Fanhai Zeng

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

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

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

(1 \pm x)^\mu P_j^{a,b}(x)

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

, where

Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions☆

{\mu>-1}

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

. 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

A tunable finite difference method for fractional differential equations with non-smooth solutions

\mu

Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes

, even for cases where we do not know explicitly the form of singularity in the solution at the boundaries.

A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities

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

Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions

(1+x)^{\mu_1}(1-x)^{\mu_2}P_{j}^{a,b}(x) \,({a,b,\mu_1,\mu_2>-1})

Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations

, 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