# THE MODIFIED DECOMPOSITION METHOD FOR SOLVING LINEAR SECOND-ORDER FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

#### Authors:

Anas Al-Haboobi,Ghassan A. Al-Guaifri,

#### DOI NO:

https://doi.org/10.26782/jmcms.2020.05.00005

#### Keywords:

MDM,Integro-differential Equations,Fredholm integral Equation,approximate solutions,

#### Abstract

This paper applies Modifed Decomposition Method (MDM) as numerical analysis linear second-order FredholmIntegro-differential Equations. The calculation of the approximate solutions are computed by mathematical package. The main aim of this paper is to demonstrate how effective this method minimizes the size of calculations and reaching the final solution in the shortest time and best result. When com paring the results with the (ADM) and with the exact solution, we will note how effective this method minimizes the size of calculations of the solution and reaches the exact solution. Accordingly, the (MDM) is the best method to be used to solve linear second-order FredholmIntegro-Differential equation. The convertion to the exact solution is notably fast and also a time saver, as it requires less computational work in solving equations. This is why the (MDM) is more efficient in solving this kind of equations.

#### Refference:

I. A. Mohsen, M. El-Gamel. “A Sinc–Collocation method for the linear Fredholmintegro-differential equations”, in Zeits chrift fürangewandte Mathematik und Physik, pp.380-390, 2007.

II. A.M Wazwaz, “A reliable modification of Adomian decomposition method”, in Applied Mathematics and Computation, pp.77-86, 1999.

III. D.D Bainov, M.B Dimitrova, A.B Dishliev, “Oscillation of the bounded solutions of impulsive differential-difference equations. of second order”, in Applied Mathematics and Computation, pp.61-68, 2000.

IV. E. Aruchunan, J. Sulaiman, “Numerical Solution of First-Order Linear FredholmIntegro-Differential Equations using Conjugate Gradient Method”, in International Symposium on Geology, pp.11-13, 2009.

V. E. Aruchunan, J. Sulaiman, “Numerical solution of second-order linear fredholmintegro-differential equation using generalized minimal residual method”, in American Journal of Applied Science, pp.780-783,2010.

VI. H. Safdari, Y.E Aghdam, “Numerical Solution of Second-Order Linear FredholmIntegro-Differetial Equations by Trigonometric Scaling Functions”, in Open Journal of Applied Sciences, pp.135-144, 2015.

VII. M. Gülsu, M. Sezer, “A Taylor polynomial approach for solving differential-difference equations”, in Journal of Computational and Applied Mathematics, pp.349-364, 2006.

VIII. M. Fathy, M. El-Gamel, M.S El-Azab, “Legendre–Galerkin method for the linear Fredholmintegro-differential equations”, in Applied Mathematics and Computation, pp.789-800, 2014.

IX. M.FKarim, M. Mohamad, M.S Rusiman, N. Che-Him, R.Roslan, K. Khalid, “ADM For Solving Linear Second-Order FredholmIntegro-Differential Equations”, in Journal of Physics: Conference Series, pp.012009, 2018.

X. S. Yalçinbaş, M. Sezer,“The approximate solution of high-order linear Volterra–Fredholmintegro-differential equations in terms of Taylor polynomials”,in Applied Mathematics and Computation, pp.291-308, 2000.

XI. S.M Hosseini, S. Shahmorad,“Tau numerical solution of Fredholmintegro-differential equations with arbitrary polynomial bases”, in Applied Mathematical Modelling,pp.145-154, 2003.