الفهرس 
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Abstract
Fractional calculus (FC) generalizes the classical integral and differential calculus by allowing arbitrary orders of integration and differentiation. Fractional partial differential equations (FPDEs) are generalizations of classical partial differential equations (PDEs). FPDEs [1] and [2] are frequently able to better model real phenomena than strictly integer ordered PDEs. So this topic has received a great deal of attention in many fields of science and engineering, including fluid flow, viscoelasticity materials, electrochemical processes, dielectric polarization, colored noise, anomalous diffusion, signal processing, biology, electrical networks, electromagnetic theory and probability.
The most vital advantage of using fractional differential equations (FDEs) in these and other applications is their non-local property. It is well known that the integer order differential operator is a local operator, but the fractional order differential operator is non-local. This means that the state of a system depends not only upon its current state, but also upon all of its historical states. This makes the study fractional order systems a lively area of research. FPDEs have proven to describe better certain dynamics of real world phenomena. They have attacked the attentions of a vast community of researchers. The drawback is that it is very difficult to find exact solutions to them in many situations, so many analytical and numerical methods are used to solve different types of FDEs. There exists a large number of analytical and numerical methods which are developed in order to be able to solve them without the need of physically unrealistic assumptions, linearization, perturbation and a lot of computational works.
The Aim of the Study
This thesis is dedicated to proposing some analytical and numerical methods to solve some physical fractional partial differential equations on a finite domain. These equations play a vital role in various physical flow problems in fluid dynamics, propagation of shallow water waves and the chemical reaction diffusion model. These models can be represented by the following equations:
(i) The time-fractional Burgers’ equation (TFBE), which is represented by the equation,
D_t^α u(x,t)+γu(x,t) D_x u(x,t)=μD_x^2 u(x,t), (1)
subject to the conditions
u(x,0)=f(x), o<x<L, (2)
and
u(0,t)=ϕ_1 (t), u(L,t)=ϕ_2 (t), o<t<T. (3)
where D_x=∂/∂x, D_t^α=∂^α/(∂t^α ), D_x^2=∂^2/(∂x^2 ), 0<α≤1, (x,t)∈[0,L]×[0,T], u,x,t and μ are the velocity, spatial, time coordinates and kinematic viscosity, respectively.
(ii) The time-space and time fractional coupled Burgers’ equations in one and two dimensional spaces respectively represent by the equations:
1. One- dimensional time-space fractional coupled Burgers’ equations (TSFCBEs)
■(D_t^(α_1 ) u(x,t)&=D_x^(β_1 ) u(x,t)+2u(x,t)D_x^(α_3 ) u(x,t)-D_x (u(x,t)v(x,t)),@D_t^(α_2 ) v(x,t)&=D_x^(β_2 ) v(x,t)+2v(x,t)D_x^(α_4 ) v(x,t)-D_x (u(x,t)v(x,t)).) (4)
2. Two- dimensional time fractional coupled Burgers’ equations (TFCBEs)
■(D_t^(α_1 ) u(x,y,t)+u(x,y,t)D_x u(x,y,t)&+v(x,y,t)D_y u(x,y,t)@=&1/Re [D_x^2 u(x,y,t)+D_y^2 u(x,y,t)],@D_t^(α_2 ) v(x,y,t)+u(x,y,t)D_x v(x,y,t)&+v(x,y,t)D_y v(x,y,t)@=&1/Re [D_x^2 v(x,y,t)+D_y^2 v(x,y,t)],) (5)
where a≤x,y≤b, t>0, D_x^2=∂^2/(∂x^2 ) and D_y^2=∂^2/(∂y^2 ). The factors α_i,β_i, 0<α_1,α_2,α_3,α_4≤1 and 1<β_1,β_2≤2, denote to the order of the fractional time and space derivatives, respectively. Re is the Reynolds number, u(x,y,t) and v(x,y,t) are the velocity components along the x- and y-axes. It is worth noting that when at least one of the factors varies, different reaction systems are obtained.
