Fourth-order alternating direction implicit difference scheme to simulate the space-time Riesz tempered fractional diffusion equation

References

• M. Abbaszadeh, Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation, Appl. Math. Lett. 88 (2019), pp. 179–185.
   Web of Science ®Google Scholar
• M. Abbaszadeh and H. Amjadian, Second-order finite difference/spectral element formulation for solving the fractional advection-diffusion equation, Commun. Appl. Math. Comput. (2020). Available at https://doi.org/10.1007/s42967-020-00060-y.
   Web of Science ®Google Scholar
• M. Abbaszadeh, M. Dehghan, and Y. Zhou, Crank-Nicolson/Galerkin spectral method for solving two-dimensional time-space distributed-order weakly singular integro-partial differential equation, J. Comput. Appl. Math. 374 (2020), pp. 112739.
   Web of Science ®Google Scholar
• B. Baeumer and M.M. Meerschaert, Tempered stable Levy motion and transient superdiffusion, J. Comput. Appl. Math. 233 (2010), pp. 2438–2448.
   Web of Science ®Google Scholar
• W. Bu, Y. Tang, Y. Wu, and J. Yang, Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations, J. Comput. Phys. 293 (2015), pp. 264–279.
   Web of Science ®Google Scholar
• C. Çelik and M. Duman, Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys. 231(4) (2012), pp. 1743–1750.
   Web of Science ®Google Scholar
• M. Chen and W. Deng, Discretized fractional substantial calculus, ESAIM: Math. Model. Numer. Anal. 49(2) (2015), pp. 373–394.
   Web of Science ®Google Scholar
• M. Chen and W. Deng, A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation, Appl. Math. Lett. 68 (2017), pp. 87–93.
   Web of Science ®Google Scholar
• M. Chen and W. Deng, High-order algorithm for the time-tempered fractional Feynman-Kac equation, J. Sci. Comput. 76(2) (2018), pp. 867–887.
   Web of Science ®Google Scholar
• X. Chen, Y. Di, J. Duan, and D.F. Li, Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations, Appl. Math. Lett. 84 (2018), pp. 160–167.
   Web of Science ®Google Scholar
• X. Cheng, J. Duan, and D.F. Li, A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equations, Appl. Math. Comput. 346 (2019), pp. 452–464.
   Web of Science ®Google Scholar
• E. Cuesta, C. Lubich, and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comput. 75(254) (2006), pp. 673–696.
   Web of Science ®Google Scholar
• M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys.228(20) (2009), pp. 7792–7804.
   Web of Science ®Google Scholar
• M. Dehghan and M. Abbaszadeh, A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation, Comput. Math. Appl. 75(8) (2018), pp. 2903–2914.
   Web of Science ®Google Scholar
• M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differ. Equ. 26 (2010), pp. 448–479.
   Web of Science ®Google Scholar
• M. Dehghan, M. Abbaszadeh, and W. Deng, Fourth-order numerical method for the space–time tempered fractional diffusion-wave equation, Appl. Math. Lett. 73 (2017), pp. 120–127.
   Web of Science ®Google Scholar
• D. del Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas 13(8) (2006), p. 082308.
   Web of Science ®Google Scholar
• H. Ding, A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation, Appl. Numer. Math. 135 (2019), pp. 30–46.
   Web of Science ®Google Scholar
• M. Donatelli, M. Mazza, and S. Serra–Capizzano, Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations, SIAM J. Sci. Comput. 40(6) (2018), pp. A4007–A4039.
   Web of Science ®Google Scholar
• V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ. 22(3) (2006), pp. 558–576.
   Web of Science ®Google Scholar
• G. Fairweather, X. Yang, D. Xu, and H. Zhang, An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput., 65 (2015), pp. 1217–1239.
   Web of Science ®Google Scholar
• K.M. Furati, M. Yousuf, and A.Q.M. Khaliq, Fourth-order methods for space fractional reaction-diffusion equations with non-smooth data, Int. J. Comput. Math. 95 (2018), pp. 1240–1256.
   Web of Science ®Google Scholar
• G.H. Gao and Z.Z. Sun, Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations, J. Sci. Comput. 66 (2016), pp. 1281–1312.
   Web of Science ®Google Scholar
• M. Garg and P. Manohar, Matrix method for numerical solution of space-time fractional diffusion-wave equations with three space variables, Afrika Mat. 25(1) (2014), pp. 161–181.
   Google Scholar
• Z.P. Hao, Z.Z. Sun, and W.R. Cao, A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys., 281 (2015), pp. 787–805.
   Web of Science ®Google Scholar
• K. Kazmi and A.Q.M. Khaliq, A split–step predictor–corrector method for space–fractional reaction–diffusion equations with nonhomogeneous boundary conditions, Commun. Appl. Math. Comput. 1 (2019), pp. 525–544.
   Web of Science ®Google Scholar
• K. Kazmi and A.Q.M. Khaliq, An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions, Appl. Numer. Math. 147 (2020), pp. 142–160.
   Web of Science ®Google Scholar
• D. Kumar and M. Kaplan, New analytical solutions of (2+1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques, Chinese J. Phys. 56 (2018), pp. 2173–2185.
   Web of Science ®Google Scholar
• C.P. Li, F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, New York, 2015.
   Google Scholar
• H.L. Liao, P. Lyu, and S. Vong, Second-order BDF time approximation for Riesz space-fractional diffusion equations, Int. J. Comput. Math. 95 (2018), pp. 144–158.
   Web of Science ®Google Scholar
• A. Lischke, J.F. Kelly, and M.M Meerschaert, Mass-conserving tempered fractional diffusion in a bounded interval, Fract. Calc. Appl. Anal. 22(6) (2019), pp. 1561–1595.
   Web of Science ®Google Scholar
• F. Liu, M.M. Meerschaert, R.J. McGough, P. Zhuang, and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal. 16(1) (2013), pp. 9–25.
   PubMed Web of Science ®Google Scholar
• F. Liu, P. Zhuang, I. Turner, V. Anh, and K. Burrage, A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys. 293 (2015), pp. 252–263.
   Web of Science ®Google Scholar
• Z. Luo and H. Wang, A highly efficient reduced-order extrapolated finite difference algorithm for time–space tempered fractional diffusion-wave equation, Appl. Math. Lett. 102 (2020), p. 106090.
   Web of Science ®Google Scholar
• M.M. Meerschaert, Y. Zhang, and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett. 35(17) (2008). doi:10.1029/2008GL034899.
   PubMed Web of Science ®Google Scholar
• R. Metzler and T.F. Nonnenmacher, Space-and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation, Chem. Phys. 284(1) (2002), pp. 67–90.
   Web of Science ®Google Scholar
• A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Vol. 23, Springer Science and Business Media, Milan, 2008.
   Google Scholar
• J.P. Roop, Variational solution of the fractional advection dispersion equation, Ph.D. thesis, Clemson University, 2004.
   Google Scholar
• K.M. Saad, E.H.F. AL-Shareef, A.K. Alomari, D. Baleanu, and J.F. Gomez-Aguilar, On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger's equations using homotopy analysis transform method, Chinese J. Phys. 63 (2020), pp. 149–162.
   Web of Science ®Google Scholar
• Z.Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56(2) (2006), pp. 193–209.
   Web of Science ®Google Scholar
• X. Sun, C. Li, and F. Zhao, Local discontinuous Galerkin methods for the time tempered fractional diffusion equation, Appl. Math. Comput. 365 (2020), pp. 124725.
   Web of Science ®Google Scholar
• W. Tian, H. Zhou, and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84(294) (2015), pp. 1703–1727.
   Web of Science ®Google Scholar
• S. Vong, P. Lyu, X. Chen, and S. Lei, High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives, Numer. Algorithms 72 (2016), pp. 195–210.
   Web of Science ®Google Scholar
• Y.M. Wang and L. Ren, Efficient compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients, Int. J. Comput. Math. 96 (2018), pp. 264–297.
   Web of Science ®Google Scholar
• Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), pp. 1–15.
   Web of Science ®Google Scholar
• Y.M. Wang and T. Wang, A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions, Comput. Math. Appl.75 (2018), pp. 721–739.
   Web of Science ®Google Scholar
• L. Wei, Analysis of a new finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Math. Comput. 304 (2017), pp. 180–189.
   Web of Science ®Google Scholar
• X. Wu, W. Deng, and E. Barkai, Tempered fractional Feynman-Kac equation, Phys. Rev. E. 93 (2016), p. 032151.
   PubMed Web of Science ®Google Scholar
• J. Yang, J. Huang, D. Liang, and Y. Tang, Numerical solution of fractional diffusion-wave equation based on fractional multistep method, Appl. Math. Model. 38(14) (2014), pp. 3652–3661.
   Web of Science ®Google Scholar
• C. Yaslan, Numerical solution of the conformable space-time fractional wave equation, Chinese J. Phys. 56 (2018), pp. 2916–2925.
   Web of Science ®Google Scholar
• M. Yousuf, K.M. Furati, A.Q.M. Khaliq, High-order time-stepping methods for two-dimensional Riesz fractional nonlinear reaction–diffusion equations. Comput. Math. Appl. 80(1) (2020), pp. 204–226. https://doi.org/10.1016/j.camwa.2020.03.010.
   Web of Science ®Google Scholar
• Y. Yu, W. Deng, Y. Wu, and J. Wu, Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations, Appl. Numer. Math.112 (2017), pp. 126–145.
   Web of Science ®Google Scholar
• M.A. Zaky, Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math. 145 (2019), pp. 429–457.
   Web of Science ®Google Scholar
• M.A. Zaky and J.A.T Machado, Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations, Comput. Math. Appl. 79 (2020), pp. 476–488.
   Web of Science ®Google Scholar
• M.A. Zaky, A.S. Hendy, and J.E. Macias-Diaz, Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion–reaction equations with smooth and nonsmooth solutions, J. Sci. Comput. 82 (2020), pp. 1–27.
   Web of Science ®Google Scholar
• Y. Zhang and Z.Z. Sun, Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput. 59 (2014), pp. 104–128.
   Web of Science ®Google Scholar
• H. Zhang, F. Liu, X. Jiang, F. Zeng, and I. Turner, A Crank–Nicolson ADI Galerkin–Legendre spectral method for the two-dimensional Riesz space distributed–order advection–diffusion equation, Comput. Math. Appl. 76 (2018), pp. 2460–2476.
   Web of Science ®Google Scholar
• Y. Zhang, Q. Li, and H. Ding, High-order numerical approximation formulas for Riemann-Liouville (Riesz) tempered fractional derivatives: Construction and application (I), Appl. Math. Comput. 329 (2018), pp. 432–443.
   Web of Science ®Google Scholar
• H. Zhang, X. Jiang, C. Wang, and S. Chen, Crank-Nicolson Fourier spectral methods for the space fractional nonlinear Schrodinger equation and its parameter estimation, Int. J. Comput. Math. 96 (2018), pp. 238–263.
   Web of Science ®Google Scholar
• Z. Zhao and C.P. Li, Fractional difference/finite element approximations for the time–space fractional telegraph equation, Appl. Math. Comput. 219 (2012), pp. 2975–2988.
   Web of Science ®Google Scholar
• X. Zhao, Z.Z. Sun, and Z.P. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation, SIAM. J. Sci. Comput.36(6) (2014), pp. A2865–A2886.
   Web of Science ®Google Scholar
• L. Zhao, W. Deng, and J.S. Hesthaven, Spectral methods for tempered fractional differential equations, preprint (2016). Available at arXiv:1603.06511.
   Google Scholar