الفهرس 
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Abstract
This thesis is devoted to study various numerical schemes for solving two dimensional first-order time-dependent parabolic partial differential models. Our study is based on using Compact Finite Differences methods and pseudo-spectral methods for discretizing the space while applying various methods for time discretization. The thesis is composed of five chapters:
The first chapter contains a short introduction to parabolic partial differential equations with a brief history of compact finite difference and spectral collocation methods with some explanatory examples.
In Chapter two: we study numerical methods for one of the classical parabolic problems: convection-diffusion equation. Two numerical methods are presented to solve a general case of this equation with variables coefficients coupled with the stability study of each method. The second method results were published in the following paper:
M. EL-Azab, R. El-Ashwah, M. Abbas, G. El-Baghdady, A highly approximate pseudo-spectral method for the solution of convection-diffusion equations, J. Math. Sci. Model., (in press)..
Chapter three is based on constructing several efficient numerical schemes for solving a model of non-linear parabolic partial differential equations. Temporal discretization in some of these schemes is based on using Runge-Kutta methods, while in other schemes is based on using collocation methods. The following paper is extracted from this chapter:
El-Baghdady, G.I., Abbas, M.M., El-Azab, M.S. El-Ashwah, R. M. Int. J. Appl. Comput. Math (2017) 3: 3333. https://doi.org/10.1007/s40819-016-0299-8.
As for Chapter Four, we examine the approach of implicit-explicit methods to obtain two new numerical solutions of cancer invasion mathematical model as an application of nonlinear parabolic partial differential systems.
Based on the results of the second method in this chapter, we prepare the following paper:
M. S. El-Azab, R. M. El-Ashwah, M. M. Abbas, A high order pseudo-spectral method for cancer invasion mathematical model, (submitted for publication).
The thesis ended with Chapter five, in which we presented some remarks and conclusions based on the obtained results during our study with some suggestions for future works.