Analysis of a finite difference method based on L1 discretization for solving multi‐term fractional differential equation involving weak singularity. (8th March 2022)

• Record Type: Journal Article
• Authors: Santra, Sudarshan; Mohapatra, Jugal
• Is Part Of: Mathematical methods in the applied sciences. Volume 45:Number 11(2022)
• Page Start: 6677
• Page End: 6690
• Publication Date: 2022-03-08
• Subjects: Caputo fractional derivative -- classical L1 scheme -- error analysis -- experimental evidences -- multi‐term fractional IVP; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8199
• Languages: English
• ISSNs: 0170-4214

Abstract

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 γ 1 where γ 1 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 O ( τ γ 1 ) 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.