### jzs-10751

The Weighted Residual Method in Solution of More General Nonlinear Integro-Fractional Differential Equation of Volterra Hammerstein Type with Variable Coefficients

Shazad Shawki Ahmed1 &   Mariwan Rashid Ahmed2

Department of Mathematics, College of Science,  Sulaimani University, Kurdistan Region, Iaq
Department of Science, College of Basic Education, Charmo University, Kurdistan Region, Iaq

Original: 19 March 2019, Revised: 5 May 2019, Accepted: 21 May 2019, Published online:  20 June 2019

Abstract

In this paper, we solve numerically Volterra-Hammerstein (V-H) Integro-Differential Equations (IDE’s) of various fractional order  in the Caputo sense by the well-known (Collocation, Sub-domain, Least-square, and Galerkin) weighted residual method. This method is utilized to converts the integro-fractional differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown polynomial coefficients. A good algorithm for treating numerically our problem by applying the above process has been developed, in order to express these solutions, programs are written in MatLab (V.8). The validity and reliability of this method are tested by several illustrative numerical examples with the known exact solution, the behaviors of the error are examined, and the obtained results reveal that the method is more accurate and efficient.

Key Words:  Integro-Fractional Differential Equation, Caputo derivative, Collocation, Sub-domain , Least square, Galerkin

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