A high-order compact alternating direction implicit method for solving the 3D time-fractional diffusion equation with the Caputo–Fabrizio operator

In this paper, a high-order compact finite difference method (CFDM) with an operator-splitting technique for solving the 3D time-fractional diffusion equation is considered. The Caputo–Fabrizio time operator is evaluated by the $$L_1$$ approximation, and the second-order space derivatives are approximated by the compact CFDM to obtain a discrete scheme. Alternating direction implicit method (ADI) is used to split the problem into three separate one-dimensional problems. The local truncation error analysis is discussed. Moreover, the convergence and stability of the numerical method are investigated. Finally, some numerical examples are presented to demonstrate the accuracy of the compact ADI method.

[1]  R. Striegl,et al.  A numerical solution for the diffusion equation in hydrogeologic systems , 1989 .

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

[3]  I. Podlubny.  Fractional differential equations , 1998 .

[4]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[5]  Santos B. Yuste,et al.  On an explicit finite difference method for fractional diffusion equations , 2003, ArXiv.

[6]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

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

[8]  O. Agrawal,et al.  A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems , 2007 .

[9]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

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

[11]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[12]  J. Seiber,et al.  Environmental Transport and Fate , 2010 .

[13]  Fawang Liu,et al.  An implicit RBF meshless approach for time fractional diffusion equations , 2011 .

[14]  Maria Alessandra Ragusa,et al.  Necessary and sufficient condition for a VMO function , 2012, Appl. Math. Comput..

[15]  Mingrong Cui.  Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation , 2012, Numerical Algorithms.

[16]  Reza Ansari,et al.  A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects , 2013, Appl. Math. Comput..

[17]  M. Kanoria,et al.  Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell , 2014 .

[18]  T. Sun,et al.  Characteristic fractional step finite difference method for nonlinear section coupled system , 2014 .

[19]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[20]  B. Guo,et al.  Fractional Calculus and Fractional Differential Equations , 2015 .

[21]  B. Guo,et al.  Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions , 2016 .

[22]  Hasan A. Fallahgoul,et al.  Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application , 2016 .

[23]  Zhengguang Liu,et al.  A second order Crank-Nicolson scheme for fractional Cattaneo equation based on new fractional derivative , 2017, Appl. Math. Comput..

[24]  M. N. Özişik,et al.  Finite Difference Methods in Heat Transfer , 2017 .

[25]  W. Zahra,et al.  Non standard finite difference method for solving variable order fractional optimal control problems , 2017 .

[26]  Dongfang Li,et al.  A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction-diffusion equations , 2018, Comput. Math. Appl..

[27]  Ziwen Jiang,et al.  Fourth-order compact finite difference method for solving two-dimensional convection–diffusion equation , 2018, Advances in Difference Equations.

[28]  Jinming Wen,et al.  Stability and convergence of compact finite difference method for parabolic problems with delay , 2018, Appl. Math. Comput..

[29]  Yanzhi Zhang,et al.  A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem , 2018, J. Comput. Phys..

[30]  S. Ray,et al.  Generalized Fractional Order Differential Equations Arising in Physical Models , 2018 .

[31]  T. Abdeljawad,et al.  Ulam stability for delay fractional differential equations with a generalized Caputo derivative , 2018 .

[32]  Fazal Ghaffar,et al.  HIGHER ORDER COMPACT FINITE DIFFERENCE METHOD FOR THE SOLUTION OF 2-D TIME FRACTIONAL DIFFUSION EQUATION , 2018 .

[33]  Delfim F. M. Torres,et al.  A finite element approximation for a class of Caputo time-fractional diffusion equations , 2019, Comput. Math. Appl..

[34]  Dongfang Li,et al.  A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations , 2019, Appl. Math. Comput..

[35]  Mingrong Cui.  Compact Difference Scheme for Time-Fractional Fourth-Order Equation with First Dirichlet Boundary Condition , 2019, East Asian Journal on Applied Mathematics.

[36]  M. Kratou.  Ground state solutions of p-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative , 2019, Filomat.

[37]  Akanksha Bhardwaj,et al.  A meshless local collocation method for time fractional diffusion wave equation , 2019, Comput. Math. Appl..

[38]  Raquel García Rubio,et al.  CMMSE: Computational and Mathematical Methods in Science and Engineering , 2020, Int. J. Comput. Math..

[39]  Dongdong Hu,et al.  A fourth-order compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equation , 2019, Int. J. Comput. Math..