(iii) The inhomogeneous linear system of time-FPDE, which is represented by the equations in the form:
■(D_t^α u(x,t)-D_x v(x,t)-u(x,t)+v(x,t)=-2,@D_t^α v(x,t)+D_x u(x,t) -u(x,t)+v(x,t)=-2,) (6)
where a≤x≤b,t>0 and the fractional derivative order α∈(0,1].
With the initial conditions
u(x,0)=1+e^x, v(x,0)=-1+e^x (7)
(iv) The inhomogeneous nonlinear system of time-FPDEs
■(D_t^α u(x,t)-v(x,t)D_x (u(x,t))-u(x,t)=1,@D_t^α v(x,t)-u(x,t)D_x (v(x,t))-v(x,t)=1,) (8)
with the initial conditions
u(x,0)=e^x, v(x,0)=e^(-x), (9)
where 0<α≤1, (x,t)∈[a,b]×[0,T].
The fractional derivatives in these equations are defined in the Caputo sense.
The proposed methods based on the Laplace Adomian decomposition method (LADM), Laplace variational iteration method (LVIM), reduced differential transform method (RDTM) and shifted Gegenbauer collocation method (SGCM). These methods provide an accurate approximate solutions. These solutions which take the form of convergent series with easily computable terms, without the need for physically unrealistic assumptions, linearization, perturbation and massive computation. The aim of this thesis is to extend the applications of the suggested practical methods to the initial-boundary value FPDEs (1) - (9) on a finite domain and introduce analytical and numerical studies to these tested problems. Comparing these methodologies with each others and with some known techniques in the existing literature is another objective of our study in this dissertation.
Thesis Structure
The thesis is planned as follows:
In Chapter 1: We provide a brief overview of the history of fractional calculus, its importance and its applications in different fields. Some necessary definitions of fractional integration, differentiation, special functions and some fractional calculus’s rules are presented. We compute the fractional differentiation and integration of some elementary functions. Also, we will discuss and give a brief survey of the suggested methods which are LADM, LVIM, RDTM and SGCM. Finally, some physical fractional partial differential equations such as the time- fractional Burgers’ equation (TFBE), the one and two- dimensional coupled Burgers’ equations with time- and space-fractional derivatives (TSFCBEs) and fractional linear and nonlinear system with time-fractional derivatives are presented.
In Chapter 2: We explain the LADM, LVIM, RDTM and SGCM and show how to use each of them to solve the nonlinear FPDE. These methods are applied to the time fractional Burgers’ equation (TFBE). Also, the convergence analysis of analytical methods and SGCM are debated. Figures and tables are used to show the efficiency as well as the accuracy of the approximate results achieved by the proposed techniques.
In Chapter 3: The solutions of LADM and LVIM are derived for the system of linear and nonlinear time FPDEs. Numerical results of the two examples are graphically illustrated and comparisons are held between the two methods and the exact solution. The figures and tables are used to show the validity, effectiveness and accuracy of the achieved approximate results.
In Chapter 4: LADM, LVIM and RDTM are directly extended to study and derive explicit analytical approximate solutions of the one and two- dimensional TSFCBEs. Numerical studies of the applications of these approaches for a number of sample problems are given and are illustrated graphically. The numerical results are shown to detect the effectiveness and the correctness of the proposed methods. Also, they compared with those previously reported in the literature.
In Chapter 5: We report a simple and accurate spectral collocation technique to solve the one and two- dimensional TSFCBEs in which the fractional derivatives are defined according to Caputo’s definition. The suggested method is based on the shifted Gegenbauer polynomials (SGPs) for approximating the solution of TSFCBEs. Numerical examples are provided to show the efficiency and accuracy of the proposed method. The obtained numerical results are compared with those previously reported in the literature.
In Chapter 6: We present some concluding remarks on the works developed in this dissertation, including some suggestions for future research.