The collocation method is very common in solving different types of differential equations. A main difficulty of this method is that its coefficient matrix becomes ill-conditioned when the degree of approximation increases. This can cause numerical troublesome and decreases the accuracy of the solution. In this study, three methods are proposed based on the combination of Bernstein collocation and optimization methods for approximate solutions of initial and boundary value problems involving linear differential equations with variable coefficients. In these methods, the approximate solution of the problem is obtained using the solution of a constrained linear least squares problem or a linear programming problem. To investigate the effectiveness of the methods, experimental problems of the different orders are considered and the results are compared with the results reported from other methods. Studies show that the proposed methods are accurate, efficient and have good numerical stability.

Keywords: Initial and Boundary Value Problems, Linear Differential Equations, Bernstein Collocation Method, Optimization

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