Compact Alternating Direction Implicit Scheme for the Two-Dimensional Fractional Diffusion-Wave Equation
Abstract
In this paper, we consider the numerical method for solving the two-dimensional fractional diffusion-wave equation with a time fractional derivative of order $\alpha$ ($1<\alpha<2$). A difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping is proposed and analyzed. The unconditional stability and $H^1$ norm convergence of the scheme are proved rigorously. The convergence order is $\mathcal {O}(\tau^{3-\alpha}+h_1^4+h^4_2),$ where $\tau$ is the temporal grid size and $h_1,h_2$ are spatial grid sizes in the $x$ and $y$ directions, respectively. In addition, a Crank--Nicolson ADI scheme is presented and the corresponding error estimates are also established. Numerical results are presented to support our theoretical analysis and indicate that the compact ADI scheme reduces the storage requirement and CPU time successfully.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
2.
R. Hilfer, ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
3.
J. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), pp. 127--293.
4.
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1--77.
5.
T. H. Solomon, E. R. Weeks, and H. L. Swinney, Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow, Phys. Rev. Lett., 71 (1993), pp. 3975--3979.
6.
W. Wyss, Fractional diffusion equation, J. Math. Phys., 27 (1986), pp. 2782--2785.
7.
W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), pp. 134--144.
8.
F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), pp. 23--28.
9.
R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dyn., 29 (2002), pp. 129--143.
10.
R. Gorenflo, F. Mainardi, and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34 (2007), pp. 87--103.
11.
O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam., 29 (2002), pp. 145--155.
12.
S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862--1874.
13.
S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006), pp. 264--274.
14.
C. M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), pp. 886--897.
15.
T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), pp. 719--736.
16.
P. Zhuang, F. Liu, V. Anh, and I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46 (2008), pp. 1079--1095.
17.
Z. Z. Sun and X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), pp. 193--209.
18.
E. Cuesta, C. Lubich, and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), pp. 673--696.
19.
M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), pp. 7792--7804.
20.
R. Du, W. R. Cao, and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), pp. 2998--3007.
21.
C. M. Chen, F. Liu, V. Anh, and I. Turner, Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32 (2010), pp. 1740--1760.
22.
G. H. Gao and Z. Z. Sun, A compact difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), pp. 586--595.
23.
P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), pp. 1760--1781.
24.
X. Zhao and Z. Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), pp. 6061--6074.
25.
Y. N. Zhang, Z. Z. Sun, and H. W. Wu, Error estimates of Crank--Nicolson-type difference schemes for the subdiffusion equation, SIAM J. Numer. Anal., 49 (2011), pp. 2302--2322.
26.
Y. Gu, P. Zhuang, and F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES Comput. Model. Eng. Sci., 56 (2010), pp. 303--333.
27.
Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533--1552.
28.
X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108-2131.
29.
Y. Lin, X. Li, and C. Xu, Finite difference/spectral approximations for the fractional cable equation, Math. Comp., 80 (2011), pp. 1369--1396.
30.
W. Deng, Finite element method for the the space and time fractional Fokker--Planck equation, SIAM J. Numer. Anal., 47 (2008), pp. 204--226.
31.
C. M. Chen, F. Liu, I. Turner, and V. Anh, Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algorithms, 54 (2010), pp. 1--21.
32.
Y. N. Zhang and Z. Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), pp. 8713--8728.
33.
H. Brunner, L. Ling, and M. Yamamoto, Numerical simulations of 2D fractional subdiffusion problems, J. Comput. Phys., 229 (2010), pp. 6613--6622.
34.
P. Zhuang and F. Liu, Finite difference approximation for two-dimensional time fractional diffusion equation, J. Algorithm Computat. Tech., 1 (2007), pp. 1--15.
35.
C. Chen, F. Liu, V. Anh, and I. Turner, Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation, Math. Comp., 81 (2011), pp. 345--366.
36.
C. Chen and F. Liu, A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation, J. Appl. Math. Comput., 30 (2009), pp. 219--236.
