A study of distributed‐order time fractional diffusion models with continuous distribution weight functions

Abstract

AbstractThe distributed‐order time fractional diffusion model with Dirichlet nonhomogeneous boundary conditions on a finite domain is considered. Four choices of continuous distribution weight functions with mean μ and standard deviation σ are investigated to study their impact on both the short‐time and long‐time solution behavior. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. Semi‐analytic solutions are also derived for these distributions to assess the accuracy of the scheme. Numerical results highlight that the four continuous distribution weight functions produce a short‐time solution behavior that is consistent with those solutions from the classical time fractional partial differential equation with fractional order γ* = μ. There are however long‐time differences in the solution behavior that become more distinguishable as σ increases. In particular, we find a smaller value of σ produces more diffuse profiles and the diffusion rate slows as σ increases. Furthermore, the asymptotic behavior of the solution may be influenced by the time‐fractional orders ranging between the smallest nonzero weight order and mean μ for the continuous uniform and raised cosine distribution weight functions, respectively. Similar findings are also observed for the truncated normal and beta distributions.