Bernstein Polynomials Method and it’s Error Analysis for Solving Nonlinear Problems in the Calculus of Variations: Convergence Analysis via Residual Function

Abstract

In this paper, Bernstein polynomials method (BPM) and their operational matrices are adopted to
obtain approximate analytical solutions of variational problems. The operational matrix of differentiation is introduced and utilized to reduce the calculus of variations problems to the solution of system of algebraic equations. The solutions are obtained in the form of rapidly convergent finite series with easily computable terms. Comparison between the present method and the homotopy perturbation method (HPM), the non-polynomial spline method and the B-spline collocation method are made to show the effectiveness and efficiency for obtaining approximate solutions of the calculus of variations problems. Moreover, convergence analysis based on residual function is investigated to verified the numerical results.