Galerkin finite element approximation of symmetric space-fractional partial differential equations

Abstract In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2 β  ∈ (0, 1) and 2 α  ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax–Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.

[1]  Om P. Agrawal,et al.  Riesz Fractional Derivatives and Fractional Dimensional Space , 2010 .

[2]  Fawang Liu,et al.  Numerical Solution of the Fractional-Order Advection-Dispersion Equation , 2002 .

[3]  J. P. Roop.  Variational Solution of the Fractional Advection Dispersion Equation , 2004 .

[4]  Bruce J. West,et al.  Fractional Langevin model of memory in financial markets. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[6]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[7]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[8]  Fawang Liu,et al.  New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation , 2008, SIAM J. Numer. Anal..

[9]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time , 1986 .

[10]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[11]  Fawang Liu,et al.  A Fourier method for the fractional diffusion equation describing sub-diffusion , 2007, J. Comput. Phys..

[12]  Fawang Liu,et al.  The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation , 2008 .

[13]  A. M. Mathai,et al.  On generalized fractional kinetic equations , 2004 .

[14]  Fawang Liu,et al.  Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method , 2007, J. Comput. Phys..

[15]  G. Fix,et al.  Least squares finite-element solution of a fractional order two-point boundary value problem , 2004 .

[16]  I. Podlubny.  Fractional differential equations , 1998 .

[17]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[18]  J. Rogers.  Chaos , 1876 .

[19]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[20]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[21]  Timothy Nigel Phillips,et al.  On the solution of the Fokker–Planck equation using a high-order reduced basis approximation , 2009 .

[22]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[23]  Katja Lindenberg,et al.  Reaction front in an A+B-->C reaction-subdiffusion process. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[25]  Santos B. Yuste,et al.  Subdiffusion-limited A+A reactions. , 2001 .

[26]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[27]  N. Leonenko,et al.  Spectral Analysis of Fractional Kinetic Equations with Random Data , 2001 .

[28]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[29]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[30]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[31]  H. K. Moffatt.  Topological aspects of the dynamics of fluids and plasmas , 1992 .