A Generalized Spectral Collocation Method with Tunable Accuracy for Fractional Differential Equations with End-Point Singularities

Abstract

We develop spectral collocation methods for fractional differential equations with variable order with two end-point singularities. Specifically, we derive three-term recurrence relations for both integrals and derivatives of the weighted Jacobi polynomials of the form $(1+x)^{\mu_1}(1-x)^{\mu_2}P_{j}^{a,b}(x) \,({a,b,\mu_1,\mu_2>-1})$, which leads to the desired differentiation matrices. We apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end-point singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters $\mu_1$ and $\mu_2$, even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method.