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Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 69, 2016 - Issue 4
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Original Articles

Investigations on several compact ADI methods for the 2D time fractional diffusion equation

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Pages 364-376 | Received 27 May 2015, Accepted 20 Jul 2015, Published online: 23 Mar 2016

References

  • M. Raberto, E. Scalas, and F. Mainardi, Waiting-Times and Returns in High-Frequency Financial Data: An Empirical Study, Physica A, vol. 314, pp. 749–755, 2002.
  • L. Sabatelli, S. Keating, J. Dudley, and P. Richmond, Waiting Time Distributions in Financial Markets, Eur. Phys. J. B, vol. 27, pp. 273–275, 2002.
  • R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in A. Carpinteri and F. Mainardi (eds.), Fractals and Fractional Calculus in Continuun Mechanics, pp. 223–276, Springer-Verlag, Wien and New York, 1997.
  • F. Mainardi, Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena, Chaos, Solitons & Fractals, vol. 7, pp. 1461–1477, 1996.
  • I. Podlubny, Fractional Differential Equations Academic Press, New York, 1999.
  • R. Y. Molliq, M. S. M. Noorani, and I. Hashim, Variational Iteration Method for Fractional Heat- and Wave-Like Equations, Nonlinear Anal. Real World Appl., vol. 10, pp. 1854–1869, 2009.
  • Y. M. Lin and C. J. Xu, Finite Difference/Spectral Approximations for the Time-Fractional Diffusion Equation, J. Comput. Phys., vol. 225, pp. 1533–1552, 2007.
  • M. R. Cui, Compact Alternating Direction Implicit Method for Two Dimensional Time Fractional Diffusion Equation, J. Comput. Phys., vol. 231, pp. 2621–2633, 2012.
  • J. C. Ren, Z. Z. Sun, and X. Zhao, Compact Difference Scheme for the Fractional Sub-diffusion Equation with Neumann Boundary Conditions, J. Comput. Phys., vol. 232, pp. 456–467, 2013.
  • S. Y. Zhai, Z. F. Weng, D. W. Gui, and X. L. Feng, High-Order Compact Operator Splitting Method for Three-Dimensional Fractional Equation with Subdiffusion, Int. J. Heat Mass Transfer, vol. 84, pp. 440–447, 2015.
  • S. Y. Zhai, X. L. Feng, and Y. N. He, An Unconditionally Stable Compact ADI Method for Three-Dimensional Time-Fractional Convection-Diffusion Equation, J. Comput. Phys., vol. 269, pp. 138–155, 2014.
  • W. Y. Tian, H. Zhou, and W. H. Deng, A Class of Second Order Difference Approximations for Solving Space Fractional Diffusion Equations, Math. Comput., vol. 84, pp. 1703–1727, 2015.
  • Z. B. Wang and S. W. Vong, Compact Difference Schemes for the Modified Anomalous Fractional Sub-Diffusion Equation and the Fractional Diffusion-Wave Equation, J. Comput. Phys., vol. 277, pp. 1–15, 2014.
  • G. H. Gao, Z. Z. Sun, and H. W. Zhang, A New Fractional Numerical Differentiation Formula to Approximate the Caputo Fractional Derivative and Its Applications, J. Comput. Phys., vol. 259, pp. 33–50, 2014.
  • A. A. Alikhanov, A New Difference Scheme for the Time Fractional Diffusion Equation, J. Comput. Phys., vol. 280, pp. 424–438, 2015.
  • H. Nasir, B. Gunawardana, and H. Abeyrathna, A Second Order Finite Difference Approximation for the Fractional Diffusion Equation, Int. J. Appl. Phys. Math., vol. 3, pp. 237–243, 2013.
  • G. H. Gao, H. W. Sun, and Z. Z. Sun, Stability and Convergence of Finite Difference Schemes For a Class of Time-Fractional Sub-diffusion Equations Based On Certain Superconvergence, J. Comput. Phys., vol. 280, pp. 510–528, 2015.
  • S. Y. Zhai, L. L. Wei, L. Y. Huang, and X. L. Feng, An Efficient Algorithm with High Accuracy for Time-Space Fractional Heat Equations, Numer. Heat Transfer B, vol. 67, pp. 550–562, 2015.
  • Y. Dimitrov, Numerical Approximations for Fractional Differential Equations, arXiv:1311.3935v1, 2013.
  • A. A. Alikhanov, A New Difference Scheme for the Time Fracional Diffusion Equation, J. Comput. Phys., vol. 280, pp. 424–438, 2015.
  • T. Abdeljawad, On Riemann and Caputo Fractional Differences, Comput. Math. Appl., vol. 62, pp. 1602–1611, 2011.
  • K. Diethelm, N. J. Ford, and A. D. Freed, Detailed Error Analysis for a Fractional Adams Method, Numer. Algorithms, vol. 36, pp. 31–52, 2004.
  • G. S. Priya, P. Prakash, J. J. Nieto, and Z. Kayar, Higher-Order Numerical Scheme for the Fractional Heat Equation with Dirichlet and Neumann Boundary Conditions, Numer. Heat Transfer B, vol. 63, pp. 540–559, 2013.
  • J. Douglas, Jr. and S. Kim, Improved Accuracy for Locally One-Dimensional Methods for Parabolic Equations, Math. Models Meth. Appl. Sci., vol. 11, pp. 1563–1579, 2001.

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