High-order numerical methods for solving time fractional partial differential equations
The authors develop a fully discrete numerical scheme to solve the time fractional partial differential equation \[ \begin{alignedat}{3} _0^C D_{t}^{\alpha } u (x, t) - \Delta u (x, t) &= f(x, t), &\qquad &x \in \Omega, \quad t \in (0, T], \\ u(x, 0) &= u_{0}, &&x \in \Omega, \\ u(x, t) &= q(x, t), &&x \in \partial \Omega, \quad t \in (0, T],\end{alignedat} \] ``where \(\Omega \subset \mathbb{R}^{d}\), \(d=1, 2, 3\) is a convex polygonal/polyhedral domain and \(\Delta\) denotes the Laplacian. Here \(f\) and \(q\) are two given functions and \(_0^C D_{t}^{\alpha } x(t)\), \(0< \alpha <1\) denotes the Caputo fractional order derivative.'' The time discretization is based the Diethelm's method and the space discretization is based on the standard finite element method. The error estimates are obtained with the convergence of second order in space and (2-3) order in time. Some numerical examples are considered to show that the numerical results are consistent with the theoretical results. Several lemmas and theorems are stated and proved.
time fractional partial differential equations
scientific article; zbMATH DE number 6849491
scientific article