A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations

Abstract

We generalize existing Jacobi--Gauss--Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form $(1 \pm x)^\mu P_j^{a,b}(x)$, where ${\mu>-1}$. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals and derivatives of these weighted Jacobi polynomials, hence extending the three-term recurrence relationship of Jacobi polynomials. The new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter $\mu$, even for cases where we do not know explicitly the form of singularity in the solution at the boundaries.