الفهرس 
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Abstract
ABSTRACT:
The study of numerical methods for the solution of partial differential equations has been on intense activity over the last 40 years both from theoretical and practical point of view. Many of the partial differential equations arising from engineering and scientific applications were previously intractable. But since after improvements in numerical techniques along with the rapid advances in computer technology, those problems can now be routinely solved
With the availability of symbolic computation packages like Maple or Mathematical, the search for obtaining exact solutions of nonlinear partial differential equations (PDEs) has become more and more stimulating for mathematicians and scientists. Having exact solutions of nonlinear PDEs makes it possible to study nonlinear physical phenomena, thoroughly and facilitates testing the numerical solvers as well as aiding the stability analysis of solutions.
In the last few decades active research efforts were focused on nonlinear dynamical systems that emerge in various fields, such as fluid mechanics, plasma physics, biology, hydrodynamics, solid-state physics and optical fibers.
These nonlinear phenomena are often related to nonlinear wave equations. In order to better understand these phenomena as.
Well, as further apply them in practical scientific research, it is important to seek their exact solutions. Many powerful methods have been developed for this purpose, such as, Hirota’s Backlund transformation, Hirota’s bilinear method, Darboux transformation, symmetry method, inverse scattering transform, tanh method, Sine–Cosine Method, Adomian Decomposition Method.
The material of this thesis is divided into three chapters as follows:
Chapter (1): collects some preliminaries, notations and definitions and basic concepts and relevant results to the thesis, as (Partial differential equation, Ordinary differential equation, Nonlinear equations, Initial value problem, Boundary value problem, Fractional differential equations, Riemann–Liouville Fractional Integral Operator, The Caputo time fractional derivative, Contraction mapping, Fixed point, Lebesgue integrable)
Chapter (2): In this chapter, we apply Adomian Decomposition Method to derive the solution of a system of fractional partial differential equations.
Theorem 1 (Uniqueness theorem). Eq. (1.3) has a unique solution whenever
0<𝛼<1,where,𝛼=(𝐿1+𝐿2)𝑡𝑘 𝑘!.
Theorem 2 (convergence theorem): The solution (1.5) of Eq. (1.3) using ADM converges if 0<𝛼<1 and |𝑢1|<∞ .
Chapter (3): In this chapter, we will examine traveling wave solution by using Sine, Cosine and Tanh Function algorithm for nonlinear partial differential equations.
The methods are used to obtain the exact solutions of differential equations.
a- The Sine-Cosine Method.
Consider the nonlinear partial differential equation in the form 𝐹(𝑢 ,𝑢𝑡 ,𝑢𝑥 ,𝑢𝑦 ,𝑢𝑡𝑡 ,𝑢𝑥𝑥 ,𝑢𝑦𝑦 ,𝑢𝑥𝑦 ,…… )=0
where 𝑢(𝑥,𝑦,𝑡) is a traveling wave solution of nonlinear partial differential equation above equation we use the transformation
𝑢(𝑥,𝑦,𝑡)=𝑓(𝜉)
where 𝜉=𝑥+𝑙𝑦+𝑐𝑡
This enables us to use the following changes
𝜕𝜕𝑥(.)=𝑑𝑑𝜉(.) ,𝜕𝜕𝑦(.)=𝑙𝑑𝑑𝜉(.) , 𝜕𝜕𝑡(.)=𝑐𝑑𝑑𝜉
To transfer the nonlinear partial differential equation to nonlinear ordinary differential equation.
𝑄(𝑓 ,𝑓′ ,𝑓′′ ,𝑓′′′ ,…..)=0
The ordinary differential equation Q is then integrated as long as all terms contain derivatives, where we neglect the integration constants. The solutions of many nonlinear equations can be expressed in the form.
