A Lagrange-quadratic spline optimal collocation method for the time tempered fractional diffusion equation

Abstract In the current paper, for the time fractional diffusion equation with an exponential tempering, we propose a numerical algorithm based on the Lagrange-quadratic spline interpolations and the optimal technique. The discretized linear systems and some properties are investigated in details. By using these properties, the coefficient matrix and the right-hand term at each time step are given to analyze the computational cost. Theoretical analyses show that this proposed method enjoys both stability and convergence order of O ( τ 2 + h 4 ) . Some numerical examples are provided to verify the practical feasibility and efficiency of the proposed scheme.

[1]  Kassem Mustapha,et al.  Time-stepping discontinuous Galerkin methods for fractional diffusion problems , 2014, Numerische Mathematik.

[2]  Minghua Chen,et al.  A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation , 2016, Appl. Math. Lett..

[3]  J. Rosínski.  Tempering stable processes , 2007 .

[4]  Yong Zhang,et al.  Moments for Tempered Fractional Advection-Diffusion Equations , 2010 .

[5]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[6]  Ali H. Bhrawy,et al.  A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation , 2015, Numerical Algorithms.

[7]  Siu-Long Lei,et al.  Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations , 2019, Numerical Algorithms.

[8]  Cui-Cui Ji,et al.  Numerical Method for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Equation with the Temperature-Jump Boundary Condition , 2018, J. Sci. Comput..

[9]  Mingrong Cui,et al.  Compact finite difference method for the fractional diffusion equation , 2009, J. Comput. Phys..

[10]  Siu-Long Lei,et al.  A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations , 2017, Numerical Algorithms.

[11]  Mehdi Dehghan,et al.  Fourth-order numerical method for the space-time tempered fractional diffusion-wave equation , 2017, Appl. Math. Lett..

[12]  Mehdi Dehghan,et al.  A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation , 2018, Comput. Math. Appl..

[13]  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..

[14]  Christina C. Christara,et al.  Fast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation , 2002 .

[15]  Meng Li,et al.  A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations , 2018, J. Comput. Phys..

[16]  D. del-Castillo-Negrete,et al.  Truncation effects in superdiffusive front propagation with Lévy flights. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  D. Schötzau,et al.  Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations , 2014 .

[18]  Guang Lin,et al.  A second-order difference scheme for the time fractional substantial diffusion equation , 2016, J. Comput. Appl. Math..

[19]  M. Dehghan,et al.  Solving nonlinear fractional partial differential equations using the homotopy analysis method , 2010 .

[20]  Á. Cartea,et al.  Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Xianjuan Li,et al.  A Space-Time Spectral Method for the Time Fractional Diffusion Equation , 2009, SIAM J. Numer. Anal..

[22]  Michael K. Ng,et al.  A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations , 2015, J. Comput. Phys..

[23]  Can Li,et al.  Local discontinuous Galerkin methods for the time tempered fractional diffusion equation , 2017, Appl. Math. Comput..

[24]  Ting-Zhu Huang,et al.  A High-Order Accurate Numerical Scheme for the Caputo Derivative with Applications to Fractional Diffusion Problems , 2018 .

[25]  Xian-Ming Gu,et al.  A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel , 2020, J. Comput. Phys..

[26]  C. C. Christara,et al.  Optimal Quadratic and Cubic Spline Collocation on Nonuniform Partitions , 2005, Computing.

[27]  Tong Chen,et al.  Quadratic spline collocation for one-dimensional linear parabolic partial differential equations , 2009, Numerical Algorithms.

[28]  Weihua Deng,et al.  Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations , 2017 .

[29]  Graeme Fairweather,et al.  Compact optimal quadratic spline collocation methods for the Helmholtz equation , 2011, J. Comput. Phys..

[30]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[31]  Mark M. Meerschaert,et al.  Tempered stable Lévy motion and transient super-diffusion , 2010, J. Comput. Appl. Math..

[32]  J. Rice,et al.  Quadratic‐spline collocation methods for two‐point boundary value problems , 1988 .

[33]  M. Meerschaert,et al.  Tempered anomalous diffusion in heterogeneous systems , 2008 .

[34]  B. Kumar,et al.  Finite element method for drifted space fractional tempered diffusion equation , 2019, J. Appl. Math. Comput..

[35]  C. C. Christara,et al.  Adaptive Techniques for Spline Collocation , 2005, Computing.

[36]  C. Christara.  Quadratic spline collocation methods for elliptic partial differential equations , 1994 .

[37]  M. Dehghan,et al.  The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations , 2016 .

[38]  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.

[39]  Weihua Deng,et al.  High order schemes for the tempered fractional diffusion equations , 2016, Adv. Comput. Math..

[40]  Cécile Piret,et al.  A Chebyshev PseudoSpectral Method to Solve the Space-Time Tempered Fractional Diffusion Equation , 2014, SIAM J. Sci. Comput..

[41]  George Em Karniadakis,et al.  Efficient Multistep Methods for Tempered Fractional Calculus: Algorithms and Simulations , 2018, SIAM J. Sci. Comput..