High order numerical treatment of the Riesz space fractional advection-dispersion equation via matrix transform method

Abdollah Borhanifar, Sohrab Valizadeh

Abstract


In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method.


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M. I. Abbas, M. A. Ragusa, Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag-Leffler functions, Applicable Analysis 260 (2020), 1–15. https://doi.org/10.1080/00036811.2020.1839645

K. Al-Khaled, Sinc-Legendre collocation method for the non-linear Burger’s fractional equation, Annals of the University of Craiova, Mathematics and Computer Science Series 41 (2014), no. 2, 234–250.

A. Borhanifar, S. Valizadeh, Numerical solution for fractional partial differential equations using Crank-Nicolson method with shifted Grunwald estimate, Walailak Journal of Science and Technology 9 (2012), no. 4, 433–444.

A. Borhanifar, S. Valizadeh, A fractional finite difference method for solving the fractional Poisson equation based on the shifted Grunwald estimate, Walailak Journal of Science and Technology 10 (2013), no. 5, 427–435.

A. Borhanifar, S. Valizadeh, Mittag-Leffler-Padé approximations for the numerical solution of space and time fractional diffusion equations, International Journal of Applied Mathematical Research 4 (2015), no. 4, 466–480.

C. Cęlik, M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, Journal of Computational Physics 231 (2012), no. 4, 1743–1750.

M. Chen, W. Deng, Y. Wu, Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation, Applied Numerical Mathematics 70 (2013), 22–41.

H. Ding, General Padé approximation method for time-space fractional diffusion equation, Journal of Computational and Applied Mathematics 299 (2016), 221–228.

H. Ding, C. Li, Y. Chen, High-order algorithms for Riesz derivative and their applications (II), Journal of Computational Physics 293 (2015), 218–237.

H. Ding, Y. Zhang, A new difference scheme with high accuracy and absolute stability for solving convection-diffusion equations, Journal of Computational and Applied Mathematics 230 (2009), no. 2, 600–606.

H. F. Ding, Y. X. Zhang, New numerical methods for the Riesz space fractional partial differential equations, Computers & Mathematics with Applications 63 (2012), no. 7, 1135– 1146.

R. Gorenflo, F. Mainardi, Approximation of Lévy-Feller diffusion by random walk models, Zeitschrift für Analysis and ihre Anwendungen 18 (1999), no. 2, 231–246.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

K. Hoffman, R. A. Kunze, Linear algebra, Prentice-Hall, New Jersey, 1961.

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, North-Holland Mathematics Studies, 2006.

R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Connecticut, 2006.

M.D. Ortigueira, Riesz potential operators and inverses via fractional centred derivatives, International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation (2006), 1–12.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

M. Popolizio, A matrix approach for partial differential equations with Riesz space fractional derivatives, The European Physical Journal Special Topics 222 (2013), no. 8, 1975–1985.

M.A. Ragusa, Operators in Morrey type spaces and applications, Eurasian Mathematical Journal 3 (2012), no. 3, 94–109.

M.A. Ragusa, A. Scapellato, Mixed Morrey spaces and their applications to partial differential equations, Nonlinear Analysis: Theory, Methods & Applications 151 (2017), 51–65.

M. Rahman, A. Mahmood, M. Younis, Improved and more feasible numerical methods for Riesz space fractional partial differential equations, Applied Mathematics and Computation 237 (2014), 264–273.

H. Safdari, H. Mesgarani, Y. Esmaeelzade, M. Javidi, The Chebyshev wavelet of the second kind for solving fractional delay differential equations, Annals of the University of Craiova, Mathematics and Computer Science Series 47 (2020), no. 1, 111–124.

E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Physica A: Statistical Mechanics and its Applications 284 (2000), no. 1-4, 376–384.

R. Schumer, D.A. Benson, M.M. Meerschaert, S.W. Wheatcraft, Eulerian derivation of the fractional advection-dispersion equation, Journal of Contaminant Hydrology 48 (2001), no. 1-2, 69–88.

R. Schumer, D.A. Benson, M.M. Meerschaert, B. Baeumer, Multiscaling fractional advection-dispersion equations and their solutions, Water Resources Research 39 (2003), no. 1, 1022–1032.

S. Shen, F. Liu, V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numerical Algorithms 56 (2011), no. 3, 383–403.

S. Shen, F. Liu, V. Anh, I. Turner, The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA Journal of Applied Mathematics 73 (2008), no. 6, 850–872.

J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, New York, 1995.

S. Valizadeh, A. Borhanifar, M.R. Abdollahpour, Optimal feedback control of fractional semilinear integro-differential equations in the Banach spaces, International Journal of Industrial Mathematics 12 (2020), no. 4, 335–343.

S. Valizadeh, A. Borhanifar, Numerical solution for Riesz fractional diffusion equation via fractional centered difference scheme, Walailak Journal of Science and Technology 18 (2021), no. 7, 9246.

Q. Yang, F. Liu, I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling 34 (2010), no. 1, 200– 218.

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics, Dover publications, New York, 1972.

Q. Yu, Numerical simulation of anomalous diffusion with application to medical imaging, Ph.D. thesis, School of Mathematical Sciences, Queensland University of Technology (2013).

A. Zada, B. Dayyan, Stability analysis for a class of implicit fractional differential equations with instantaneous impulses and Riemann–Liouville boundary conditions, Annals of the University of Craiova, Mathematics and Computer Science Series 47 (2020), no. 1, 88–110.

Y. Zhang, [3,3] Padé approximation method for solving space fractional Fokker-Planck equations, Applied Mathematics Letters 35 (2014), 109–114.

Y. Zhang, H. Ding, Improved matrix transform method for the Riesz space fractional reaction dispersion equation, Journal of Computational and Applied Mathematics 260 (2014), 266–280.




DOI: https://doi.org/10.52846/ami.v51i1.1496