الفهرس 
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Abstract
Abstract
This thesis consists of an introduction (English and Arabic) and four chapters. The objective of this thesis is discussing the existence and the uniqueness of the solutions of some fractional and linear partial integro differential equations with its applications in some mathematical physics problems as heat equations and wave equations. Besides the famous numerical methods, Laplace homotopy perturbation method and Adomian decomposition method are used to calculate the numerical solutions.
Chapter 1:
In this chapter, some famous definitions and theorems will be needed in other chapters. In addition, basic concepts of fractional calculus and partial differential equations are considered. Moreover, the continuing and boundedness of differential and integral operators are discussed. Finally, classification of PDEs and its some analytic methods for the following PDEs are considered.
F(x,y,...,u,ux,uy,...,uxx,uxy,...) = G(x,y).
Chapter 2:
In this chapter, existence and uniqueness of the following fractional linear integro partial differential equation with evolution kernel is discussed using Modified Bielecki method.
,
under initial condition
u(x,0) = u0(x).
In addition, LHPM is used to obtain the solution numerically in the space CE(E × [0,T]). Moreover, Numerical results are calculated and the error is computed.
Chapter 3:
In this chapter, existence and uniqueness of solutions for the following initial fractional and partial integro differential equations (F-PIDEs) of heat type are discussed using semi-group method.
First: we study the following initial value problem of fractional partial integro differential equation
,
with condition
u(x,0) = u0(x).
Second: we study the following initial value problem of partial integro differential equation of heat type
,
with condition
u(x,0) = g(x).
Also, stability of solutions for F-PIDEs of heat type are discussed. In addition, ADM is used in solving F-PIDEs of heat type numerically.
Chapter 4:
In this chapter, existence and uniqueness of solutions for the following initial value problem for F- PIDEs of wave type are discussed. First: we study the following initial value problem of fractional partial integro differential wave equation
,
with conditions
u(x,0) = f(x), ut(x,0) = g(x).
Second: we study the following initial value problem of partial integro differential wave equation
with conditions
u(x,0) = r(x), ut(x,0) = ϕ(x).
In addition, ADM is used to calculate the numerical solutions of FPIDEs of wave type on a finite domain.
Our results in this thesis are concerned in four papers, two had accepted [5,6] and others are submitted [2,3].