الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
The primary objective of this thesis is to propose some modifications of the classical homotopy perturbation method (HPM) for solving partial differential equations of arbitrary (fractional) order. These modifications are proposed to overcome some of the drawbacks in the use of the classical HPM.
Firstly, an effective algorithm is implemented to reduce the computational complexity of the HPM when dealing with nonlinear problems.
Secondly, when dealing with nonhomogeneous problems, an improved HPM is suggested to cancel the noise terms that appear in the classical HPM solution.
Finally, when dealing with initial-boundary value problems (IBVPs) over finite domains, a new technique that incorporates both types of conditions (boundary and initial) in the solution scheme is suggested. For this technique, we deduced the conditions that guarantee the convergence of the solution and presented an error estimate for the series solution.
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, properties and relations of some of the main types of fractional derivatives and integrals are listed. Also, we present a literature review that includes recent publications for approximate series solutions methods for FPDEs. Finally, the HPM is illustrated with its main technique for solving linear and nonlinear problems.
In chapter two, a reliable treatment of the HPM for solving nonlinear FPDE problems is implemented. This chapter is concluded by some numerical experiments to show the behavior of the algorithm.
Chapter three, an improved HPM that deals with nonhomogeneous problems is suggested. We propose a formula for series components that replace the nonhomogeneous term in the HPM system of equations. This technique is proposed first for integer order partial differential equations (PDEs), then it is generalized for FPDEs. This chapter is concluded by some numerical experiments to show the effectiveness of the new algorithm.
In chapter four, a new homotopy perturbation technique is suggested for solving fractional-order IBVPs over finite domains. The convergence analysis and error estimate of the new technique are discussed in abstract space.
In chapter five, we present conclusion remarks and suggest some points for future work.