Solution of Fractional Differential Equations: Transform and Iterative Methods Approach

Abstract

Fractional calculus is a novel and highly active area of research in the literature as it has a wide spectrum of applications in the physical sciences and engineering. In this thesis, we study the numerical solution of fractional differential equations subject to initial or boundary conditions. We employ two iterative methods for the solution of such equations. First, we implement a Laplace decomposition method (LDM) which is a combination of two methods: Laplace transform and a decomposition scheme. The nonlinear term is decomposed and a recursive algorithm is composed for the determination of the proposed infinite series solution. Second, we present another iterative technique that is based on the incorporation of Green’s functions into well-established fixed-point iterations, including Krasnoselskii-Mann’s and Picard’s schemes. Numerical experiments are conducted to demonstrate the efficiency, accuracy and applicability of the proposed methods, and then, to compare them with other schemes. In addition, we present graphs to understand the behavior of the numerical solutions. A patching strategy, based on domain decomposition, is suggested to overcome a deficiency of the LDM. Fractional differential equations are widely investigated by several researchers.