Numerical Solutions of Fractional Volterra-Fredholm Integro-Differential Equations

Abstract

Fractional Volterra-Fredholm integro-differential equations (FVFIDEs) have gained increasing attention among researchers over the last decades because of their usefulness in modelling physical phenomena in engineering and technology. Although collocation, Homotopy Perturbation, Adomian Decomposition and related methods have been applied to these problems, many existing approaches involve high computational cost or reduced accuracy for difficult cases. This study therefore applied the Standard Collocation Method (SCM), Least Squares Method (LSM) and Akbari-Ganji Method (A-GM) to obtain numerical solutions of FVFIDEs. The general problem considered is given by:where   denotes the fractional-order derivative of y(x). Orthogonal polynomials were adopted as basic functions for the general class of FVFIDEs. Approximate solutions were assumed in polynomial form and substituted into the governing equations. The resulting equations were collocated at equally spaced interior points, producing systems of linear algebraic equations. These systems were solved for the unknown coefficients, which were then substituted back into the trial solutions to obtain the required approximations. Absolute estimated errors were computed to assess the accuracy and comparative performance of the methods. The findings showed that SCM, LSM and A-GM successfully solved both linear and nonlinear FVFIDEs. The computed absolute errors were small, indicating close agreement between the numerical and exact solutions. LSM and A-GM performed better than SCM, while the results obtained compared favourably with those reported in the literature. The study concluded that SCM, LSM and A-GM are reliable alternative methods for solving FVFIDEs and recommended their adoption for related problems and future studies.