Numerical techniques for the variable order time fractional diffusion equation

Abstract In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient.

[1]  Carlos F.M. Coimbra,et al.  On the Selection and Meaning of Variable Order Operators for Dynamic Modeling , 2010 .

[2]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[3]  Carlos F.M. Coimbra,et al.  Mechanics with variable‐order differential operators , 2003 .

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

[5]  C. Coimbra,et al.  Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation , 2009 .

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

[7]  B. Ross,et al.  Integration and differentiation to a variable fractional order , 1993 .

[8]  Stefan Samko,et al.  Fractional integration and differentiation of variable order , 1995 .

[9]  Zaid M. Odibat,et al.  Approximations of fractional integrals and Caputo fractional derivatives , 2006, Appl. Math. Comput..

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

[11]  Carlos F.M. Coimbra,et al.  The variable viscoelasticity oscillator , 2005 .

[12]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

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

[14]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[15]  M. Ciesielski,et al.  Numerical solution of fractional oscillator equation , 2011, Appl. Math. Comput..

[16]  Dov Ingman,et al.  Control of damping oscillations by fractional differential operator with time-dependent order , 2004 .

[17]  Fawang Liu,et al.  Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation , 2009, Appl. Math. Comput..

[18]  Noel C. Perkins,et al.  Electrostatics and Self-Contact in an Elastic Rod Approximation for DNA , 2009 .

[19]  Fawang Liu,et al.  Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation , 2010, SIAM J. Sci. Comput..

[20]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[21]  Carlos F.M. Coimbra,et al.  Variable Order Modeling of Diffusive-convective Effects on the Oscillatory Flow Past a Sphere , 2008 .

[22]  Fawang Liu,et al.  Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term , 2009, J. Comput. Appl. Math..

[23]  Fawang Liu,et al.  Novel techniques in parameter estimation for fractional dynamical models arising from biological systems , 2011, Comput. Math. Appl..

[24]  T. Hartley,et al.  Initialization, conceptualization, and application in the generalized (fractional) calculus. , 2007, Critical reviews in biomedical engineering.

[25]  Shyam L. Kalla,et al.  Fractional extensions of the temperature field problems in oil strata , 2007, Appl. Math. Comput..

[26]  Fawang Liu,et al.  Numerical methods and analysis for a class of fractional advection-dispersion models , 2012, Comput. Math. Appl..

[27]  I. Turner,et al.  Two New Implicit Numerical Methods for the Fractional Cable Equation , 2011 .

[28]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[29]  Dov Ingman,et al.  Constitutive Dynamic-Order Model for Nonlinear Contact Phenomena , 2000 .