A novel finite volume method for the Riesz space distributed-order advection–diffusion equation

Abstract In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) equation. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional model. Next, the transformed multi-term fractional equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 γ

[1]  Fawang Liu,et al.  A numerical investigation of the time distributed-order diffusion model , 2014 .

[2]  Fawang Liu,et al.  Stability and convergence of a finite volume method for the space fractional advection-dispersion equation , 2014, J. Comput. Appl. Math..

[3]  Michele Caputo,et al.  Wave simulation in dissipative media described by distributed-order fractional time derivatives , 2011 .

[4]  Francesco Mainardi,et al.  The Two Forms of Fractional Relaxation of Distributed Order , 2007 .

[5]  Zhi-Zhong Sun,et al.  Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations , 2015, Comput. Math. Appl..

[6]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[7]  Changpin Li,et al.  Finite difference Methods for fractional differential equations , 2012, Int. J. Bifurc. Chaos.

[8]  M. L. Morgado,et al.  Numerical approximation of distributed order reaction-diffusion equations , 2015, J. Comput. Appl. Math..

[9]  Michele Caputo,et al.  Diffusion with space memory modelled with distributed order space fractional differential equations , 2003 .

[10]  Stanly Steinberg,et al.  Random walk models associated with distributed fractional order differential equations , 2006 .

[11]  P. N. Nelson,et al.  The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests , 2015 .

[12]  I M Sokolov,et al.  Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  M. Meerschaert,et al.  Stochastic model for ultraslow diffusion , 2006 .

[14]  M. Naber.  DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION , 2003, math-ph/0311047.

[15]  R J McGough,et al.  THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN. , 2013, Electronic journal of mathematical analysis and applications.

[16]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[17]  S. C. Lim,et al.  Fractional Langevin equations of distributed order. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Michele Caputo,et al.  Mean fractional-order-derivatives differential equations and filters , 1995, ANNALI DELL UNIVERSITA DI FERRARA.

[19]  YangQuan Chen,et al.  On distributed order low-pass filter , 2010, Proceedings of 2010 IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications.

[20]  Fawang Liu,et al.  Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains , 2015 .

[21]  Anatoly N. Kochubei,et al.  Distributed order calculus and equations of ultraslow diffusion , 2008 .

[22]  R. Gorenflo,et al.  Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density , 2013 .

[23]  Fawang Liu,et al.  Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation , 2015, Appl. Math. Comput..

[24]  Sabir Umarov,et al.  Continuous time random walk models associated with distributed order diffusion equations , 2015 .

[25]  Holger Kantz,et al.  Distributed-order diffusion equations and multifractality: Models and solutions. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Fawang Liu,et al.  Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains , 2015, J. Comput. Phys..

[27]  Zhi-Zhong Sun,et al.  Some high-order difference schemes for the distributed-order differential equations , 2015, J. Comput. Phys..

[28]  Zhi‐zhong Sun,et al.  Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions , 2016 .

[29]  Xueke Pu,et al.  Fractional Partial Differential Equations and their Numerical Solutions , 2015 .

[30]  Yury Luchko,et al.  BOUNDARY VALUE PROBLEMS FOR THE GENERALIZED TIME-FRACTIONAL DIFFUSION EQUATION OF DISTRIBUTED ORDER , 2009 .

[31]  Mathematical modeling of the dynamics of anomalous migration fields within the framework of the model of distributed order , 2013 .

[32]  K. Burrage,et al.  A new fractional finite volume method for solving the fractional diffusion equation , 2014 .

[33]  X. Y. Li,et al.  A numerical method for solving distributed order diffusion equations , 2016, Appl. Math. Lett..

[34]  Teodor M. Atanackovic,et al.  Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod , 2010, 1005.3379.

[35]  M. Meerschaert,et al.  Numerical methods for solving the multi-term time-fractional wave-diffusion equation , 2012, Fractional calculus & applied analysis.

[36]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

[37]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[38]  Mark M. Meerschaert,et al.  Distributed-order fractional diffusions on bounded domains , 2009, 0912.2521.

[39]  I. Turner,et al.  A novel numerical approximation for the space fractional advection-dispersion equation , 2014 .

[40]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[41]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[42]  Fawang Liu,et al.  Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations , 2015 .

[43]  Fawang Liu,et al.  An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation , 2015, Numerical Algorithms.

[44]  Yangquan Chen,et al.  Distributed-Order Dynamic Systems - Stability, Simulation, Applications and Perspectives , 2012, Springer Briefs in Electrical and Computer Engineering.

[45]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .