Solving fractional differential equations using collocation method based on hybrid of block-pulse functions and Taylor polynomials

In this paper, a novel approach is proposed to solve fractional differential equations (FDEs) based on hybrid functions. The hybrid functions consist of block-pulse functions and Taylor polynomials. The exact formula for the Riemann--Liouville fractional integral of the hybrid functions is derived via Laplace transform. The FDE under consideration is converted into an algebraic equation with undetermined coefficients by using this formula. A set of linear or nonlinear equations are obtained through collocating the algebraic equation at Newton-Cotes nodes. The numerical solution of the FDE is achieved by solving the linear or nonlinear equations. Error analysis is performed on the proposed method. Several numerical examples are given, and the results have proven that the proposed method is effective.

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