37.
Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu, Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions, ANZIAM J. Electron. Suppl., 52 (2010), pp. C395--C409.
38.
Q. Yang, I. Turner, F. Liu, and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), pp. 1159--1180.
39.
Q. Liu and F. Liu, Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives in uniform media, Math. Numer. Sin., 31 (2009), pp. 179--194.
40.
Q. Liu, F. Liu, I. Turner, and V. Anh, Numerical simulation for the three-dimensional seepage flow with fractional derivatives in porous media, IMA J. Appl. Math., 74 (2009), pp. 201--229.
41.
S. Chen and F. Liu, ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation, J. Appl. Math. Comput., 26 (2008), pp. 295--311.
42.
J. C. López-Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal., 27 (1990), pp. 20--31.
43.
T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11 (1993), pp. 309--319.
44.
A. A. Samarskii and V. B. Andreev, Difference Methods for Elliptic Equation, Nauka, Moscow, 1976.
45.
Z. Z. Sun, The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.
Information & Authors
Information
Published In
Copyright
© 2012, Society for Industrial and Applied Mathematics.
History
Submitted: 14 July 2011
Accepted: 16 April 2012
Published online: 5 June 2012
Keywords
MSC codes
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equationApplied Mathematics and Computation, Vol. 488 | 1 Mar 2025
- A comprehensive Approach of Caputo Space Fractional Bioheat Model During Hyperthermia Based on Fractional Chebyshev Collocation SchemeIranian Journal of Science, Vol. 49, No. 1 | 24 September 2024
- An efficient computational approach and its analysis for the Caputo time‐fractional convection reaction–diffusion equation in two‐dimensional spaceMathematical Methods in the Applied Sciences, Vol. 48, No. 1 | 5 July 2024
- Integrated Lucas and Fibonacci polynomial approach for time fractional fast and slow diffusion modelsInternational Journal of Modern Physics C, Vol. 11 | 10 January 2025
- A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equationComputers & Mathematics with Applications, Vol. 176 | 1 Dec 2024
- Sinc-Galerkin method and a higher-order method for a 1D and 2D time-fractional diffusion equationsBoundary Value Problems, Vol. 2024, No. 1 | 6 September 2024
- Convergence analysis of a L1‐ADI scheme for two‐dimensional multiterm reaction‐subdiffusion equationNumerical Methods for Partial Differential Equations, Vol. 40, No. 6 | 21 May 2024
- An efficient temporal approximation for weakly singular time-fractional semilinear diffusion-wave equation with variable coefficientsNumerical Algorithms, Vol. 62 | 18 October 2024
- A meshless particle method for solving time-fractional wave equationsComputational Particle Mechanics, Vol. 11, No. 5 | 30 May 2024
- A fourth-order compact ADI scheme for solving a two-dimensional time-fractional reaction-subdiffusion equationJournal of Mathematical Chemistry, Vol. 62, No. 8 | 25 June 2024
- Convergence analysis of a fully discrete scheme for diffusion-wave equation forced by tempered fractional Brownian motionComputers & Mathematics with Applications, Vol. 169 | 1 Sep 2024
- Numerical Approach of Cattaneo Equation with Time Caputo-Fabrizio Fractional DerivativeIranian Journal of Mathematical Sciences and Informatics, Vol. 19, No. 2 | 1 September 2024
- Fast and Accurate Computation of the Regime-Switching Jump-Diffusion Option Prices Using Laplace Transform and Compact Difference with Convergence GuaranteeComputational Economics, Vol. 64, No. 1 | 22 July 2023
- An Efficient IMEX Compact Scheme for the Coupled Time Fractional Integro-Differential Equations Arising from Option Pricing with JumpsComputational Economics, Vol. 95 | 5 June 2024
- Compact IMEX scheme for a moving boundary PIDE system of the regime-switching jump-diffusion Asian option pricingNumerical Algorithms, Vol. 95, No. 3 | 4 July 2023
- A Bound-Preserving Numerical Scheme for Space–Time Fractional Advection EquationsFractal and Fractional, Vol. 8, No. 2 | 30 January 2024
