Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order

Abstract

In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply C1-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about O(hmin{4−α,p}) while the interpolating function belongs to Cp(p≥1), where h is the mesh size and α the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases.