الفهرس 
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Abstract
The main objectives of this work is to introduce and develop new efficient spec- tral algorithms based on orthogonal polynomials for solving special types of linear or nonlinear ordinary differential equations subject to initial and homogeneous or non- homogeneous boundary conditions. Also, we extend this work to include some types of partial differential equations.
In Chapter 1, we give a brief introduction to the spectral methods and their ad- vantages. Also, we clarify the differences between the three most commonly used spectral methods, namely, the Galerkin, collocation, and tau methods. A brief ac- count of orthogonal polynomials, their properties and expansion of functions in terms of them are given. General and useful needed properties of Legendre, Chebyshev; in particular first, third and fourth kinds; and Laguerre polynomials are also considered.
In Chapter 2, we developed the shifted third and fourth kinds Chebyshev opera- tional matrices of derivatives for choices of Galerkin/tau basis functions. Therefore, effective numerical algorithms for solving Lane-Emdan type equations are introduced. We have ascertained that the introduced truncated shifted third and fourth Cheby- shev expansions of a function u(x) converges uniformly to u(x).