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Abstract Summary In recent years, numerical methods for partial differential equations have enjoyed intense interest and development. This is due to the rapid progress of high speed computers and also due to that many of most significant problems for engineers and scientists can be formulated as partial differential equations which can be solved in a satisfactory manner by modern numerical methods. The method of finite differences is one of most popular methods for the numerical solutions of partial differential equations. In this thesis, new finite difference schemes are constructed to approximate the solutions of different problems in parabolic partial differential equations in two-dimensional space. Also, new finite difference schemes are constructed to approximate the solutions of parabolic partial differential equations in the three-dimensional space. These schemes are derived using the restrictive Pade and the restrictive Taylor’s approximations which give high accurate results and have many other advantage features. The stability conditions are studied fir the suggested schemes. Comparisons of the numerical results obtained with the classical methods are done . Three papers have been extracted from this thesis which were presented in national conferences held in Banha University and Military Technical College as outlined below:: [1] Hassan N. A. Ismail and Ahmed R.A. Shafay ”Implicit Restrictive Pade method for Almost Exact Solution for IBVPs for Two- Dimensional Parabolic PDEs”, The 31’x’ International Conference for Statistics, Computer Sciences and its Applications, Cairo, Vol. 2, pp.37- 47 (2006). [2] Hassan N. A. Ismail and Ahmed R.A. Shafay ”Explicit Restrictive Taylor’s method For Almost Exact Solution for IBVPs for two- Dimensional Parabolic PDEs”, fhe 31 th International Conference for Statistics, Computer Sciences and its Applications, Cairo, Vol. 2, pp.49- 59 (2006). [3] Hassan N. A. Ismail and Ahmed R.A. Shafay ”Restrictive Taylor’s Approximation for Higher Dimension IBV Problem for Parabolic PDE”, The 3rd International Conference for Mathematics and Engineering Physics ICMEP-3, M.T.C., Cairo, (2006), and sent for possible publications in lnt. J. Computer Mathematics. |