An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

Automated summary

This paper introduces two new methods for solving systems of ordinary differential equations, one based on generalized Bernstein functions and the other on the tau method. By applying the residual correction procedure, one can estimate the absolute errors for both methods and obtain more accurate results. The numerical tests show that the methods are efficient and work well, with an increase in accuracy as m increases.