Fractional Differential Equations

Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets

Convergence analysis of moving finite element methods for space fractional differential equations

https://onlinelibrary.wiley.com/doi/pdf/10.1002/cmm4.1055

A numerical approach for solving variable order differential equations based on Bernstein polynomials

A novel fully discrete Crank–Nicolson finite element method, which is obtained by finite difference in time and finite element in space, is presented to approximate the fractional Cable equation. Compared to the L1-formula for discretizing fractional derivatives at time $$t_{n+1}$$
 , the proposed approximate scheme is directly obtained at time $$t_{n+\frac{1}{2}}$$
 , in which some new coefficients $$(k+\frac{1}{2})^{1-\alpha }-(k-\frac{1}{2})^{1-\alpha }$$
 instead of $$ (k+1)^{1-\alpha }-k^{1-\alpha }$$
 are derived. Based on the new approximate formula, the stability and error estimate are analyzed in detail and the optimal convergence rate $$O(\tau ^{\min \{1+\alpha _{1},1+\alpha _{2}\}}+h^{r+1})$$
 is obtained. Numerical examples in one-dimensional and two-dimensional spaces are shown to illustrate the effectiveness and feasibility of the studied algorithm.

A new fully discrete finite difference/element approximation for fractional cable equation

High order finite difference WENO schemes for fractional differential equations

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High order finite difference WENO schemes for fractional differential equations