Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T06:00:01.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Moving Finite Element Methods for a System of Semi-Linear Fractional Diffusion Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Integral equations, integral transforms Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  19 September 2016

Jingtang Ma  and
Zhiqiang Zhou
Jingtang Ma*
Affiliation:
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
Zhiqiang Zhou*
Affiliation:
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
*
*Corresponding author. Email:[email protected] (J. T. Ma), [email protected] (Z. Q. Zhou)
*Corresponding author. Email:[email protected] (J. T. Ma), [email protected] (Z. Q. Zhou)
Get access

Abstract

This paper studies a system of semi-linear fractional diffusion equations which arise in competitive predator-prey models by replacing the second-order derivatives in the spatial variables with fractional derivatives of order less than two. Moving finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are proved. Numerical examples are carried out to confirm the theoretical findings. Some applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-prey models are studied.

Keywords

Finite element methodsfractional differential equationspredator-prey models

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods 35S10: Initial value problems for pseudodifferential operators 65R20: Integral equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 911 - 931
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baeumer, B., Kovács, M. and Meerschaert, M. M., Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bull. Math. Biol., 69 (2007), pp. 2281–2297.CrossRefGoogle ScholarPubMed
[2] Baeumer, B., Kovács, M. and Meerschaert, M. M., Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl., 55 (2008), pp. 2212–2226.Google Scholar
[3] Bank, R. E. and Santos, R. F., Analysis of some moving space-time finite element methods, SIAM J. Numer. Anal., 30 (1993), pp. 1–18.CrossRefGoogle Scholar
[4] Brauer, F. and Castillo-Chavez, C., Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, Heidelberg, 2000.Google Scholar
[5] Cavani, M. and Farkas, M., Bifurcation in a predator-prey model with memory and diffusion, II: Turing bifurcation, Acta Math. Hungar., 63 (1994), pp. 375–393.Google Scholar
[6] Chen, C., Liu, F., Anh, V. and Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32 (2010), pp. 1740–1760.CrossRefGoogle Scholar
[7] Chen, H. and Xu, D., A compact difference scheme for an evolution equation with weakly singular kernel, Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 559–572.Google Scholar
[8] Chen, H., Xu, D. and Peng, Y., An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation, Int. J. Comput. Math., 92 (2015), pp. 2178–2197.Google Scholar
[9] Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), pp. 204–226.Google Scholar
[10] Deng, W., Du, S. and Wu, Y., High order finite difference WENO schemes for fractional differential equations, Appl. Math. Lett., 26 (2013), pp. 362–366.Google Scholar
[11] Dupont, T., Mesh modification for evolution equations, Math. Comput., 39 (1982), pp. 85–107.CrossRefGoogle Scholar
[12] Dupont, T. and Liu, Y., Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations, SIAM J. Numer. Anal., 40 (2002), pp. 914–927.CrossRefGoogle Scholar
[13] Ervin, V. J. and Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. PDEs, 22 (2006), pp. 558–576.Google Scholar
[14] Ervin, V. J. and Roop, J. P., Variational solution of fractional advection dispersion equations on bounded domains in ℝd , Numer. Meth. PDEs, 23 (2007), pp. 256–281.Google Scholar
[15] Ervin, V. J., Heuer, N. and Roop, J. P., Numerical approximation of a time dependent nonlinear space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007), pp. 572–591.CrossRefGoogle Scholar
[16] Fix, G. J. and Roop, J. P., Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 48 (2004), pp. 1017–1033.Google Scholar
[17] Ford, N. J., Xiao, J. and Yan, Y., A finite element method for time fractional partial differential equations, Fract. Cal. Appl. Anal., 14 (2011), pp. 454–474.Google Scholar
[18] Gao, G. and Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), pp. 586–595.CrossRefGoogle Scholar
[19] Gao, G., Sun, Z. and Zhang, Y., A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions, J. Comput. Phys., 231 (2012), pp. 2865–2879.Google Scholar
[20] Huang, W. and Russell, R. D., Adaptive Moving Mesh Methods, Springer, New York, 2011.Google Scholar
[21] Ilic, M., Liu, F., Turner, I. and Anh, V., Numerical approximation of a fractional-in-space diffusion equation I, Frac. Calc. Appl. Anal., 8 (2005), pp. 323–341.Google Scholar