- Analysis of two conservative fourth-order compact finite difference schemes for the Klein-Gordon-Zakharov system in the subsonic limit regimeApplied Mathematics and Computation, Vol. 460 | 1 Jan 2024
- Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equationAIMS Mathematics, Vol. 9, No. 1 | 1 Jan 2024
- Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equationComputers & Mathematics with Applications, Vol. 151 | 1 Dec 2023
- A class of improved fractional physics informed neural networksNeurocomputing, Vol. 562 | 1 Dec 2023
- A high-order numerical scheme for multidimensional convection-diffusion-reaction equation with time-fractional derivativeNumerical Algorithms, Vol. 94, No. 2 | 4 March 2023
- An iterative method based on Nesterov acceleration for identifying space-dependent source term in a time-fractional diffusion-wave equationJournal of Computational and Applied Mathematics, Vol. 429 | 1 Sep 2023
- Second‐order difference scheme for the time fractional mixed diffusion‐wave equation with initial weak regularityMathematical Methods in the Applied Sciences, Vol. 281 | 16 August 2023
- The transformed differential quadrature method for solving time-dependent partial differential equations: Framework and examplesComputers & Mathematics with Applications, Vol. 140 | 1 Jun 2023
- High-order orthogonal spline collocation ADI scheme for a new complex two-dimensional distributed-order fractional integro-differential equation with two weakly singular kernelsInternational Journal of Computer Mathematics, Vol. 100, No. 4 | 23 November 2022
- Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutionsMathematical Methods in the Applied Sciences, Vol. 46, No. 5 | 11 November 2022
- Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensionsMathematics and Computers in Simulation, Vol. 205 | 1 Mar 2023
- On Solutions of Fractional Integrodifferential Systems Involving Ψ-Caputo Derivative and Ψ-Riemann–Liouville Fractional IntegralMathematics, Vol. 11, No. 6 | 17 March 2023
- An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional spaceJournal of Applied Mathematics and Computing, Vol. 69, No. 1 | 21 June 2022
- An ADI orthogonal spline collocation method for a new two-dimensional distributed-order fractional integro-differential equationComputers & Mathematics with Applications, Vol. 132 | 1 Feb 2023
- The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanicsDiscrete and Continuous Dynamical Systems - B, Vol. 28, No. 5 | 1 Jan 2023
- An ADI finite difference method for the two-dimensional Volterra integro-differential equation with weakly singular kernelInternational Journal of Computer Mathematics, Vol. 99, No. 12 | 10 May 2022
- A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equationComputers & Mathematics with Applications, Vol. 126 | 1 Nov 2022
- A novel high-order numerical scheme and its analysis for the two-dimensional time-fractional reaction-subdiffusion equationNumerical Algorithms, Vol. 90, No. 4 | 24 January 2022
- ADI Galerkin finite element scheme for the two-dimensional semilinear partial intergro-differential equation with a weakly singular kernelJournal of Applied Mathematics and Computing, Vol. 68, No. 4 | 10 September 2021
- A second-order L2-1 Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficientsComputers & Mathematics with Applications, Vol. 118 | 1 Jul 2022
- A novel meshless collocation solver for solving multi-term variable-order time fractional PDEsEngineering with Computers, Vol. 38, No. S2 | 2 February 2021
- Compact implicit difference approximation for time-fractional diffusion-wave equationAlexandria Engineering Journal, Vol. 61, No. 5 | 1 May 2022
- Fast ADI difference/compact difference schemes for the nonlocal evolution equation with weakly singular kernels in three dimensionsMathematics and Computers in Simulation, Vol. 194 | 1 Apr 2022
- A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equationInverse Problems, Vol. 38, No. 4 | 24 February 2022
- Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional spaceApplied Numerical Mathematics, Vol. 172 | 1 Feb 2022
