High precision algorithm for spatial distributed-order time fractional diffusion equation
[Objective]Effectively,distributed-order differential equations can describe anomalous diffusion phenomena in multifractal media.Regarding the solution of spatial distributed-order equations,we find that the integer order is mostly studied in the time domain,whereas the spatial convergence order is mostly second and third orders,resulting in difficulties to achieve fourth order.To improve the spatial convergence accuracy in solving the Riesz spatial distributed-order Caputo time fractional diffusion equation,herein we propose a high-order finite difference method.[Methods]By using the composite Simpson quadrature formula,the distributed differential equation is transformed into a multi-term fractional derivative differential equation in Riesz space,and the fractional derivative of Caputo time is approximated by LI interpolation.Next,the high-order numerical discrete scheme of the equation is constructed and the stability and convergence of the scheme are proved by the matrix analysis.Finally,a numerical example is given to demonstrate the stability and convergence of the scheme.[Results]In solving this distributed-order differential equation,our numerical method achieves a spatial convergence order of four for both spatial and distributed orders,and a time convergence order of 2-a.[Conclusions]We construct a high-order difference format for the Riesz spatial distributed-order Caputo time fractional diffusion equation and achieve a spatial convergence order of four.Because distributed-order equations with spatial convergence orders have been studied mostly at the second or third order,the effectiveness of the proposed method is observed.
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