A Modified Semi-Analytical Method Using the Shehu Transform for Solving Fractional Volterra Integro-differential Equations

Background: Finding an analytical solution to Volterra integro-differential equations (VIDEs), especially nonlinear types, often poses serious difficulties and is many times impossible, thus the need to provide a semi-analytical solution. Objective: This research focuses on the solutions of systems of linear and nonlinear fractional-order integro-differential equations with difference kernels. To achieve this, we exploited the advantage of integral transforms and one of the existing semi-analytical methods to develop the desired method of solution. Methods: One of the recently developed integral transforms, the Shehu transform, which generalizes Laplace and Sumudu transforms, is systematically integrated into the well-known Adomian Decomposition Method (ADM) to obtain a simplified approach to solving the class of problems considered. The Shehu transform is first applied to both sides of the given VIDEs with difference kernels, followed by the application of the convolution theorem. The ADM is then employed to handle the nonlinearities encountered. Results: The proposed method, the Modified Semi-analytical Method (MSM), is applied to selected problems in the literature and produces comparatively good results. The method also produces the exact solution whenever the solution is in closed form. The results are presented in tabular and 2D or 3D graphical forms for easy comparison. All computations are carried out using Mathematica 13.3, with the fractional-order derivative interpreted in the Caputo sense. Conclusions: Since MSM has been successfully used to solve linear and nonlinear VIDEs with difference kernels, the scope of the method can be expanded to cover Volterra-Fredholm integro-differential equations (VFIDEs) in future studies.