Some second-order \(\theta\) schemes combined with finite element method for nonlinear fractional cable equation
The authors investigate numerical methods for solving fractional partial differential equations. Specifically, they develop a second-order numerical method for the fractional cable equation \[ \frac{\partial u}{\partial t}+\mathcal K D^\alpha_tu-D^\beta_t\Delta u+g(u)=f, \] where \(\mathcal K\) is a constant and \(D^\alpha_t\) denotes the Riemann-Liouville fractional derivative with respect to time. The authors approximate the fractional time derivatives using the weighted and shifted Grünwald difference (WSGD) operators developed in [\textit{W. Tian} et al., Math. Comput. 84, No. 294, 1703--1727 (2015; Zbl 1318.65058)]. For the space discretization, they use standard finite element methods. For the integer time derivative \(\frac{\partial u}{\partial t}\), the authors use the \(\theta\) method, a generalization of the backwards difference method and the Crank-Nicolson method. Combining these three ingredients, the authors develop their new numerical method and prove stability and error estimates. The authors illustrate their method with numerical examples in one and two dimensions, verifying the rates of convergence.
fractional partial differential equation
scientific article; zbMATH DE number 7015337
scientific article