136
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

&
Pages 1-20 | Received 07 Jul 2023, Accepted 26 Oct 2023, Published online: 13 Nov 2023

References

  • P. Brenner, M. Crouzeix, and V. Thomée, Single step methods for inhomogeneous linear differential equations in banach space, Rairo. Anal. Numer. 16 (1982), pp. 5–26.
  • C. Çelik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys. 231 (2012), pp. 1743–1750.
  • A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Retarding subdiffusion and accelerating super diffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002), Article ID 046129.
  • K. Diethelm and N.J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math. 225 (2009), pp. 96–104.
  • H.F. Ding, C.P. Li, and Y.Q. Chen, High-order algorithms for Riesz derivative and their applications (I), Abstr. Appl. Anal. 2014 (2014), Article ID 653797.
  • H.F. Ding, C.P. Li, and Y.Q. Chen, High-order algorithms for Riesz derivative and their applications (II), J. Comput. Phys. 293 (2015), pp. 218–237.
  • W. Ding, S. Patnaik, S. Sidhardh, and F. Semperlotti, Applications of distributed-order fractional operators: A review, Entropy 23 (2021), pp. 110.
  • W. Fan and F. Li, A numerical method for solving the two-dimensional distributed-order space-fractional diffusion equation on an irregular convex domain, Appl. Math. Lett. 77 (2018), pp. 114–121.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, Springer-Verlag, Berlin, 1991.
  • J.H. Jeon, N. Leijnse, L.B. Oddershede, and R. Metzler, Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions, New. J. Phys. 15 (2013), Article ID 045011. https://doi.org/10.1088/1367-2630/15/4/045011.
  • J. Jia and H.A. Wang, Fast finite difference method for distributed-order space-fractional partial differential equations on convex domains, Comput. Math. Appl. 75 (2018), pp. 2031–2041.
  • A.Q.M. Khaliq, E.H. Twizell, and D.A. Voss, On parallel algorithms for semi discretized parabolic partial differential equations based on sub diagonal Padé approximations, Numer. Methods Partial Differ. Equ. 9 (1993), pp. 107–116.
  • A.A. Kilbas, H.M. Srivastave, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, 2006.
  • J. Li, F. Liu, L. Feng, and I. Turner, A novel finite volume method for the Riesz space distributed-order diffusion equation, Comput. Math. Appl. 74 (2017), pp. 772–784.
  • C.P. Li and F.H. Zeng, Numerical Methods for Fractional Differential Calculus, Chapman and Hall/CRC, Boca Raton, USA, 2015.
  • R. Metzler, J.H. Jeon, A.G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and aging at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16 (2014), pp. 24128–24164.
  • S.P. Norsett and A. Wolfbrandt, Attainable order of rational approximations to the exponential function with only real poles, BIT Numer. Math. 17 (1977), pp. 200–208.
  • M.D. Ortigueira, Riesz potential operators and inverses via fractional centered derivatives, Int. J. Math. Sci. 2006 (2006), Article ID 048391.
  • S. Patnaik and F. Semperlotti, Application of variable- and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators, Nonlinear Dyn. 100 (2020), pp. 561–580.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999.
  • Y.L. Qiao, X.P. Wang, H.Y. Xu, and H.T. Qi, Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models, Appl. Math. Mech. (Eng. Ed.)42 (2021), pp. 1771–1786.
  • I. Sokolov, A. Chechkin, and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Pol. B 35 (2004), pp. 123–134.
  • V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational MathematicsSpringer-Verlag, Berlin, 2006.
  • X. Wang, F. Liu, and X. Chen, Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv. Math. Phys. 2015 (2015), Article ID 590435.
  • Z. Yang, Z. Yuan, Y. Nie, J. Wang, X. Zhu, and F. Liu, Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains, J. Comput. Phys. 330 (2017), pp. 863–883.
  • H. Zhang, F. Liu, X. Jiang, F. Zeng, and T. Turner, A Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation, Comput. Math. Appl. 76 (2018), pp. 2460–2476.
  • Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.