الفهرس 
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Abstract
This thesis is devoted in the study of the Compact Finite Differences as a generalization of (classical) finite differences, which is the most popular method for solving both ordinary and partial differential equations numerically.
Also, this thesis is interested in studying how to apply the method to some examples of parabolic partial differential equations in one dimension.
The thesis consists of four chapters:
Chapter one:
This chapter contains a short introduction to the history of compact finite differences and some of the methods used before for some examples in parabolic partial differential equations, which are considered here.
Chapter two:
In this chapter, different formulae of compact finite differences are presented in a computational domain using uniform step, simple staggered step and non-uniform step for all points, whether internal or boundary, together with the study of stability for some of these formulae, also the study of how to make general formulae with uniform steps using Padé approximation.
Chapter three:
This chapter is interested in studying a famous formula known as combined compact finite differences in a computational domain using both uniform step and non-uniform step for all the points, internal or boundary, also in calculating new formulae for the boundary points.
Chapter four:
In this chapter, new methods are presented for solving some examples of linear and nonlinear parabolic partial differential equations in one dimension, such as the heat equation, convection-diffusion equation, Burgers’ equation, and Burgers’-Fisher equation in both finite and semi-finite domains.
The following paper is extracted from the previous chapter:
M. H. Mousa, A. A. Abadeer and M. M. Abbas, ”Combined compact finite difference treatment of Burgers’ equation”, International Journal of Pure and applied Mathematics, Vol. 75 (2), 2012, 169-184.