Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials

Abstract

In this paper, a collocation method based on the Bernstein polynomials is presented for the fractional Riccati type differential equations. By writing t→tα (0<α<1) in the truncated Bernstein series, the truncated fractional Bernstein series is obtained and then it is transformed into the matrix form. By using Caputo fractional derivative, the matrix forms of the fractional derivatives are constructed for the truncated fractional Bernstein series. We convert each term of the problem to the matrix form by means of the truncated fractional Bernstein series. By using the collocation points, we have the basic matrix equation which corresponds to a system of nonlinear algebraic equations. Lastly, a new system of nonlinear algebraic equations is obtained by using the matrix forms of the conditions and the basic matrix equation. The solution of this system gives the approximate solution for the truncated limited N. Error analysis included the residual error estimation and the upper bound of the absolute errors is introduced for this method. The technique and the error analysis are applied to some problems to demonstrate the validity and applicability of the proposed method.