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Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 6
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Articles

Compact scheme for fractional diffusion-wave equation with spatial variable coefficient and delays

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Pages 1911-1932 | Received 23 Mar 2020, Accepted 14 Jun 2020, Published online: 07 Jul 2020

References

  • Li TY, Zhang Q, Niazi W, et al. An effective algorithm for delay fractional convection–diffusion wave equation based on reversible exponential recovery method. IEEE Access. 2019;7:5554–5563.
  • Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Physics Reports. 2000;339:1–77.
  • Oldham KB, Spanier J. The Fractional Calculus. New York: Academic Press; 1974.
  • Deng WH, Zhang ZJ. High Accuracy Algorithm for the Differential Equations Governing Anomalous Diffusion: Algorithm and Models for Anomalous Diffusion. Singapore: World Scientific; 2019.
  • Deng WH, Hou R, Wang WL. Modeling Anomalous Diffusion from Statistics to Mathematics. Singapore: World Scientific; 2020.
  • Podlubny I. Fractional Differential Equations. New York: Academic Press; 1999.
  • Sun ZZ, Gao GH. Difference Schemes for Fractional Order Differential Equations. Beijing: Science Press; 2015.
  • Kochubei AN. Cauchy problem for fractional diffusion-wave equations with variable coefficients. Appl Anal. 2014;93(10):2211–2242.
  • Kima I, Kimb KH, Limb S. An Lq(Lp)-theory for the time fractional evolution equations with variable coefficients. Adv Math. 2017;306:123–176.
  • Li ZY, Huang XC, Yamamoto M. Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients. ArXiv:1802.06269. 2018.
  • Momani S. Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl Math Comput. 2005;165:459–472.
  • Chen YM, Wu YB, Cui YH, et al. Wavelet method for a class of fractional convection–diffusion equation with variable coefficients. J Comput Sci. 2010;1:146–149.
  • Garra R. Analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods. Commun Nonlinear Sci. 2012;17:1549–1554.
  • Hesameddini E, Rahimi A. Solving fractional partial differential equations with variable coefficients by the reconstruction of variational iteration method. Z für Naturforschung A. 2015;70:375–382.
  • Zhang S, Zhu R, Zhang LY. Exact solutions of time fractional heat-like and wave-like equations with variable coefficients. Therm Sci. 2016;20(3):S689–S693.
  • Jafari H, Khalique CM, Nazari M. Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations. Appl Math Lett. 2011;24:1799–1805.
  • Jafari H, Momani S. Solving fractional diffusion and wave equations by modified homotopy perturbation method. Phys Lett A. 2007;370:388–396.
  • Wang WS, Li DF. Stability analysis of Runge–Kutta methods for nonlinear neutral Volterra delay-integro-differential equations. Numer Math Theor Meth Appl. 2011;4(4):537–561.
  • Zhou BY, Li DF. Newton linearized methods for semilinear parabolic equations. Numer Math Theor Meth Appl. 2020;13(4):928–945.
  • Zhang Q, Li T. Asymptotic stability of compact and linear θ-methods for space fractional delay generalized diffusion equation. J Sci Comput. 2019;81:2413–2446.
  • Zhang Q, Ran MH, Xu DH. Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay. Appl Anal. 2017;96(11):1867–1884.
  • Ren L, Liu L. Efficient compact finite difference method for variable coefficient fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions in conservative form. Comput Math Appl. 2018;37:6252–6269.
  • Ren L, Wang YM. A fourth-order extrapolated compact difference method for time-fractional convection–reaction–diffusion equations with spatially variable coefficients. Appl Math Comput. 2017;312:1–22.
  • Wang YM. A compact finite difference method for a class of time fractional convection–diffusion-wave equations with variable coefficients. Numer Algorithms. 2015;70(3):625–651.
  • Wang YM. A compact finite difference method for solving a class of time fractional convection–subdiffusion equations. BIT Numer Math. 2015;55(4):1187–1217.
  • Wang YM, Ren L. A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients. Appl Math Comput. 2019;342:71–93.
