求解时间分布阶扩散方程的两个高阶有限差分格式
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摘要
基于复化Simpson公式和复化两点Gauss-Legendre公式,构造了两个求解时间分布阶扩散方程的高阶有限差分格式.不同于以往文献中提出的时间一阶或二阶格式,这两种格式在时间方向都具有三阶精度,而在分布阶和空间方向可达到四阶精度.数值结果表明,两种算法都是稳定且收敛的,从而是有效的.两种格式的收敛速率也通过数值实验进行了验证,并且通过和文献中的算法对比可以得出其更为高效.
Based on the composite Simpson's quadrature rule and the composite 2-point Gauss-Legendre quadrature rule, 2 high-order finite difference schemes were proposed for solving time distributed-order diffusion equations. Other than the existing methods whose convergence rates are only 1 st-order or 2 nd-order in the temporal domain, the proposed 2 schemes both have 3 rd-order convergence rates in the temporal domain, and 4 th-order rates in the spatial domain and the distributed order, respectively. Such high-order convergence rates were further verified with numerical examples. The results show that, both of the proposed 2 schemes are stable, and have higher accuracy and efficiency compared with existing algorithms.
Based on the composite Simpson's quadrature rule and the composite 2-point Gauss-Legendre quadrature rule, 2 high-order finite difference schemes were proposed for solving time distributed-order diffusion equations. Other than the existing methods whose convergence rates are only 1 st-order or 2 nd-order in the temporal domain, the proposed 2 schemes both have 3 rd-order convergence rates in the temporal domain, and 4 th-order rates in the spatial domain and the distributed order, respectively. Such high-order convergence rates were further verified with numerical examples. The results show that, both of the proposed 2 schemes are stable, and have higher accuracy and efficiency compared with existing algorithms.
引文
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[18] ZHU Y,SUN Z Z.A high-order difference scheme for the space and time fractional Bloch-Torrey equation[J].Computational Methods in Applied Mathematics,2018,18(1):147-164.
[2] 张平奎,杨绪君.基于激励滑模控制的分数阶神经网络的修正投影同步研究[J].应用数学和力学,2018,39(3):343-354.(ZHANG Pingkui,YANG Xujun.Modified projective synchronization of a class of fractional-order neural networks based on active sliding mode control[J].Applied Mathematics and Mechanics,2018,39(3):343-354.(in Chinese))
[3] 杨旭,梁英杰,孙洪广,等.空间分数阶非Newton流体本构及圆管流动规律研究[J].应用数学和力学,2018,39(11):1213-1226.(YANG Xu,LIANG Yingjie,SUN Hongguang,et al.A study on the constitutive relation and the flow of spatial fractional non-Newtonian fluid in circular pipes[J].Applied Mathematics and Mechanics,2018,39(11):1213-1226.(in Chinese))
[4] CAPUTO M.Elasticità e Dissipazione[M].Bologna:Zanichelli,1969.
[5] SINAI Y G.The limiting behavior of a one-dimensional random walk in a random medium[J].Theory of Probability & Its Applications,1983,27(2):256-268.
[6] CHECHKIN A V,KLAFTER J,SOKOLOV I M.Fractional Fokker-Planck equation for ultraslow kinetics[J].Europhysics Letters,2003,63(3):326-332.
[7] KOCHUBEI A N.Distributed order calculus and equations of ultraslow diffusion[J].Journal of Mathematical Analysis and Applications,2008,340(1):252-281.
[8] JIAO Z,CHEN Y,PODLUBNY I.Distributed-Order Dynamic Systems:Stability,Simulation,Applications and Perspectives[M].London:Springer,2012.
[9] CAPUTO M.Distributed order differential equations modelling dielectric induction and diffusion[J].Fractional Calculus and Applied Analysis,2001,4(4):421-442.
[10] HARTLEY T T,LORENZO C F.Fractional system identification:an approach using continuous order-distributions:NASA/TM-1999-209640[R].USA:NASA,1999.
[11] FORD N J,MORGADO M L,REBELO M.An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time[J].Electronic Transactions on Numerical Analysis,2015,44:289-305.
[12] GAO G H,SUN Z Z.Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations[J].Numerical Methods for Partial Differential Equations,2016,32(2):591-615.
[13] GAO G H,SUN H W,SUN Z Z.Some high-order difference schemes for the distributed-order differential equations[J].Journal of Computational Physics,2015,298:337-359.
[14] HU J H,WANG J G,NIE Y F.Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term[J].Advances in Difference Equations,2018(1):352.DOI:10.1186/s13662-018-1817-2.
[15] 郭晓斌,尚德泉.复化两点Gauss-Legendre公式及其误差分析[J].数学教学研究,2010,29(4):49-51.(GUO Xiaobin,SHANG Dequan.Composite two-point Gauss-Legendre formula and the error analysis[J].Research of Mathematic Teaching-Learning,2010,29(4):49-51.(in Chinese))
[16] TIAN W,ZHOU H,DENG W.A class of second order difference approximations for solving space fractional diffusion equations[J].Mathematics of Computation,2015,84(294):1703-1727.
[17] SUN Z Z.The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations[M].Beijing:Science Press,2009.
[18] ZHU Y,SUN Z Z.A high-order difference scheme for the space and time fractional Bloch-Torrey equation[J].Computational Methods in Applied Mathematics,2018,18(1):147-164.