الفهرس 
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Abstract
Ordinary differential equations have a great merit of modeling some of the most important phenomenon that encounter in problems in science and engineering. Numerical methods overcome the difficulty of solving ordinary differential equations when analytical solutions are hard to find. Chebyshev polynomial is one of the most important orthogonal polynomials. Chebyshev collocation method gives the highest accuracy for the solution of ordinary differential equations.The thesis is organized as follows: In chapter one, an overview of some ordinary differential equations , their applications and solution methods are presented. Moreover, an abstract on Chebyshev method. This is followed by an outline of the most important researches that indicated this method and contributed in its development. In addition, work objectives are introduced.In chapter two, sheds light on the improvement of Chebyshev collocation method to gain the approximate solution for singular linear and non-linear higher-order ordinary differential equations. This solution demonstrates the effectiveness of Chebyshev-collocation algorithm via a comparison of the results that is made with B-spline functions.In chapter three, this chapter of the thesis gives illustration about promotion on the presented method in solution of two-point BVP in modeling viscoelastic flows. A comparisons with other methods in literature proved the accuracy of the proposed scheme.In chapter four, in this part of the thesis, we exhibit a novel development in Chebyshev collocation method for solution nonlinear higher-order systems of Fredholm integro-differential equations. We apply some numerical examples to reveal the efficaciousness and precision of this method in comparison with reproducing kernel Hilbert space method( RKHSM). Concluding remarks and suggested future work are given in chapter five.