الفهرس 
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Abstract
The aim of this thesis is to study a special type of PDEs called ”Sine Gordon (SG) and Double Sine Gordon (DSG) equations”. For this purpose, a nonuniform implicit Finite Difference scheme with variable spatial step and constant time step is proposed. This proposed scheme is investigated theoretically as well as numerically by solving some specific different problems of wave equations that exhibit ”Soliton” solutions, e.g. the SG and DSG equations with known exact solutions named as ”Kink and Anti-kink”. This results in a tridiagonal system, which is solved by using Thomas Algorithm. Also, the same problems are also solved numerically by applying the uniform classical scheme, to investigate how far the proposed nonuniform scheme is adequate to the SG and DSG problems.
Emphasis is made on the finite difference method with the different classical and proposed schemes, such as their stability, convergence and consistency.
The interaction of Solitary waves has been discussed in addition to the study of the movement of single Kink and Anti-kink solutions.
Significantly, on one hand, this work shows that the use of a well derived more applicable nonuniform scheme leads to a great qualitative dealing with solving nonlinear wave equations- the scheme can also be extended for solving more general nonlinear PDEs.
On the other hand, for application, a complete computer program is well established to meet the requirements for solving SG and DSG equations, by using either the uniform classical scheme or the proposed nonuniform one, moreover it can be slightly modified to solve more general PDEs. Keywords: Sine Gordon equation, Double Sine Gordon equation,
Solitary waves, Solitons, Numerical solution of PDEs, Finite
diflerence methods.