An analytical solution to the fractional Fredholm–Volterra Integro-differential equation using the limit residual function technique

Abstract

The primary objective of this study is to develop an analytic solution for mixed integrodifferential equations of fractional order, which are commonly applied in the mathematical modeling of various physical phenomena. This work introduces a novel approach based on the limit of the residual function, resulting in a convergent series expansion for the solutions. The new method has the advantage of quickly determining the coefficients of the series solution and the limited calculations required compared to other methods. The article presents and discusses various applications to validate the theoretical findings. These applications cover three types of fractional integrodifferential equations: the Fredholm integrodifferential equation, the Volterra integrodifferential equation, and the Fredholm-Volterra integrodifferential equation. The results demonstrate a strong agreement between the exact and approximate solutions. A key feature of this proposed method is that it does not necessitate any unreasonably restrictive assumptions, such as perturbation, linearization, or guessing initial data. This makes it a practical tool for directly solving nonlinear fractional problems.