الفهرس | Only 14 pages are availabe for public view |
Abstract The main objective of this thesis is to introduce an analytical and numerical treatments based on the Adomain decomposition method (ADM), Homotopy analysis method (HAM) and Variational iteration method (VIM) for linear and non-linear fractional partial differential equations. The linear time – fractional Fokker-Planck equation represents the linear fractional partial differential equations and the non-linear time-fractional Huxley equation represents the non-linear fractional partial differential equations. The obtained solutions are calculated in the form of rapidly convergent series with easily computable components.The accuracy of the proposed method is demonstrated by several test problems and we make comparison between these methods. Our studies in this thesis are considered in the following four chapters as follow: In chapter one, the historical survey for fractional calculus is considered. Some basic definitions and properties of the fractional calculus theory are considered and studied. Also, the general properties of fractional derivatives and integrals are illustrated and studied. Some fractional integrals and derivatives of few elementary functions are calculated. Finally, the analysis of the three methods which are ADM, HAM and VIM respectively are presented and some physical fractional partial differential equations such as Fokker-Planck equation and Huxley equation are illustrated and studied. In chapter two, we propose ADM and VIM to obtain analytic solution for linear Fokker-Planck equation with time-fractional derivative. The obtained solutions are calculated in the form of rapidly convergent series with easily v computable components. The ADM is extended to derive an analytical solution of linear time – fractional Fokker-Planck equation. Also an analytical approximation solution of linear time - fractional Fokker-Planck equation is obtained by using the VIM. Two examples are presented to show the efficiency and simplicity of these methods. Figures are used to show the efficiency as well as the accuracy of the approximate results achieved, finally the conclusions of the obtained results. In chapter three, ADM and VIM are directly extended to study the nonlinear Huxley equation with time-fractional derivative. As a result, the explicit and numerical solutions are obtained in the form of rapidly convergent series with easily computable components. The application of ADM is extended to derive an analytic solution of non-linear time – fractional Huxley equation. Also the application of VIM is extended to derive an analytic solution nonlinear time – fractional Huxley equation. The numerical solution of nonlinear time – fractional Huxley equation obtained is to show the efficiency and the simplicity of these methods and figures are used to show the efficiency as well as the accuracy of the approximate results achieved, finally the conclusions of the obtained results are followed. In chapter four, we adopt HAM to obtain analytic solutions of linear fractional partial differential equations (Fokker-Planck equation) and nonlinear fractional partial differential equations (Huxley equation) with timefractional derivative. As a result the realistic numerical solutions are obtained in the form of rapidly convergent infinite series with easily computable components. The HAM is applied for solving both linear time –fractional Fokker-Planck equation and non-linear time – fractional Huxley equation. The numerical results of linear time – fractional Fokker-Planck equation and non-linear time – fractional Huxley equation is used to show the efficiency and the simplicity of these methods. Also, figures show the effectiveness and accuracy of the proposed method, finally the conclusions of the obtained results are followed. |