A Novel Lagrange Operational Matrix and Tau-Collocation Method for Solving Variable-Order Fractional Differential Equations

The main result achieved in this paper is an operational Tau-Collocation method based on a class of Lagrange polynomials. The proposed method is applied to approximate the solution of variable-order fractional differential equations (VOFDEs). We achieve operational matrix of the Caputo's variable-order derivative for the Lagrange polynomials. This matrix and Tau-Collocation method are utilized to transform the initial equation into a system of algebraic equations. Also, we discuss the numerical solvability of the Lagrange-Tau algebraic system in the case of a variable-order linear equation. Error estimates are presented. Some examples are provided to illustrate the accuracy and computational efficiency of the present method to solve VOFDEs. Moreover, one of the numerical examples is concerned with the shape-memory polymer model.