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ABSTRACT In this study,we apply a new implicit method of high accuracy and the number of linear systems which to be solved are smaller than that for many famous known implicit methods of small step length. Therefore our required machine time is less than that for the other methods. The convection - diffusion equation applies to problems in such areas a mass transport, momentum transport, energy transport and neutron transport. The problem is solving t he convection - diffusion equation by a method related to the Restrictive Pade Approximation(RPA) is considered. This method will exhibit several advantageous features. For example the accuracy has not been lost when the value of the exact solution is suffic iently large the absolute error is sufficiently small whenever the exact solution is relatively large.The choice of time step length k is sufficiently large compared with that can be used for the classical schemes,this allows us to have the solution at hig h level of time. Restrictive pade approximation for parabolic partial differential equation and partial difference equation is a new technique done by İsmail and Elbarbary. In addition they studied numerical solution of the convection - diffusion equation us ing restrictive Taylor approximation [1]. The advantage is that it has the exact value at certain r. This method will exhibit several advantages for example highly accurate,fast and good results, etc. The absolutely error is still very small.The compute d results are compared with the exact solution and the other methods.
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Restrictive Pade Approximation, Convection - Diffusion Equation, Finite Difference
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In this study,we will deal with the problem of applying cubic parametric spline functions to investigate a numerical method for obtaining approximate solution of the linear time fractional Klein-Gordon Equation. We will prove the solvibility of the proposed method. A numerical example is included to illustrate the practical implementation of the proposed method. In recent years ,there has been a growing interest in the field of fractional calculus. Fractional differential equations have attracted increasing attention because they have applications in various fields of science and engineering. Many phenomena in fluid mechanics ,chemistry ,physics, finance and other sciences can be described very succesfully by using mathematical tools from fractional calculus. Most of the applications are given in the book of Oldham and Spanier, the book of Podulbny, and the paper of Metzler and Klafter, Bagley and Torvik. There are several definitions of a fractional derivative of order α>0. Two most commonly used are the Riemann-Louville and Caputo. The difference between the two definitions is in the order of evaluation.In this work we will use the Caputo fractional derivative. The truncation error of the method will be theorically analyzed. Computed results will compared with the other methods.
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Time Fractional Klein-Gordon, Spline Functions, Stability
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