Abstract
Background:Finding an analytical solution to Volterraintegro-differential equations (VIDEs), especially nonlinear types, often posesserious difficulties and is many times impossible, thus the need to provide asemi-analytical solution.Objective:This research focuses on the solutionsof systems of linear and nonlinear fractional-order integro-differential equa-tions with difference kernels. To achieve this, we exploited the advantage ofintegral transforms and one of the existing semi-analytical methods to developthe desired method of solution.Methods:One of the recently developed inte-gral transforms, the Shehu transform, which generalizes Laplace and Sumudutransforms, is systematically integrated into the well-known Adomian Decom-position Method (ADM) to obtain a simplified approach to solving the classof problems considered. The Shehu transform is first applied to both sidesof the given VIDEs with difference kernels, followed by the application of theconvolution theorem. The ADM is then employed to handle the nonlinearitiesencountered.Results:The proposed method, the Modified Semi-analyticalMethod (MSM), is applied to selected problems in the literature and producescomparatively good results. The method also produces the exact solution when-ever the solution is in closed form. The results are presented in tabular and2D or 3D graphical forms for easy comparison. All computations are carriedout using Mathematica 13.3, with the fractional-order derivative interpretedin the Caputo sense.Conclusions:Since MSM has been successfully usedto solve linear and nonlinear VIDEs with difference kernels, the scope of themethod can be expanded to cover Volterra-Fredholm integro-differential equa-tions (VFIDEs) in future studies
Keywords
Shehu transform; Integro-differential equations; Fractionalderivatives; Difference kernel; Adomian polynomials