2023/09/23 by Shweta Kumari, Kumari, Shweta, Abhishek Kumar Singh +5 · 2 citations
Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2309.13316
openalex publication_date 2023/09/23 · openalex created_date 2023/09/27 · openalex updated_date 2026/07/28
In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the non-uniform mesh. Then truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as min\4-α,rα\ and \frac4-αα, respectively, where α∈(0,1) is the order of fractional derivative and r≥ 1 is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general r. Then we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for r=1, is analyzed for non-smooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.