A windowing waveform relaxation method for time-fractional differential equations

Abstract This paper presents a new windowing waveform relaxation method for time-fractional differential equations. Unlike the classical case, the proposed windowing method uses the history part of the solution at each window. Second, it is the first time that a multi-domain finite difference scheme together with a windowing method has been used for the time-fractional differential equations, which makes the numerical scheme very efficient. Third, the paper provides an effective estimation on window length. Numerical results are given to further illustrate the theoretical analysis.

[1]  Jan S. Hesthaven,et al.  A parareal method for time-fractional differential equations , 2015, J. Comput. Phys..

[2]  Jingjun Zhao,et al.  Collocation methods for fractional integro-differential equations with weakly singular kernels , 2013, Numerical Algorithms.

[3]  Gianni Pagnini,et al.  Short note on the emergence of fractional kinetics , 2014, 1404.0215.

[4]  Yao-Lin Jiang,et al.  Waveform relaxation methods for fractional functional differential equations , 2013 .

[5]  Jürgen Geiser,et al.  An iterative splitting method via waveform relaxation , 2011, Int. J. Comput. Math..

[6]  Yao-Lin Jiang.  Windowing Waveform Relaxation of Initial Value Problems , 2006 .

[7]  T. Barrick,et al.  From diffusion‐weighted MRI to anomalous diffusion imaging , 2008, Magnetic resonance in medicine.

[8]  Chunye Gong,et al.  An efficient parallel solution for Caputo fractional reaction–diffusion equation , 2014, The Journal of Supercomputing.

[9]  D. K. Salkuyeh,et al.  Two-stage waveform relaxation method for the initial value problems with non-constant coefficients , 2014 .

[10]  Barbara Zubik-Kowal,et al.  Waveform Relaxation for Functional-Differential Equations , 1999, SIAM J. Sci. Comput..

[11]  Yao-Lin Jiang,et al.  Waveform relaxation methods for fractional differential equations with the Caputo derivatives , 2013, J. Comput. Appl. Math..

[12]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[13]  Andrey G. Cherstvy,et al.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. , 2014, Physical chemistry chemical physics : PCCP.

[14]  Yaolin Jiang,et al.  Semilinear fractional differential equations based on a new integral operator approach , 2012 .

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

[16]  Jan S. Hesthaven,et al.  Stable multi-domain spectral penalty methods for fractional partial differential equations , 2014, J. Comput. Phys..

[17]  Delfim F. M. Torres,et al.  Towards a combined fractional mechanics and quantization , 2012, 1206.0864.

[18]  Mehdi Dehghan,et al.  Application of the collocation method for solving nonlinear fractional integro-differential equations , 2014, J. Comput. Appl. Math..

[19]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[20]  Wayne Read,et al.  Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart , 2014, J. Comput. Phys..

[21]  Alberto L. Sangiovanni-Vincentelli,et al.  The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits , 1982, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[22]  Hong Zhang.  A NOTE ON WINDOWING FOR THE WAVEFORM RELAXATION , 1994 .

[23]  Yao-Lin Jiang,et al.  A general approach to waveform relaxation solutions of nonlinear differential-algebraic equations: the continuous-time and discrete-time cases , 2004, IEEE Trans. Circuits Syst. I Regul. Pap..

[24]  Benedict Leimkuhler,et al.  Rapid convergence of waveform relaxation , 1993 .