ABSTRACT
𝑓(𝜉)={𝜆 𝑠𝑖𝑛𝛽(𝜇𝜉) , |𝜉|≤𝜋2𝜇0 , 𝑜𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒}
Or in the form 𝑓(𝜉)={𝜆 𝑐𝑜𝑠𝛽(𝜇𝜉) , |𝜉|≤𝜋2𝜇0 , 𝑜𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒}
where 𝜆 ,𝜇 and 𝛽 are parameters to be determined, 𝜇 and 𝑐 are the wave number and the wave speed, respectively [63], we use 𝑓(𝜉) =𝜆 𝑠𝑖𝑛𝛽 (𝜇𝜉) 𝑓′(𝜉)= 𝑑𝑓(𝜉)𝑑𝜉=𝜆𝛽𝜇 𝑠𝑖𝑛𝛽−1(𝜇𝜉)𝑐𝑜𝑠(𝜇𝜉) 𝑓′′(𝜉)= 𝑑2𝑓(𝜉)𝑑𝜉2=𝜆𝜇2𝛽 (𝛽−1)𝑠𝑖𝑛𝛽−2(𝜇𝜉)−𝜆𝜇2𝛽2 𝑠𝑖𝑛𝛽(𝜇𝜉)
And their derivative…,
Or use 𝑓(𝜉)=𝜆 𝑐𝑜𝑠𝛽 (𝜇𝜉)
𝑓′(𝜉)= 𝑑𝑓(𝜉)𝑑𝜉=−𝜆𝛽𝜇 𝑐𝑜𝑠𝛽−1(𝜇𝜉)𝑠𝑖𝑛(𝜇𝜉)
𝑓′′(𝜉)= 𝑑2𝑓(𝜉)𝑑𝜉2=−𝜆𝛽𝜇2 𝑐𝑜𝑠𝛽(𝜇𝜉)+𝜆𝜇2𝛽(𝛽−1) 𝑐𝑜𝑠𝛽−2(𝜇𝜉)
−𝜆𝜇2𝛽(𝛽−1)𝑐𝑜𝑠𝛽(𝜇𝜉)
⋮
and so on.
b- Tanh method.
The Tanh method introduced by Malfiet [69], and Wazwaz [5,6]
ABSTRACT
The method is applied to find out an exact solution of a coupled system of nonlinear, any partial differential equation (PDE) with unknown:
We consider a nonlinear equation, say in two variables 𝑃(𝑢(𝑥,𝑡) ,𝑢𝑡 ,𝑢𝑥𝑥 ,𝑢𝑡𝑡 ,𝑢𝑥𝑥𝑥 ,…….)=0
where 𝑃 is a polynomial of the variable u and its derivatives.
If we consider 𝑢(𝑥,𝑡)=𝑓(𝜉), and 𝜉=𝑥−𝑐𝑡,
where c is the speed of the wave, we can use the following changes:
𝜕𝜕𝑥=𝑑𝑑𝜉 ,𝜕 𝜕𝑡=−𝑐𝑑𝑑𝜉
And so on, then becomes an ordinary differential equation (ODE) 𝑄 (𝑓(𝜉),𝑓′,𝑓′′,𝑓′′′,……)=0
With Q being another polynomial form, which will be called the reduced ordinary differential equation.
It is based on a priori assumption that the traveling wave solutions can be expressed in terms of the Tanh function to solve a nonlinear equation.
For the Tanh method, we introduce the new independent variable
𝑌(𝑥,𝑡)=𝑡𝑎𝑛ℎ (𝜉)
That leads to the change of variables:
𝑑𝑑𝜉=(1−𝑌2)𝑑𝑑𝑌
𝑑2𝑑𝜉2=−2𝑌(1−𝑌2)𝑑𝑑𝑌+(1−𝑌2)2𝑑2𝑑𝑌2
𝑑3𝑑𝜉3=2(1−𝑌2)(3𝑌2−1)𝑑𝑑𝑌−6𝑌(1−𝑌2)𝑑2𝑑𝑌2+(1−𝑌2)3𝑑3𝑑𝑌3
𝑑4𝑑𝜉4=8𝑌(1−𝑌2)(2−3𝑌2)𝑑𝑑𝑌−4(1−𝑌2)2(2−9𝑌2)𝑑2𝑑𝑌2+(1−𝑌2)4𝑑4𝑑𝑌4− 12𝑌(1−𝑌2)3𝑑3𝑑𝑌3
And the remaining derivatives may be derived similarity.
ABSTRACT
Then the solution is expressed in the form of Tanh Method.
𝑢(𝑥,𝑡)=𝑓(𝜉)=Σ𝑎𝑖𝑌𝑖𝑚𝑖=0=𝑎0+𝑎1𝑌+𝑎2𝑌2+⋯+𝑎𝑚𝑌𝑚
In order to construct more general, it is reasonable to introduce the following ansatz. Extended Tanh Method.
𝑓(𝜉)=Σ𝑎𝑖𝑌𝑖𝑚𝑖=−𝑚=𝑎−𝑚𝑌−𝑚+⋯+𝑎0+𝑎1𝑌+⋯.+𝑎𝑚𝑌𝑚
where m is a positive integer which is unknown to be later determined.
𝑎𝑖 is unknown constants and where m can be found by balancing the highest-order linear term with the nonlinear terms in the above equation and 𝜆, 𝑎0 , 𝑎1 ,…, 𝑎𝑚 are to be determined.