الفهرس 
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Abstract
Differential equations lie in the heart of mathematics bringing together the best of pure mathematics to deal with some of the most important models, both classical and modern real life phenomenon in applied mathematics. Numerical methods play a very important role in solving differential equations both ordinary and partial when analytical solutions are hard to find. One of the numerical techniques that is used to solve differential equations is collocation methods. One of the most important non orthogonal polynomials to be used in this is the Bernoulli polynomials. Euler and Bernoulli polynomials are widely used in boundary value problems of non orthogonality nature on the interval [0, 1]. Chapter one is an introduction. In chapter two, preliminaries of Euler and Bernoulli bases are introducedalong with the differentiation matrices. High-order boundary value problems of linear and nonlinear types are solved in chapter three using both Euler and Bernoulli-collocation methods. In chapter four, the singular two point and time dependent equations are solved using Bernoulli bases along with Bratu’s equation which is special type of two point problems. The fourth order eigenvalue problem is considered in chapter five, where the Bernoulli-collocation method is applied to determine the eigenvalue and compute the Eigenfunctions of different problems. Chapter six gives a comparison between the wavelet-Galerkin and Bernoulli-collocation methods for solving linear partial differential equations. Chapter seven gives a conclusion and future work.