A Comparative Study for Numerical and Analytical Solutions of Differential and Integro-Differential Equations of Fractional Order Using Different Techniques

Abstract

The use of numerical algorithms is an important area of research in pure mathematics (numerical analysis) for solving differential and integral equations of both ordinary and fractional orders. The purpose of this thesis, which consists of seven chapters, is to present a new modification and analysis of some numerical and analytical methods in order to solve differential and integral equations. Two principal developments are presented. First, a novel framework is introduced for solving differential and integral equations of ordinary order. This framework has the distinct advantage that it can handle a large variety of well- known differential and integral equations. This is accomplished using new modifications to some well-known numerical and analytical methods. The resulting framework admits, in a general fashion, the construction of those methods after modification gave better results and higher accuracy. Numerical examples involving differential and integral equations are introduced. Second, the manipulation of some numerical methods has resulted in algorithms that can actually solve fractional order differential equations. This approach has attached great importance to the modified methods by widening its effect in numerical analysis. A unified analytical framework is developed for establishing the convergence properties of a wide class of numerical methods for fractional differential equations. The accuracy of the proposed numerical methods, for the considered problems, is found to be in good agreement with the exact solutions. Due to the great effect of those modifications, some of these methods have approached the exact solution itself.