2015/04/07 by Bangti Jin, Raytcho Lazarov, Jin, Bangti +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.1504.01529
openalex publication_date 2015/04/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in some physical problems, whose solution decays only logarithmically as the time t tends to infinity. We develop a space semidiscrete scheme based on the standard Galerkin finite element method, and establish error estimates optimal with respect to data regularity in L2(D) and H1(D) norms for both smooth and nonsmooth initial data. Further, we propose two fully discrete schemes, based on the Laplace transform and convolution quadrature generated by the backward Euler method, respectively, and provide optimal convergence rates in the L2(D) norm, which exhibits exponential convergence and first-order convergence in time, respectively. Extensive numerical experiments are provided to verify the error estimates for both smooth and nonsmooth initial data, and to examine the asymptotic behavior of the solution.