الفهرس 
approxiImation of functionsOnly 14 pages are availabe for public view
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Abstract
The Padé approximation of functions is type of functions approximations, which takes the form as Rational function and could be used to find approximate solutions of ordinary differential equations as well as partial differential equations, which describing of the problem in the different branches of mathematics as arithmetic applications.
Starting from 1995, Dr. Hassan N.A.Ismail added a parameter (or more) in the definition of Padé approximation, which can be set to find out the solution at a point (or more), for this definition, the error for numerical solution of some type of initial boundary value problems for partial differential equations are equal to zero, this types of approximations takes the name “Restrictive Padé Approximations.”
This type of approximations is used to obtain finite difference equations with good accuracy and high efficiency to approximate the solutions of partial differential equations.
In 1999, by the same way, he put the same limitation on the functions that approximated by Taylor series, this appears the definition named “Restrictive Taylor Approximation” which can also be used for solving partial differential equations.
In each of the restrictive Taylor and restrictive Padé approximations by knowing an accurate solution at one point (or more), we can find the desired parameter.
Chapter 1:
In this chapter, we introduce some famous approximations used in numerical solutions, such as the method of least squares (fitting of curves) and the interpolation by Newton, Lagrange and Rational approximation, which are used in many fields.
We presented a simple explanation of the Taylor, Padé approximations and some examples with a comparison between the two approximations. Then we give an explanation of the restrictive Taylor and restrictive Padé approximations, noted the accuracy of these methods, which depend on finding a parameter proposed, when we put the proposed parameter equal to zero, the restrictive Taylor and restrictive Padé approximations become the Taylor and Padé approximations in their traditional forms.
Chapter 2:
In this chapter, we obtain a numerical solution Convection-Diffusion equation by using the restrictive Taylor approximation, comparing the results of absolute error by one of finite differences method (FTCS), and we observed how accurate our restrictive Taylor.
III
We study the conditions of the stability of finite-difference equation containing the restrictive parameter.
Chapter 3:
In this chapter, we introduce the numerical solution by using the restrictive Taylor approximation for Gardner equation, which is a mixed of Korteweg and de Vries and modified Korteweg de Vries equation .We obtain an accurate numerical solution and also discuss the stability of the way. This has been published in the Bulletin of 38th International Conference for Statistics, Computer Science and its Applications from page 49-60.
As well as the general Korteweg and de Vries equation has been resolved by restrictive Taylor approximation, and compared with the previous results. Also solving the Burger - Korteweg de Vries equation by using restrictive Taylor approximation and it is compared with the standard error with the previous methods.
Chapter 4:
In this chapter, we introduce a new finite difference technique to solve Kuramoto–Sivashinsky equation, finding the absolute error and studying the stability. We observe the high accurate results by using restrictive Taylor method which the absolute error is equal to zero in some points. This solution is published in the Bulletin of 38th International Conference for Statistics, Computer Science and its Applications pages 61-69. Also we introduce the solutions of the Inviscid Burger Equation and Viscous Burger Equation.
Chapter 5:
In this chapter, we use the restrictive Taylor method to solve the Modified Burger equation, the Burger’s–Fisher equation and The general Burger’s–Huxley equation.