Application of the Exact Operational Matrices Based on the Bernstein Polynomials

This paper aims to develop a new category of operational matrices. Exact operational matrices (EOMs) are matrices which integrate, differentiate and product the vector(s) of basis functions without any error. Some suggestions are offered to overcome the difficulties of this idea (including being forced to change the basis size and having more equations than unknown variables in the final system of algebraic equations). The proposed idea is implemented on the Bernstein basis functions. By both of the newly extracted Bernstein EOMs and ordinary operational matrices (OOMs) of the Bernstein functions, one linear and one nonlinear ODE is solved. Special attention is given to the comparison of numerical results obtained by the new algorithm with those found by OOMs.