Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory

Computing numerical solutions to fractional differential equations can be computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In general, numerical approaches that depend on truncating part of the system history while efficient, can suffer from high degrees of error and inaccuracy. Here we present an adaptive time step memory method for smooth functions applied to the Grunwald-Letnikov fractional diffusion derivative. This method is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system's history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points backward in time, a temporally 'weighted' history is computed that includes contributions from the entire past of the system, maintaining accuracy, but with fewer points actually calculated, greatly improving computational efficiency.

[1]  Fawang Liu,et al.  Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation , 2010, Numerical Algorithms.

[2]  Albert Compte.  CONTINUOUS TIME RANDOM WALKS ON MOVING FLUIDS , 1997 .

[3]  Vito Volterra,et al.  Leçons sur la théorie mathématique de la lutte pour la vie , 1931 .

[4]  David A. Benson,et al.  Subordinated advection‐dispersion equation for contaminant transport , 2001 .

[5]  Fawang Liu,et al.  Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation , 2008, Appl. Math. Comput..

[6]  Francesco Mainardi,et al.  Sub-diffusion equations of fractional order and their fundamental solutions , 2007 .

[7]  William H. Press,et al.  Numerical recipes in C , 2002 .

[8]  Jianer Chen,et al.  An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem , 2007, Algorithmica.

[9]  J. Klafter,et al.  Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach , 1999 .

[10]  R. Gorenflo,et al.  Time Fractional Diffusion: A Discrete Random Walk Approach , 2002 .

[11]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

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

[13]  S. Wearne,et al.  Fractional cable models for spiny neuronal dendrites. , 2008, Physical review letters.

[14]  Christian Lubich,et al.  Adaptive, Fast, and Oblivious Convolution in Evolution Equations with Memory , 2006, SIAM J. Sci. Comput..

[15]  Alain Oustaloup,et al.  Fractional differentiation for edge detection , 2003, Signal Process..

[16]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[17]  C. Lubich.  Convolution quadrature and discretized operational calculus. I , 1988 .

[18]  Christian Lubich,et al.  Fast and Oblivious Convolution Quadrature , 2006, SIAM J. Sci. Comput..

[19]  Timothy R. Ginn,et al.  Fractional advection‐dispersion equation: A classical mass balance with convolution‐Fickian Flux , 2000 .

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

[21]  Jing-Rebecca Li,et al.  A Fast Time Stepping Method for Evaluating Fractional Integrals , 2009, SIAM J. Sci. Comput..

[22]  Santos B. Yuste,et al.  On an explicit finite difference method for fractional diffusion equations , 2003, ArXiv.

[23]  Zakary S. Singer,et al.  Characterization of Calcium-Mediated Intracellular and Intercellular Signaling in the rMC-1 Glial Cell Line , 2009, Cellular and molecular bioengineering.

[24]  B. West.  Fractional Calculus in Bioengineering , 2007 .

[25]  E. Soczkiewicz,et al.  Application of Fractional Calculus in the Theory of Viscoelasticity , 2002 .

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

[27]  Pernille Yde,et al.  Variety in intracellular diffusion during the cell cycle , 2009, Physical biology.

[28]  Richard L. Magin,et al.  Modeling the cardiac tissue electrode interface using fractional calculus , 2006 .

[29]  William McLean,et al.  Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation , 2009, Numerical Algorithms.

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

[31]  E. Newman,et al.  Calcium signaling in specialized glial cells , 2006, Glia.

[32]  E. Newman,et al.  Propagation of Intercellular Calcium Waves in Retinal Astrocytes and Müller Cells , 2001, The Journal of Neuroscience.

[33]  Kazuhiko Seki,et al.  Fractional reaction-diffusion equation , 2003 .

[34]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[35]  C. Lubich.  Discretized fractional calculus , 1986 .

[36]  Walter Lauriks,et al.  Application of fractional calculus to ultrasonic wave propagation in human cancellous bone , 2006, Signal Process..

[37]  Christian Lubich,et al.  Fast Convolution for Nonreflecting Boundary Conditions , 2002, SIAM J. Sci. Comput..