A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations

Abstract Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations. Firstly, the considered problem is equivalently transformed into its partial integro-differential form with the Riemann-Liouville integral and multi-term Caputo derivatives. Secondly, the ADI scheme is constructed by using the first-order approximations and L1 approximations to approximate the terms in time, and using the fractional centered differences to discretize the multi-term Riesz fractional derivatives in space. Furthermore, the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals. Then, the solvability, unconditional stability and convergence of the proposed ADI scheme are strictly established. Finally, two numerical examples are given to support our theoretical results, and demonstrate the computational performances of the fast ADI scheme.

[1]  Zhimin Zhang,et al.  Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations , 2015, 1511.03453.

[2]  J. C. López-Marcos.  A difference scheme for a nonlinear partial integrodifferential equation , 1990 .

[3]  Hao Guo,et al.  A superlinear convergence scheme for nonlinear fractional differential equations and its fast implement , 2019, Int. J. Model. Simul. Sci. Comput..

[4]  Fawang Liu,et al.  The Unstructured Mesh Finite Element Method for the Two-Dimensional Multi-term Time–Space Fractional Diffusion-Wave Equation on an Irregular Convex Domain , 2018, J. Sci. Comput..

[5]  Xuan Zhao,et al.  Compact Alternating Direction Implicit Scheme for the Two-Dimensional Fractional Diffusion-Wave Equation , 2012, SIAM J. Numer. Anal..

[6]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[7]  K. Burrage,et al.  Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain , 2012 .

[8]  Cui-Cui Ji,et al.  Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation , 2016, J. Sci. Comput..

[9]  Zhibo Wang,et al.  A high-order ADI scheme for the two-dimensional time fractional diffusion-wave equation , 2013, Int. J. Comput. Math..

[10]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[11]  Fawang Liu,et al.  Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains , 2019, Commun. Nonlinear Sci. Numer. Simul..

[12]  Fawang Liu,et al.  Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation , 2016 .

[13]  Sachin Bhalekar,et al.  Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method , 2008, Appl. Math. Comput..

[14]  Mehdi Dehghan,et al.  Spectral element technique for nonlinear fractional evolution equation, stability and convergence analysis , 2017 .

[15]  Chengming Huang,et al.  An energy conservative difference scheme for the nonlinear fractional Schrödinger equations , 2015, J. Comput. Phys..

[16]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[17]  Wenping Chen,et al.  A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients , 2018, J. Comput. Appl. Math..

[18]  Emilia G. Bazhlekova,et al.  Subordination approach to multi-term time-fractional diffusion-wave equations , 2017, J. Comput. Appl. Math..

[19]  K. N. Rai,et al.  A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues , 2010, Math. Comput. Model..

[20]  Jianfei Huang,et al.  Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations , 2020, Numerical Algorithms.

[21]  Cem Çelik,et al.  Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative , 2012, J. Comput. Phys..

[22]  Jincheng Ren,et al.  Efficient numerical approximation of the multi-term time fractional diffusion-wave equations , 2015 .

[23]  Jiwei Zhang,et al.  Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations , 2018, SIAM J. Numer. Anal..

[24]  Fawang Liu,et al.  The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation , 2019, J. Comput. Appl. Math..

[25]  C. Lubich.  Discretized fractional calculus , 1986 .

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

[27]  Limei Li,et al.  Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation , 2013, J. Comput. Phys..

[28]  Fawang Liu,et al.  Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model , 2019, Numerical Methods for Partial Differential Equations.

[29]  Mehdi Dehghan,et al.  Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations , 2015, J. Comput. Appl. Math..

[30]  Fawang Liu,et al.  An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations , 2018, Applied Numerical Mathematics.

[31]  Mehdi Dehghan,et al.  An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate , 2017, Numerical Algorithms.

[32]  D. Benson,et al.  Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications , 2009 .

[33]  A. Aminataei,et al.  An algorithm for solving multi-term diffusion-wave equations of fractional order , 2011, Comput. Math. Appl..

[34]  I. Turner,et al.  A high-order spectral method for the multi-term time-fractional diffusion equations , 2016 .

[35]  Da Xu,et al.  Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel , 2020, Comput. Math. Appl..

[36]  Ahmed M. A. El-Sayed,et al.  Simulation of the approximate solutions of the time-fractional multi-term wave equations , 2017, Comput. Math. Appl..

[37]  Jianfei Huang.  Alternating Direction Implicit Schemes for the Two-Dimensional Time Fractional Nonlinear Super-Diffusion Equations , 2019, Journal of Computational Mathematics.

[38]  A. Aminataei,et al.  Sixth-order non-uniform combined compact difference scheme for multi-term time fractional diffusion-wave equation , 2018, Applied Numerical Mathematics.

[39]  Wen Chen,et al.  Coupling of the improved singular boundary method and dual reciprocity method for multi-term time-fractional mixed diffusion-wave equations , 2019, Comput. Math. Appl..

[40]  Mostafa Abbaszadeh,et al.  Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method , 2019, Applied Numerical Mathematics.

[41]  Fawang Liu,et al.  Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations , 2014, Appl. Math. Comput..

[42]  Fawang Liu,et al.  Multi-term time-fractional Bloch equations and application in magnetic resonance imaging , 2017, J. Comput. Appl. Math..

[43]  Mehdi Dehghan,et al.  An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch–Torrey equations , 2018, Applied Numerical Mathematics.

[44]  Rezvan Salehi,et al.  A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation , 2017, Numerical Algorithms.

[45]  Ali H. Bhrawy,et al.  A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations , 2015, J. Comput. Phys..

[46]  Mostafa Abbaszadeh,et al.  Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation , 2019, Appl. Math. Lett..

[47]  L. Greengard,et al.  Spectral Deferred Correction Methods for Ordinary Differential Equations , 2000 .

[48]  Tingzhu Huang,et al.  FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS , 2018, Journal of Applied Analysis & Computation.