- Locally One-Dimensional Schemes for Quasilinear Parabolic Equations with Time Fractional DerivativeMesh Methods for Boundary-Value Problems and Applications | 21 October 2021
- A fast element-free Galerkin method for the fractional diffusion-wave equationApplied Mathematics Letters, Vol. 122 | 1 Dec 2021
- A local meshless method to approximate the time-fractional telegraph equationEngineering with Computers, Vol. 37, No. 4 | 23 March 2020
- A compact scheme for two-dimensional nonlinear time fractional wave equationsInternational Journal of Modeling, Simulation, and Scientific Computing, Vol. 12, No. 05 | 31 March 2021
- Numerical methods for time-fractional convection-diffusion problems with high-order accuracyOpen Mathematics, Vol. 19, No. 1 | 16 August 2021
- Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equationsAdvances in Computational Mathematics, Vol. 47, No. 4 | 6 August 2021
- Finite difference schemes for the two-dimensional multi-term time-fractional diffusion equations with variable coefficientsComputational and Applied Mathematics, Vol. 40, No. 5 | 15 June 2021
- The Regularized Mesh Scheme to Solve Quasilinear Parabolic Equation with Time-Fractional DerivativeLobachevskii Journal of Mathematics, Vol. 42, No. 7 | 9 August 2021
- A linearized high-order Galerkin finite element approach for two-dimensional nonlinear time fractional Klein-Gordon equationsNumerical Algorithms, Vol. 87, No. 2 | 19 August 2020
- Analysis of a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equationComputational and Applied Mathematics, Vol. 40, No. 4 | 17 April 2021
- Optimal point-wise error estimate of two conservative fourth-order compact finite difference schemes for the nonlinear Dirac equationApplied Numerical Mathematics, Vol. 162 | 1 Apr 2021
- A fast temporal second-order compact ADI difference scheme for the 2D multi-term fractional wave equationNumerical Algorithms, Vol. 86, No. 2 | 12 March 2020
- A Crank-Nicolson ADI quadratic spline collocation method for two-dimensional Riemann-Liouville space-fractional diffusion equationsApplied Numerical Mathematics, Vol. 160 | 1 Feb 2021
- A New Numerical Approach for Solving 1D Fractional Diffusion-Wave EquationJournal of Function Spaces, Vol. 2021 | 6 Jan 2021
- Analytic and Numerical Solutions of Space-Time Fractional Diffusion Wave Equations with Different Fractional OrderComputational Science – ICCS 2021 | 9 June 2021
- Stability and convergence of multistep schemes for 1D and 2D fractional model with nonlinear source termApplied Mathematical Modelling, Vol. 89 | 1 Jan 2021
- A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equationsApplied Numerical Mathematics, Vol. 159 | 1 Jan 2021
- A high order numerical method for solving Caputo nonlinear fractional ordinary differential equationsAIMS Mathematics, Vol. 6, No. 12 | 1 Jan 2021
- An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile–immobile equationComputational and Applied Mathematics, Vol. 39, No. 4 | 6 October 2020
- A numerical treatment of the two-dimensional multi-term time-fractional mixed sub-diffusion and diffusion-wave equationCommunications in Nonlinear Science and Numerical Simulation, Vol. 91 | 1 Dec 2020
- Numerical Solution of a Quasilinear Parabolic Equation with a Fractional Time DerivativeLobachevskii Journal of Mathematics, Vol. 41, No. 12 | 4 February 2021
- Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source TermMathematics, Vol. 8, No. 11 | 29 October 2020
- Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementationsNumerical Algorithms, Vol. 85, No. 2 | 17 January 2020
- Ulam’s stability of multi-point implicit boundary value problems with non-instantaneous impulsesBollettino dell'Unione Matematica Italiana, Vol. 13, No. 3 | 14 March 2020
- A generalized-Jacobi-function spectral method for space-time fractional reaction-diffusion equations with viscosity termsApplied Numerical Mathematics, Vol. 152 | 1 Jun 2020
- High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problemMathematical Methods in the Applied Sciences, Vol. 43, No. 8 | 5 February 2020
