الفهرس 
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Abstract
The primary objective of this thesis is to outline the application of Adomian decomposition method in tackling nonlinear fractional partial differential equations of Riesz type on unbounded domains. First contribution made in this thesis is presenting a new algorithm for the Adomian method when dealing with nonlinear terms. Convergence and error analysis are established. Adomian method is then applied to linear space-time fractional equations in Riesz sense. With the aid of Fourier integral, the exact solutions of the diffusion and wave problems are obtained. Theories of continuation of the solution to the time-fractional and integer order equations are proved, respectively. Finally, the method is applied to nonlinear case where applications are focused on problems having initial condition in the Fourier series form. Figures are presented for the series solution obtained. This thesis is arranged a follows. In chapter one, a brief look is taken to the history of fractional calculus and how it was developed. Definitions of main types of fractional derivatives and integrals are listed. A section is devoted for the definitions and relations of Weyl and Riesz type fractional derivatives with a literature review that includes recent publications. Chapter one is concluded with the main objective of this thesis. In chapter two, the Adomain decomposition method is illustrated. A new algorithm for solving nonlinear problems is suggested and proved to converge. The error estimate when the algorithm is applied to differential equation is deduced. The chapter is concluded by some numerical experiments to show the behavior of the new algorithm and compare it with standard one. Chapter three is devoted for the application of Adomian method to Linear fractional partial differential equations in Riesz sense. A definition for Riesz fractional derivative in Caputo sense on unbounded domains is presented. Afterwards, lemma is proved to illustrate the use of Fourier series or integral to obtain the exact solution. Theorems are proved for the exact solution continuation to the solution of fractional time and integer order cases, respectively. In chapter four, Adomian method is applied to nonlinear cases. Nonlinear diffusion problems, KS equation and SG equation are generalized to fractional. Figures indicate the efficiency of the method when comparing the solution obtained with the known analytic one for the second order case. In chapter five, we present conclusion remarks and suggest some points for future work.