الفهرس 
Chapter 1: IntroductionOnly 14 pages are availabe for public view
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Abstract
In this thesis, we study some important methods such as the Lie symmetry and the invariant subspace method (ISM) to solve various type of differential difference equations (DΔEs). This thesis is organized as follows : In chapter one, an introduction is briefly presented about the basic concepts needed in our study. Firstly, the Lie symmetry method is introduced to solve ordinary differential equations (ODEs) and ordinary difference equations (OΔEs). Secondly, some definitions of fractional derivatives are showed with some basic important special functions and transformations that needed to solve fractional differential difference equation (FDΔEs). Finally, the ISM is submitted to solve partial differential equations (PDEs), differential-difference equations (DΔEs) and FDΔEs. In chapter two, the nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index is investigated. Using Lie symmetry method, some exact soliton solutions are generally obtained in the case when the coefficients of the Schrödinger’s equation are non-zero constants. Some new dark and bright soliton solutions are obtained. Also, we obtained some new peakon, kink and soliton-like solutions. Also, Lakshmanan-Porsezian-Daniel model (LPD) is investigated. Using travelling wave transformation, the LPD is transformed into two nonlinear ordinary differential equations which are proved to be equivalent under certain condition. The obtained solutions for LPD model are in the form of Jacobi elliptic functions. In Chapter three, simple transformation is used to reduce the order of the two fifth order OΔEs then we applied the Lie symmetry method to obtain exact solutions. Also, the Lie symmetry methods are considered to find exact solutions of the second order difference system in the general form. The compatible canonical coordinates are obtained to reduce the system to single equation that can be solved by a simple transformation. In the third application, the separation of variables and the Lie symmetry methods are combined as a first approach to find some new exact solutions of the fractional discrete KdV equation. After separation of variables, the fractional discrete KdV equation is decomposed into two simpler equations. The first equation is a OΔE which is solved using the Lie point symmetry method. The second equation is a FDE with 𝜓−Riemann-Liouville fractional derivative which can be solved using the compositional method. In chapter four, the ISM is used to solve FDΔEs. Exact solutions are obtained for fractional discrete KdV by using ISM. Also, Semi-discrete M-fractional derivative complex coupled dispersionless system is suggested. The properties of M-fractional derivative are utilized to convert discrete M-fractional derivative system to classical discrete differential system. Then ISM is applied to find dark, bright, kink and W-shaped soliton solutions. In chapter five, the conclusion and suggested future work are presented.