023

Share

How to cite

Statistics