[22] Ilic, M., Liu, F., Turner, I. and Anh, V., Numerical approximation of a fractional-in-space diffusion equation II: with nonhomogeneous boundary conditions, Frac. Calc. Appl. Anal., 9 (2006), pp. 333–349.Google Scholar
[23] Ji, X. and Tang, H., High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one- and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 333–358.Google Scholar
[24] Jiang, Y. and Ma, J., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), pp. 3285–3290.Google Scholar
[25] Jiang, Y. and Ma, J., Moving finite element methods for time fractional partial differential equations, Sci. China Math., 56 (2013), pp. 1287–1300.CrossRefGoogle Scholar
[26] Kovács, S., Kiss, K. and Farkas, M., Qualitative behaviour of a ratio-dependent predator-prey system, Nonlinear Anal. RWA, 10 (2009), pp. 1627–1642.CrossRefGoogle Scholar
[27] Li, X. J. and Xu, C. J., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108–2131.Google Scholar
[28] Li, X. J. and Xu, C. J., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), pp. 1016–1051.Google Scholar
[29] Ma, J., Convergence analysis of moving Godunov methods for dynamical boundary layers, Comput. Math. Appl., 59 (2010), pp. 80–93.Google Scholar
[30] Ma, J., Jiang, Y. and K. Xiang, , On a moving mesh method for solving partial integro-differential equations, J. Comput. Math., 27 (2009), pp. 713–728.Google Scholar
[31] Ma, J. and Jiang, Y., Analysis of an adaptive remeshing algorithm for reaction-diffusion equations with traveling heat source, Scientia Sinica Math., 41 (2011), pp. 235–251.Google Scholar
[32] Ma, J., Huang, W. and Russell, R. D., Analysis of a moving collocation method for one-dimensional partial differential equations, Sci. China Math., 55 (2012), pp. 827–840.Google Scholar
[33] Ma, J. and Jiang, Y., Moving collocation methods for time fractional differential equations and simulation of blowup, Sci. China Math., 54 (2011), pp. 611–622.Google Scholar
[34] Ma, J., Liu, J. and Zhou, Z., Convergence analysis of moving finite element methods for space fractional differential equations, J. Comput. Appl. Math., 255 (2014), pp. 661–670.Google Scholar
[35] MacKenzie, J. A. and Mekwi, W. R., An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh, IMA J. Numer. Anal., 27 (2007), pp. 507–528.Google Scholar
[36] McLean, W. and Mustapha, K., Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithm, 5 (2009), pp. 69–88.Google Scholar
[37] Meerschaert, M. and Tadjeran, C., Finite difference approximations for fractional advection dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65–77.Google Scholar
[38] Meerschaert, M., Scheffler, H. and Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), pp. 249–261.Google Scholar
[39] Mustapha, K. and McLean, M., Piecewise-linear, discontinous Galerkin method for a fractional diffusion equation, Numer. Algorithm, 56 (2011), pp. 159–184.Google Scholar
[40] Ren, J., Sun, Z. and Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 232 (2013), pp. 456–467.Google Scholar
[41] Shen, S., Liu, F., Anh, V. and Turner, I., The fundemental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA J. Appl. Math., 73 (2008), pp. 850–872.Google Scholar
[42] Tadjeran, C., Meerschaert, M. and Scheffler, H., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), pp. 205–213.Google Scholar
[43] Tadjeran, C. and Meerschaert, M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), pp. 813–823.Google Scholar
[44] Tang, T. and Xu, J., Adaptive Computations: Theory and Algorithms, Science Press, Beijing, 2007.Google Scholar
[45] Thomée, V., Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 2006.Google Scholar
[46] Yang, Q., Liu, F. and Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34 (2010), pp. 200–218.Google Scholar
[47] Yu, Y., Deng, W. and Wu, Y., Positivity and boundedness preserving schemes for the fractional reaction-diffusion equation, Sci. China Math., 56 (2013), pp. 2161–2178.CrossRefGoogle Scholar
[48] Zhao, X. and Sun, Z., A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), pp. 6061–6074.Google Scholar
[49] Zhang, H., Liu, F. and Anh, V., Galerkin finite element approximations of symmetric space-fractional partial differential equations, Appl. Math. Comput., 217 (2010), pp. 2534–2545.Google Scholar
[50] Zhang, N., Deng, W. and Wu, Y., Finite difference/ element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4 (2012), pp. 496–518.CrossRefGoogle Scholar
[51] Zhang, Y., Sun, Z. and Wu, H., Error estimate of Crank-Nicolson-type difference for subdiffusion equation, SIAM J. Numer. Anal., 49 (2011), pp. 2302–2322.CrossRefGoogle Scholar
[52] Zhuang, P., Liu, F., Anh, V. and Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), pp. 1760–1781.Google Scholar