Mathematics and Computational Sciences

On the numerical solution of Fredholm-type integro-differential equations using an efficient modified Adomian decomposition method

Document Type : Original Article

Authors

Kabiru Oyeleye Kareem 1
Morufu Oyedunsi Olayiwola 2
Muideen Odunayo Ogunniran 2

1 Department of Mathematics, Federal College of Education Iwo, Nigeria

2 Department of Mathematical Sciences, Osun State University Osogbo, Nigeria

https://doi.org/10.30511/mcs.2023.2009757.1137
Abstract
The efficiency of the Adomian decomposition method in the solution of integro-differential equations cannot be overemphasized. However, improvement of the method is needed as its drawbacks have been analyzed and reported in recent literature. This present work develops a new modification of the method and its implementation on linear Fredholm type of integro-differential equations. The approach is based on the modification of the traditional Adomian decomposition method. The idea employs the Taylor series expansion of the source term whose resulting functions were combined in two terms for predicting the solution in each iteration. This approach yields a very high accuracy degree when compared to related methods in literature. The newly proposed method is said to accelerates and converges faster than the standard Adomian Decomposition Method. The procedure proves to be concise, effective and converges faster to the true solution of linear Fredholm Integro-differential problems.

Keywords

Source term
Adomian decomposition method
Fredholm Integro-differential equations
Taylor series
infinite series

Subjects

[1] G. Adomian, R. Rach, On the solution of nonlinear differential equations with convolution product nonlinearities. Journal of Mathematics Analysis and Applications, 114(1) 1986, 171-175.
[2] G. Adomian, Solving frontier problem of Physics: The decomposition method, Kluwer Academic Publishers, Boston, MA 1994.
[3] Y. Aygar, E. Bairamov, Series Expansion, Asymptotic Behavior and Computation of the Values of the Schwarzschild-Milne Integrals Arising in a Radiative Transfer, Appl. Comput. Math., 20(2) 2021, 236-246.
[4] Chriscella Jalius, Zamariah Abdul Majid. Numerical solution of second order Fredholm integro-differential equations with Boundary conditions by Quadrature-difference method. Hindawi Journal of Applied Mathematics, 2, Article ID 2645097, 2017, 1–5.
[5] A. Gollbabai, M.A. Javidi, numerical approach for solving two-dimensional linear and non-linear Volterra-Fredholm integral equations. Appl. Math. Comput. 190(2) 2007, 1053-1061.
[6] C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux transformation in solitons theory and geometry application., Shanghai Science Technology Press, Shanghai, 1994.
[7] R. Hirota, Direct methods in solition theory, Springer, Berlin, 1980.
[8] D.A. Juraev, A. Shokri, D. Marian, On an approximate solution of the Cauchy problem for systems of equations of elliptic type of the first order. Entropy, 24(7) 2022, 1-18.
[9] K. Kareem, M. Olayiwola, A. Oladapo, A. Yunus, K. Adedokun, I. Alaje,, On the solution of Volterra integro differential equations using a modified Adomian decomposition method, Jambura Journal of Mathematics, 5(1) 2023. 133-138.
[10] A. U. Keskin, Adomoian Decomposition Method (ADM), Boundary Value Problems for Engineers, 2019, 311-359.
[11] D. Marian, S.A. Ciplea, N. Lungu, On Ulam–Hyers stability for a system of partial differential equations of first order, Symmetry 2020, 12, 1060.
[12] F. Mohd-Karim, Mahathir Mohamad, Mohd Saifullah Rusiman, Norziha Che-Him, Rozaini Roslam, Kamil Khalid, Adomian Decomposition method for solving linear second order Fredholm integro-differential equations, Journal of Physics: Conference Series, 2018, 995 012009.
[13] H.K. Musaev, The Cauchy problem for degenerate parabolic convolution equation, TWMS J. Pure Appl. Math., 12(2) 2021, 278- 288.
[14] M.O. Olayiwola, K. O. Kareem, Efficient Decomposition Method for Integro differential Equations. J. Math. Comput. Sci., 2022, 1-15.
[15] P.J. Oliver, Application of Lie group to differential equation. Springer, 1986. New York.
[16] P.S Pankov, Z. K. Zheentaeva, T. Shirinov, Asymptotic reduction of solution space dimension for dynamical systems, TWMS J. Pure Appl. Math., 12(2) 2021, 243-253.
[17] Rohul Amin, Ibrahim Mahariq, Kamal Shah, Muhammad Awais, Fahim El-Sayed, Numerical Solution of the Second Order Linear and Non-Linear Integro-Differential Equations using Haar Wavelet Method, Arab Journal of Basic and Applied Sciences, 28(1) 2021, 11–19.
[18] J.C. Rosales, M.B. Branco. Algorithms for Calculating the Set of Integers Contained in a Monoid Finitely Generated by Positive Rational Numbers, Appl. Comput. Math., 20(3) 2021, 381-389.
[19] A. Shokri, H. Saadat, A.R, Khodadadi, A new high Order closed Newton-Cotes trigonometrically fitted formulae for the numerical solution of the Schrödinger equation, Iran. J. Math. Sci. Inform., 13(1) 2018, 111-129.
[20] A. Shokri, H. Saadat, A.R. Khodadadi, A new high order closed Newton-Cotes trigonometrically fitted formulae for the numerical solution of the Schorodinger equation, Iran J. Math. Sci. Inform, 13(1) 2018, 111 – 129.
[21] J. Sunday, A. Shokri, J.A. Kwanamu, K. Nonlaopon, Numerical integration of stiff differential systems using non-fixed step-size strategy. Symmetry, 14(8) 2022, 1575.
[22] J. Sunday, A. Shokri, R.O. Akinola, K.V. Joshua, K. Nonlaopon, A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations, Results in Physics, 42 2022, 106031.
[23] A.M. Wazwaz, S.M. El-Sayed, A new modification of Adomian decomposition for linear and nonlinear operators, Applied Mathematics and Computation, 122(3) 2001, 393-405.
Mathematics and Computational Sciences
Volume 4, Issue 4
Autumn 2023
Pages 39-52

  • Receive Date 22 August 2023
  • Revise Date 11 December 2023
  • Accept Date 22 December 2023

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