Legendre polynomials method for solving a class of variable order fractional differential equation

In this paper, a numerical method based on the Legendre polynomials is presented for a class of variable order fractional differential equation. We adopt the Coimbra variable order fractional operator, which can be viewed as a Caputo-type definition. Three different kinds of operational matrixes with Legendre polynomials are derived. A truncated the Legendre polynomials series together with the products of several dependent matrixes are utilized to reduce the variable order fractional differential equation to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited n. An error analysis technique is also given. Some examples are included to demonstrate the validity and applicability of the approach.