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

The 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.