  • Wang YM, Ren L. Efficient compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients. Int J Comput Math. 2019;96(2):264–297.
  • Wang YM, Ren L. High-order compact difference methods for Caputo-type variable coefficient fractional sub-diffusion equations in conservative form. J Sci Comput. 2018;76(2):1007–1043.
  • Saadatmandi A, Dehghan M, Azizi MR. The Sinc–Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun Nonlinear Sci. 2012;17:4125–4136.
  • Bhrawy AH. A new numerical algorithm for solving a class of fractional advection–dispersion equation with variable coefficients using Jacobi polynomials. Abst Appl Anal. 2013;954983:9. doi:https://doi.org/10.1155/2013/954983.
  • Yi MX, Huang J. Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl Math Comput. 2014;230:383–394.
  • Vong S, Lyu P, Wang ZB. A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under Neumann boundary conditions. J Sci Comput. 2016;66:725–739.
  • Zhao X, Xu QW. Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl Math Model. 2014;38:3848–3859.
  • Chen H, Lü SJ, Chen WP. A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients. J Comput Appl Math. 2018;330:380–397.
  • Cui MR. Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficients. J Comput Phys. 2015;280:143–163.
  • Bhrawya A, Zakyc M. A fractional-order Jacobi Tau method for a class of time-fractional PDEs with variable coefficients. Math Method Appl Sci. 2016;39:1765–1779.
  • Si XH, Wang C, Shen YN, et al. Numerical method to initial-boundary value problems for fractional partial differential equations with time-space variable coefficients. Appl Math Model. 2016;40:4397–4411.
  • Mustapha K, Abdallah B, Furati KM, et al. A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients. Numer. Algorithms. 2016;7:517–534.
  • McLean W, Mustapha K. A second-order accurate numerical method for a fractional wave equation. Numer Math. 2007;105:481–510.
  • Dahaghin MS, Hassani H. A new optimization method for a class of time fractional convection–diffusion-wave equations with variable coefficients. Eur. Phys. J. Plus.. 2017;132. doi:https://doi.org/10.1140/epjp/i2017-11407-y.
  • Mardani A, Hooshmandasl MR, Heydari MH, et al. A meshless method for solving the time fractional advection–diffusion equation with variable coefficients. Comput Math Appl. 2018;75:122–133.
  • Vong S, Lyu P. On a second order scheme for space fractional diffusion equations with variable coefficients. Appl Numer Math. 2019;137:34–48.
  • Wang FL, Liu F, Zhao YM, et al. A novel approach of high accuracy analysis of anisotropic bilinear finite element for time-fractional diffusion equations with variable coefficient. Comput Math Appl. 2018;75:3786–3800.
  • Jackiewicz Z, Zubik-Kowal B. Discrete variable methods for delay-differential equations with threshold-type delays. J Comput App Math. 2009;228(2):514–523.
  • Zubik-Kowal B. Stability in the numerical solution of linear parabolic equations with a delay term. BIT Numer Math. 2006;41:191–206.
  • Liao WY. An implicit fourth-order compact finite difference scheme for one-dimensional Burgers' equation. Appl Math Comput. 2008;206:755–764.
  • Zhang Q, Zhang C. A new linearized compact multisplitting scheme for the nonlinear convection–reaction-diffusion equations with delay. Commun Nonlinear Sci. 2013;18:3278–3288.
  • Du R, Cao WR, Sun ZZ. A compact difference scheme for the fractional diffusion-wave equation. Appl Math Model. 2010;34:2998–3007.
  • Li DF, Liao HL, Sun WW, et al. Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun Comput Phys. 2018;24(1):86–103.
  • Liao HL, Li DF, Zhang JW. Sharp error estimate of the nonuniform L1 formula for linear reaction–subdiffusion equations. SIAM J Numer Anal. 2018;56(2):1112–1133.
  • Sun ZZ. An unconditionally stable and O(τ2+h4) order l∞ convergent difference scheme for linear parabolic equations with variable coefficients. Numer Meth Part D E. 2001;17(6):619–631.
  • Sun ZZ, Wu XN. A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math. 2006;56:193–209.
  • Gao GH, Sun ZZ. A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys. 2011;230:586–595.
  • Zhao X, Sun ZZ. Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J Sci Comput. 2015;62(3):747–771.

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