- A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutionsNumerical Methods for Partial Differential Equations, Vol. 36, No. 3 | 18 November 2019
- Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernelComputers & Mathematics with Applications, Vol. 79, No. 2 | 1 Jan 2020
- Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadratureAIMS Mathematics, Vol. 5, No. 6 | 1 Jan 2020
- Approximate Solutions of Time Fractional Diffusion Wave ModelsMathematics, Vol. 7, No. 10 | 3 October 2019
- A meshless local collocation method for time fractional diffusion wave equationComputers & Mathematics with Applications, Vol. 78, No. 6 | 1 Sep 2019
- Numerical algorithm for two dimensional fractional Stokes’ first problem for a heated generalized second grade fluid with smooth and non-smooth solutionComputers & Mathematics with Applications, Vol. 78, No. 5 | 1 Sep 2019
- An ADI difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernelApplied Mathematics and Computation, Vol. 354 | 1 Aug 2019
- A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domainsApplied Mathematics Letters, Vol. 94 | 1 Aug 2019
- Application of meshless local Petrov–Galerkin technique to simulate two-dimensional time-fractional Tricomi-type problemJournal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 41, No. 6 | 16 May 2019
- Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equationApplied Mathematics and Computation, Vol. 349 | 1 May 2019
- Efficient Iterative Method for Solving Optimal Control Problem Governed by Diffusion Equation with Time Fractional DerivativeLobachevskii Journal of Mathematics, Vol. 40, No. 4 | 7 June 2019
- Energy estimates for two-dimensional space-Riesz fractional wave equationNumerical Algorithms, Vol. 80, No. 3 | 22 March 2018
- A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equationEngineering with Computers, Vol. 35, No. 1 | 9 February 2018
- The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for the Time Multi-term Fractional Wave EquationJournal of Scientific Computing, Vol. 78, No. 1 | 17 September 2018
- A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equationsEngineering Analysis with Boundary Elements, Vol. 98 | 1 Jan 2019
- Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshesApplied Mathematics and Computation, Vol. 338 | 1 Dec 2018
- A novel finite difference discrete scheme for the time fractional diffusion-wave equationApplied Numerical Mathematics, Vol. 134 | 1 Dec 2018
- A Time-Spectral Algorithm for Fractional Wave ProblemsJournal of Scientific Computing, Vol. 77, No. 2 | 20 June 2018
- Alternating direction implicit OSC scheme for the two-dimensional fractional evolution equation with a weakly singular kernelActa Mathematica Scientia, Vol. 38, No. 6 | 1 Nov 2018
- Quenching Phenomenon of a Time-Fractional Kawarada EquationJournal of Computational and Nonlinear Dynamics, Vol. 13, No. 10 | 22 August 2018
- Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE ApproachFractional Calculus and Applied Analysis, Vol. 21, No. 5 | 30 January 2019
- High Order Algorithm for the Time-Tempered Fractional Feynman–Kac EquationJournal of Scientific Computing, Vol. 76, No. 2 | 17 January 2018
- Numerical methods for fractional partial differential equationsInternational Journal of Computer Mathematics, Vol. 95, No. 6-7 | 4 July 2017
- Compact Alternating Direction Implicit Scheme for Integro-Differential Equations of Parabolic TypeJournal of Scientific Computing, Vol. 76, No. 1 | 20 January 2018
- Compact finite difference schemes for the time fractional diffusion equation with nonlocal boundary conditionsComputational and Applied Mathematics, Vol. 37, No. 3 | 16 December 2017
- A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equationNumerical Methods for Partial Differential Equations, Vol. 34, No. 3 | 18 December 2017
- Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equationComputational and Applied Mathematics, Vol. 37, No. 2 | 15 May 2017
- An efficient and conservative compact finite difference scheme for the coupled Gross–Pitaevskii equations describing spin-1 Bose–Einstein condensateApplied Mathematics and Computation, Vol. 323 | 1 Apr 2018
