Analysis of a finite difference method based on L1 discretization for solving multi‐term fractional differential equation involving weak singularity

Abstract

In this article, we consider a multi‐term fractional initial value problem which has a weak singularity at the initial time t=0. The fractional derivatives are defined in Caputo sense. Due to such singular behavior, an initial layer occurs near t=0 which is sharper for small values of γ1 where γ1 is the highest order among all fractional differential operators. In addition, the analytical properties of the solution are provided. The classical L1 scheme is introduced on a uniform mesh to approximate the fractional derivatives. The error analysis is carried out, and it is shown that the numerical solution converges to the exact solution. Further analysis proves that the scheme is of order Ofalse(τγ1false) over the entire region, but it is of order O(τ) on any subdomain away from the origin. τ denotes the mesh parameter. To show the efficiency of the proposed scheme, this method is tested on several model problems, and the results are in agreement with the theoretical findings.

A novel higher-order numerical method for parabolic integro-fractional differential equations based on wavelets and $L2$-$1_\sigma$ scheme

“…It is also noteworthy that the singular behavior of the solution of a fractional order derivative system not only differs from the classical integer order systems but also poses a greater challenge for solving them. For a detailed study on fractional order problems having initial singularities, we direct the reader's attention to the work due to Chen et al [7], Santra and Mohapatra [30]. In general, finding the analytical solution to fractional differential equations involving integral operators is impractical, since a lot of iterations are required and that makes the process more time-consuming.…”

“…Among them, the L1 discretization is a good approximation, which has been used widely in [12,29], for the fractional problems having initial weak singularities. The method gives a first order convergence on any subdomain away from the origin whereas, it produces less rate of convergence over the entire region (for example have a look at [28,30]). Even though the nonuniform mesh is more effective than the uniform mesh to capture the initial layer, it fails to occur with second-order accuracy [36].…”