Design and analysis of efficient computational techniques for solving a temporal‐fractional partial differential equation with the weakly singular solution

Abstract

This work deals with the construction of robust numerical schemes for solving a time‐fractional convection‐diffusion (TFCD) equation with variable coefficients subject to weakly singular solution. The solution of the underlying problem exhibits a weak singularity at time . The methods are designed on both graded and adapted grids, in which the temporal derivative is approximated on the nonuniform grid, in order to tackle the singularity. In the case of adaptive mesh technique, the mesh is constructed adaptively via equidistribution of a positive monitor function. The resulting semi‐discretized equation is further discretized by a high‐order compact finite difference (CFD) scheme to obtain a fully discrete scheme. Stability and convergence of the method on graded mesh are analyzed. Numerical results for one test problem subjected to nonsmooth analytical solution are presented which shows the robustness and efficiency of the proposed numerical schemes. The elapsed computational time (CPU time) for the proposed methods are provided. Numerical results obtained from the methods on graded and adapted grids are compared with those from the scheme on uniform grid. Finally, as an application, the TFCD model and the suggested numerical techniques were employed to price options of a European butterfly spread. The effect of fractional order (FO) derivative on the option price profile is investigated.