- High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshesInternational Journal of Computer Mathematics, Vol. 95, No. 1 | 20 November 2017
- Trapezoidal scheme for time–space fractional diffusion equation with Riesz derivativeJournal of Computational Physics, Vol. 350 | 1 Dec 2017
- An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction–diffusion-wave equationComputers & Mathematics with Applications, Vol. 74, No. 10 | 1 Nov 2017
- ADI Galerkin FEMs for the 2D nonlinear time-space fractional diffusion-wave equationInternational Journal of Modeling, Simulation, and Scientific Computing, Vol. 08, No. 03 | 20 September 2017
- Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshesApplicable Analysis, Vol. 96, No. 8 | 3 June 2016
- An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimateNumerical Algorithms, Vol. 75, No. 1 | 14 September 2016
- Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave EquationsSIAM Journal on Numerical Analysis, Vol. 55, No. 6 | 19 December 2017
- A second order BDF alternating direction implicit difference scheme for the two-dimensional fractional evolution equationApplied Mathematical Modelling, Vol. 41 | 1 Jan 2017
- A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equationsJournal of Applied Mathematics and Computing, Vol. 52, No. 1-2 | 22 October 2015
- The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equationsMathematical Methods in the Applied Sciences, Vol. 39, No. 14 | 1 February 2016
- A second-order BDF compact difference scheme for fractional-order Volterra equationInternational Journal of Computer Mathematics, Vol. 93, No. 7 | 17 March 2015
- A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensionsInternational Journal of Computer Mathematics, Vol. 93, No. 6 | 23 February 2015
- Some temporal second order difference schemes for fractional wave equationsNumerical Methods for Partial Differential Equations, Vol. 32, No. 3 | 21 December 2015
- Spectral methods for the time fractional diffusion–wave equation in a semi-infinite channelComputers & Mathematics with Applications, Vol. 71, No. 9 | 1 May 2016
- Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with dampingJournal of Computational Physics, Vol. 312 | 1 May 2016
- Finite difference schemes for two-dimensional time-space fractional differential equationsInternational Journal of Computer Mathematics, Vol. 93, No. 3 | 10 February 2015
- A Second-Order Finite Difference Method for Two-Dimensional Fractional Percolation EquationsCommunications in Computational Physics, Vol. 19, No. 3 | 16 March 2016
- Fast difference schemes for solving high-dimensional time-fractional subdiffusion equationsJournal of Computational Physics, Vol. 307 | 1 Feb 2016
- Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel ComputingFractional Calculus and Applied Analysis, Vol. 19, No. 1 | 9 March 2016
- A compact LOD method and its extrapolation for two-dimensional modified anomalous fractional sub-diffusion equationsComputers & Mathematics with Applications, Vol. 71, No. 1 | 1 Jan 2016
- ADI Finite Element Method for 2D Nonlinear Time Fractional Reaction-Subdiffusion EquationAmerican Journal of Computational Mathematics, Vol. 06, No. 04 | 1 Jan 2016
- An ADI Crank–Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave EquationJournal of Scientific Computing, Vol. 65, No. 3 | 18 March 2015
- Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equationsJournal of Computational and Applied Mathematics, Vol. 290 | 1 Dec 2015
- A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) methodComputers & Mathematics with Applications, Vol. 70, No. 10 | 1 Nov 2015
- An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equationInternational Journal of Computer Mathematics, Vol. 92, No. 10 | 25 November 2014
- Second-Order Stable Finite Difference Schemes for the Time-Fractional Diffusion-Wave EquationJournal of Scientific Computing, Vol. 65, No. 1 | 12 December 2014
- A compact locally one-dimensional method for fractional diffusion-wave equationsJournal of Applied Mathematics and Computing, Vol. 49, No. 1-2 | 29 August 2014
