الفهرس 
Some Semi-Analytic MethodsOnly 14 pages are availabe for public view
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Abstract
A differential equation is a mathematical equation for an unknown function of one or
several variables that relates to the values of the function itself and its derivatives of various
orders. Differential equations play a prominent role in engineering, physics, economics, and
other disciplines. Differential equations are classified into two main classes: Ordinary differential
equations (ODES) and partial differential equations (PDES). Both ordinary and partial
differential equations are broadly classified as linear and nonlinear. In the analytical sense we
can only solve a small class of differential problems. Often a solution cannot be found in an
explicit or implicit form. In particular, the nonlinear differential equations, for that we can apply
numerical and semi analytic methods to obtain an approximate solution to the differential
problem. But the numerical methods give discontinuous points of a curve and thus it is often
costly and time consuming to get a complete curve of results and in some applications, we need
to get unknown function and its derivatives. Numerical difficulties additionally appear if
nonlinear differential equations contain singularities or have multiple solutions. For these reasons
we study in this thesis the semi-analytic methods to obtain an analytic approximate solutions
(series) to the differential equations.
The perturbation methods provide the most versatile tools available in nonlinear analysis
of engineering problems, but its limitations restrict its application [1, 2]:
• Perturbation method is based on assuming a small parameter. The majority of nonlinear
problems, especially those having strong nonlinearity, have no small parameters at all.
• The approximate solutions obtained by the perturbation methods, in most cases, are valid only
for small values of the small parameter.
Generally, the perturbation solutions are uniformly valid as long as a scientific system parameter
is small. The so-called non-perturbation techniques, such as differential transform method
(DTM)[3-5], Adomian’s decomposition method (ADM)[6-8], Variational iteration method
(VIM)[9] and others, are formally independent of small/large physical parameters. But, all of
these traditional non-perturbation methods cannot ensure the convergence of series solution: in
fact they are only valid for weak nonlinear problems [10].
The homotopy analysis method (HAM) is a general analytic approach to get series
solutions of various types of nonlinear equations, the validity of the HAM is independent of
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whether or not there exist small parameters in the considered equation. More important, it
provides a simple way to adjust and control the convergence of series solution. Therefore, the
HAM provides a more viable alternative to non-perturbation techniques such as the Adomian
decomposition method (ADM) and other techniques that cannot guarantee the convergence of
the series solution and may be only valid for weakly nonlinear problems [10]. In recent years;
this method has been successfully employed to solve many problems in science and engineering.
The variational iteration method (VIM) [9] cannot provide us with a simple way to adjust and
control the convergence region and the rate of obtain approximate series. For this reason there
strong motivation for authors to construct the variational iteration algorithms with an auxiliary
parameter, which proves an effective way to control the convergence region of an approximate
solution as [11-15] and others. This thesis consists of five chapters. A summary of the contents
of each one of these chapters is given below.
In chapter one, the procedures of some semi-analytical methods are explained and many
examples are solved to demonstrate the difference between different methods and demonstrate
that the homotpy analysis method (HAM) is a general analytic approach to get series solutions of
various types of nonlinear equations, the validity of the HAM is independent of whether or not
there exist small parameters in the considered equation. More important, it provides a simple
way to adjust and control the convergence of series solution.
In chapter two, an approach based on the homotopy analysis method (HAM) is proposed
to solve the second order two point boundary value problems for large/small values of important
parameters and finding the optimal value for the convergence-controller parameter. The
proposed approach has succeeded in solving the strongly nonlinear Bratu’s equation for different
value of important parameter, the nonlinear problems arising in heat transfer in a straight fins and
also for the problems with Neumann conditions.
In chapter three, an approach is proposed to reduce time consumption in the homotopy
analysis method for nonlinear initial and boundary value problems with strong nonlinear terms
like (sqrt root, exp, sinh, cos,…). The proposed approach has succeeded in some illustrative
examples and the sine-gordon equation. Also, the proposed approach is capable to predict and
find all branches of the solutions simultaneously for the first extension of Bratu problem.
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In chapter four, a combination between homotopy analysis method and Padé
approximation (HAM Padé) is introduced to solve the nonlinear boundary value problem with
one boundary condition at infinity. The proposed method is more general as compared to some
other methods such as HPM- Padé and ADM- Padé techniques. The proposed approach has
succeeded in solving some of the fluid mechanics problems and Thomas–Fermi equation.
In chapter five, an approach based on the variational iteration method (VIM) with an
auxiliary parameter is proposed to predict and find the multiple solutions of homogeneous
nonlinear ordinary differential equations with boundary conditions. The proposed approach is
capable of predicting and finding all branches of the solutions simultaneously. The proposed
approach successfully detects multiple solutions to a problem arising in mixed convection flows
in a vertical channel, the Bratu’s problem, the nonlinear model of diffusion and reaction in
porous catalysts and the nonlinear reactive transport model.
The numerical computation has been done by Mathematica program by PC, CPU
G620@2.60 GHz and 4GB of RAM.