Numerical algorithm for the variable-order Caputo fractional functional differential equation

While several high-order methods have been extensively developed for fixed-order fractional differential equations (FDEs), there are no such methods for variable-order FDEs. In this paper, we propose an accurate and robust approach to approximate the solution of functional Dirichlet boundary value problem with a type of variable-order Caputo fractional derivative. The proposed method is principally based on the shifted Chebyshev polynomials as basis functions and the matrix representation of variable-order fractional derivative of such polynomials. The underline variable-order FDE is then reduced to a system of algebraic equations, which greatly simplifies the solution process. Through numerical results, we confirm that the proposed scheme is very efficient and accurate for handling such problem.