- Some high-order difference schemes for the distributed-order differential equationsJournal of Computational Physics, Vol. 298 | 1 Oct 2015
- A backward euler orthogonal spline collocation method for the time-fractional Fokker-Planck equationNumerical Methods for Partial Differential Equations, Vol. 31, No. 5 | 28 December 2014
- Efficient and stable numerical methods for the two-dimensional fractional Cattaneo equationNumerical Algorithms, Vol. 69, No. 4 | 15 October 2014
- Method of approximate particular solutions for constant- and variable-order fractional diffusion modelsEngineering Analysis with Boundary Elements, Vol. 57 | 1 Aug 2015
- A high-order ADI scheme for the two-dimensional time fractional diffusion-wave equationInternational Journal of Computer Mathematics, Vol. 92, No. 5 | 21 May 2014
- A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equationNumerical Algorithms, Vol. 68, No. 4 | 7 June 2014
- A center Box method for radially symmetric solution of fractional subdiffusion equationApplied Mathematics and Computation, Vol. 257 | 1 Apr 2015
- Compact Crank–Nicolson Schemes for a Class of Fractional Cattaneo Equation in Inhomogeneous MediumJournal of Scientific Computing, Vol. 62, No. 3 | 13 June 2014
- Local integration of 2-D fractional telegraph equation via local radial point interpolant approximationThe European Physical Journal Plus, Vol. 130, No. 2 | 25 February 2015
- Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave EquationEast Asian Journal on Applied Mathematics, Vol. 5, No. 1 | 6 March 2015
- High Order Algorithms for the Fractional Substantial Diffusion Equation with Truncated Lévy FlightsSIAM Journal on Scientific Computing, Vol. 37, No. 2 | 16 April 2015
- Time-Splitting Schemes for Fractional Differential Equations I: Smooth SolutionsSIAM Journal on Scientific Computing, Vol. 37, No. 4 | 16 July 2015
- Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficientsJournal of Computational Physics, Vol. 280 | 1 Jan 2015
- Stability and Convergence of Modified Du Fort–Frankel Schemes for Solving Time-Fractional Subdiffusion EquationsJournal of Scientific Computing, Vol. 61, No. 3 | 21 March 2014
- A new implicit compact difference scheme for the fourth-order fractional diffusion-wave systemInternational Journal of Computer Mathematics, Vol. 91, No. 10 | 26 March 2014
- High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary ConditionsEast Asian Journal on Applied Mathematics, Vol. 4, No. 3 | 28 May 2015
- An efficient parallel solution for Caputo fractional reaction–diffusion equationThe Journal of Supercomputing, Vol. 68, No. 3 | 24 February 2014
- Finite difference methods for the time fractional diffusion equation on non-uniform meshesJournal of Computational Physics, Vol. 265 | 1 May 2014
- Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion EquationJournal of Scientific Computing, Vol. 59, No. 1 | 25 July 2013
- Optimal Point-Wise Error Estimate of a Compact Difference Scheme for the Coupled Gross–Pitaevskii Equations in One DimensionJournal of Scientific Computing, Vol. 59, No. 1 | 31 July 2013
- Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equationJournal of Mathematical Analysis and Applications, Vol. 412, No. 1 | 1 Apr 2014
- Numerical Analysis of an H1 -Galerkin Mixed Finite Element Method for Time Fractional Telegraph EquationThe Scientific World Journal, Vol. 2014 | 1 Jan 2014
- Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equationJournal of Computational Physics, Vol. 255 | 1 Dec 2013
- Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary ConditionsJournal of Scientific Computing, Vol. 56, No. 2 | 20 January 2013
- Alternating direction implicit-Euler method for the two-dimensional fractional evolution equationJournal of Computational Physics, Vol. 236 | 1 Mar 2013
- The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion EquationSIAM Journal on Scientific Computing, Vol. 35, No. 6 | 17 December 2013
- Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion EquationAdvances in Mathematical Physics, Vol. 2013 | 1 Jan 2013
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].