An Adaptive Memory method for Accurate and Efficient Computation of the Caputo Fractional Derivative

A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation is simulated to demonstrate the accuracy as well as the computational efficiency of the present method.

[1]  Santos B. Yuste,et al.  A finite difference method with non-uniform timesteps for fractional diffusion equations , 2011, Comput. Phys. Commun..

[2]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

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

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

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

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

[7]  Christopher L. MacDonald,et al.  Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory , 2015, J. Comput. Phys..

[8]  C. Lim,et al.  Mechanics of the human red blood cell deformed by optical tweezers , 2003 .

[9]  Subra Suresh,et al.  Viscoelasticity of the human red blood cell , 2006, American journal of physiology. Cell physiology.

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

[11]  Fanhai Zeng,et al.  Numerical Methods for Fractional Calculus , 2015 .

[12]  Arak M. Mathai,et al.  Special Functions for Applied Scientists , 2008 .

[13]  Changpin Li,et al.  Theory and Numerical Approximations of Fractional Integrals and Derivatives , 2019 .

[14]  Aaron U. Kaleta.  Fractional Differential Equations An Introduction To Fractional Derivatives Fractional Differential Equations To Methods Of Their Solution And Some Mathematics In Science And Engineering , 2020 .

[15]  Donghyun You,et al.  Continuum modeling of deformation and aggregation of red blood cells. , 2016, Journal of biomechanics.

[16]  Damian Craiem,et al.  Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics , 2010, Physical biology.

[17]  Chwee Teck Lim,et al.  Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria. , 2005, Acta biomaterialia.

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

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

[20]  Alan D. Freed,et al.  An efficient algorithm for the evaluation of convolution integrals , 2006, Comput. Math. Appl..

[21]  Jurij Kotar,et al.  The nonlinear mechanical response of the red blood cell , 2007, Physical biology.

[22]  S. Suresh,et al.  Nonlinear elastic and viscoelastic deformation of the human red blood cell with optical tweezers. , 2004, Mechanics & chemistry of biosystems : MCB.

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