Fast high order difference schemes for the time fractional telegraph equation

Abstract

In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ2‐1σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order Oτ2+h2 and Oτ2+h4, respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ2‐1σ method.

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

“…The SOE technique is an efficient way to reduce the computational complexity caused by the non-locality of fractional derivatives [13,19]. In [13], Yan et al presented the fast L2-1 σ (denoted by FL2-1 σ ) formula by employing the SOE technique with the kernel function in the Caputo derivative.…”

“…In [13], Yan et al presented the fast L2-1 σ (denoted by FL2-1 σ ) formula by employing the SOE technique with the kernel function in the Caputo derivative. Subsequently, some numerical studies of time-fractional models based on this approach have emerged (see [10,19] and the corresponding references therein). In [19], Liang et al proposed a fast difference scheme for solving the one-dimensional time-fractional telegraph equation based on the FL2-1 σ formula.…”

Efficient numerical algorithms of time fractional telegraph‐type equations involving Hadamard derivatives

“…In this paper, we consider the numerical algorithms for the following time fractional telegraph equation: $$$$ {\displaystyle \begin{array}{cc}\hfill {}_{CH}{D}_{a,t}^{\gamma -1}u\left(x,t\right)+{}_{CH}{D}_{a,t}^{\gamma }u\left(x,t\right)=& \kern2pt \frac{\partial^2u\left(x,t\right)}{\partial {x}^2}+g\left(x,t\right),\kern0.30em 0<x<L,\kern0.30em a<t\le T,\hfill \\ {}\hfill u\left(x,a\right)=& \kern2pt \phi (x),\kern0.30em {u}_t\left(x,a\right)=\psi (x),\kern0.30em 0\le x\le L,\hfill \\ {}\hfill u\left(0,t\right)=& \kern2pt 0,u\left(L,t\right)=0,\kern0.30em a<t\le T,\hfill \end{array}} $$$$ where $$$ 1<\gamma <2,\phi (x),\psi (x),g\left(x,t\right) $$$ are given functions. Up to now, most of literatures of fractional telegraph equations concern the Caputo or Riemann–Liouville derivatives,…”

“…where 1 < 𝛾 < 2, 𝜙(x), 𝜓(x), g(x, t) are given functions. Up to now, most of literatures of fractional telegraph equations concern the Caputo or Riemann-Liouville derivatives, [27][28][29][30] and the studies of numerical methods for fractional telegraph equations involving the Hadamard derivative are still limited. In Saxena et al, 31 analytical solution of space-time fractional telegraph equations with Hilfer and Hadamard derivatives was studied.…”

Numerical simulation of telegraph and Cattaneo fractional‐type models using adaptive reproducing kernel framework

“…Analytic, numeric, and asymptotic solutions of various classes of the fractional telegraph and Cattaneo equations have been obtained recently by several authors using developed analytical and numerical methods as well as difference schemes, namely, Chen et al, 8 Comptedag and Metzlerddag, 10 Huang, 12 Jiang and Lin, 13 Hosseini et al, 14 Modanli and Akgül, 15,16 and many others. Based on finite difference methods, the readers can refer to Wang and Mei 17 and Liang et al 18 for further details and applications. Although many methods have been proposed and developed for handling such models, the challenge remains to create high-level numerical methods for solving these equations.…”