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Abstract Thesis title: “The Numerical Treatment of Fractional order Partial Differential Equations” Fractional order partial differential equations (FPDE) are considered as one of the recent subjects although the concepts of fractional calculus are old as the beginning of classical calculus with Newton’s and Leibentz. The thesis in general considers two models of the main categories: the elliptic type class (Poisson’s equation) and the parabolic class (diffusion equation). In this thesis we consider the fractional order counterparts of these models with addition of an integral coefficient in the parabolic case (integro-differential equation). Also, we considered the finite difference method to approximate the partial differential equations by an algebraic systems. Because of the special properties of the resultant algebraic systems many authors like Young and Evans have considered the algebraic structures in the integer order case; we have considered these structures in the fractional order Poisson’s equation. Also, we considered the method of lines combined with a spectral method based on shifted Chebyshev polynomials. This thesis consists of four chapters as follows: Chapter1:Fractional Calculus: Fundamentals and Motivations In this chapter, a brief historical review about the starting and the developments of fractional calculus and its applications in many fields is considered. Also a review of important definitions and theorems to provide subsequent chapters are established. General formulas which have been modified and used in the numerical solution of fractional order differential equations are given. Chapter2: Boundary Value Problems in the Plane from Integer to Fractional Orders: In this chapter the algebraic systems which are the final step in the finite difference treatment of fractional elliptic differential equations in two independent variables are deeply studied. Poisson’s equation and its fractional counterpart represents the model problem for this chapter. The structures of the coefficient matrices of the algebraic systems corresponding to two different methods of labeling grid points (Lexicographical ordering and the Chequerboard ordering) are given. Differences between the systems of algebraic equations resulting from the finite difference treatment of the classical integer case and the corresponding fractional order are considered. from the structures of the coefficient matrices of the fractional orders we illustrated the memory and hereditary properties of the fractional order derivatives. The spectral properties of the iteration matrices of the successive overrelaxation (SOR) and its variant KSOR are considered. Precondition technique is discussed with application to the resultant algebraic systems. Two types of precondition are applied to the resultant algebraic systems appear in the Lexicographical ordering. Also, we introduced a precondition for the system appears in the Chequer-board ordering. Comparison of the spectral radius of the iteration matrices and the selection of the relaxation parameters for the SOR and the KSOR in the two systems after and before applying the precondition is made. To assist in clarifying what has been reached, programs are designed with (MATLAB 7.10), the results are introduced in terms of tables and figures illustrating the relations between the fractional orders and the relaxation parameters. Chapter3: Fractional Diffusion Equation The numerical solution of the simple parabolic PDE or integro-differential equation is considered with the essential properties of the numerical method (Finite difference method). Fractional derivatives with respect to time only, space only, space and time, and space and time with existence of integral terms are considered. Explicit and implicit finite difference schemes with their conditions of stability are established. We used the direct way method and the shifted Grünwald-Letnikov (G-L) approach in approximating the fractional derivatives. Also, we used the trapezoidal method in approximating the integral terms. The matrix structures for the implicit and explicit schemes are established. Also, the conditions of stability are discussed in comparison with the classical integer cases. The results of running Matlab programs (MATLAB 7.10) for three examples are illustrated in terms of tables and graphs. Chapter4: The Method of Lines Combined With Chebyshev Spectral Method In this chapter one of the weighted residual methods (WRM), the collocation method, is used with the shifted Chebyshev polynomials of the first kind in approximating the sequential fractional space boundary value problems results from using the finite difference method in approximating the fractional time derivatives. Also, an algorithm is introduced. Application of the algorithmic steps for solving fractional partial differential and integro-differential equations is considered. Graphs and tables to illustrate the behavior of the solution are introduced. |