الفهرس 
Chapter1: General IntroductionOnly 14 pages are availabe for public view
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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.