An Operational Matrix Method for Fractional Advection-Diffusion Equations With Variable Coefficients

ZHU Xiaogang; NIE Yufeng

doi:10.21656/1000-0887.380041

Volume 39 Issue 1

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2018 > 39(1): 104-112

ZHU Xiaogang, NIE Yufeng. An Operational Matrix Method for Fractional Advection-Diffusion Equations With Variable Coefficients[J]. Applied Mathematics and Mechanics, 2018, 39(1): 104-112. doi: 10.21656/1000-0887.380041

Citation:

PDF( 385 KB)

An Operational Matrix Method for Fractional Advection-Diffusion Equations With Variable Coefficients

doi: 10.21656/1000-0887.380041

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, P.R.China

Funds: The National Natural Science Foundation of China（11471262； 11501450）

Received Date: 2017-02-23
Rev Recd Date: 2047-03-19
Publish Date: 2018-01-15

Abstract

Abstract

A numerical method for the Caputo-fractional advection-diffusion equations with variable coefficients was investigated. Based on Chebyshev cardinal functions, an effective operational matrix was derived for Riemann-Liouville fractional integral, and with it, a new operational matrix method was proposed for the fractional advection-diffusion equations with variable coefficients. This method reduces the equation to an algebraic system and is characterized by small computing cost and easy programming. The numerical results and the comparisons with some existing methods illustrate that it is convergent and possesses advantages in accuracy.
- fractional calculus,
- Chebyshev cardinal functions,
- fractional advection-diffusion equation,
- operational matrix method

FullText(HTML)

References(21)

References

[1]	ICHISE M, NAGAYANAGI Y, KOJIMA T. An analog simulation of non-integer order transfer functions for analysis of electrode processes[J]. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry,1971,33(2): 253-265.
[2]	COLE K S. Electric conductance of biological systems[J]. Cold Spring Harbor Symposia on Quantitative Biology,1933,1: 107-116.
[3]	ZHUANG P, LIU F. Implicit difference approximation for the time fractional diffusion equation[J]. Journal of Applied Mathematic and Computing,2006,22(3): 87-99.
[4]	JIANG Yingjun, MA Jingtang. High-order finite element methods for time-fractional partial differential equations[J]. Journal of Computational and Applied Mathematics,2011,235(11): 3285-3290.
[5]	LIN Yumin, XU Chuanjun. Finite difference/spectral approximations for the time-fractional diffusion equation[J]. Journal of Computational Physics,2007,225(2): 1533-1552.
[6]	MUSTAPHA K, ABDALLAH B, FURATI K M, et al. A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients[J]. Numerical Algorithms,2016,73(2): 517-534.
[7]	YASEEN M, ABBAS M, ISMAIL A I, et al. A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations[J]. Applied Mathematics and Computation,2017,293: 311-319.
[8]	CUI Ming-rong. Compact finite difference method for the fractional diffusion equation[J]. Journal of Computational Physics,2009,228(20): 7792-7804.
[9]	LIU Q, GU Y T, ZHUANG P, et al. An implicit RBF meshless approach for time fractional diffusion equations[J]. Computational Mechanics,2011,48(1): 1-12.
[10]	IZADKHAH M M, SABERI-NADJAFI J. Gegenbauer spectral method for time-fractional convection-diffusion equations with variable coefficients[J]. Mathematical Methods in the Applied Sciences,2015,38(15): 3183-3194.
[11]	SAADATMANDI A, DEHGHAN M, AZIZI M-R. The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients[J]. Communications in Nonlinear Science and Numerical Simulation,2012,17(11): 4125-4136.
[12]	GHANDEHARI M A M, RANJBAR M. A numerical method for solving a fractional partial differential equation through converting it into an NLP problem[J]. Computers & Mathematics With Applications,2013,65(7): 975-982.
[13]	CHEN Y M, WU Y B, CUI Y H, et al. Wavelet method for a class of fractional convection-diffusion equation with variable coefficients[J]. Journal of Computational Science,2010,1(3): 146-149.
[14]	BHRAWY A H, ZAKY M. A fractional-order Jacobi Tau method for a class of time-fractional PDEs with variable coefficients[J]. Mathematical Methods in the Applied Sciences,2016,39(7): 1765-1779.
[15]	NEMATI S, SEDAGHAT S. Matrix method based on the second kind Chebyshev polynomials for solving time fractional diffusion-wave equations[J]. Journal of Applied Mathematics and Computing,2016,51(1/2): 189-207.
[16]	DOHA E H, BHRAWY A H, EZZ-ELDIEN S S. An efficient Legendre spectral Tau matrix formulation for solving fractional subdiffusion and reaction subdiffusion equations[J]. Journal of Computational and Nonlinear Dynamics,2015,10(2): 021019. DOI: 10.1115/1.4027944.
[17]	Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations [M]. Amsterdam: Elsevier Science, 2006.
[18]	Boyd J P. Chebyshev and Fourier Spectral Methods [M]. Mineola: Dover Publications Inc, 2001.
[19]	PANG Guofei, CHEN Wen, FU Zhoujia. Space-fractional advection—dispersion equations by the Kansa method[J].Journal of Computational Physics,2015,293: 280-296.
[20]	ELHAY S, KAUTSKY J. Algorithm 655: IQPACK: FORTRAN subroutines for the weights of interpolatory quadratures[J]. ACM Transactions on Mathematical Software,1987,13(4): 399-415.
[21]	GAUTSCHI W. High-order Gauss-Lobatto formulae[J]. Numerical Algorithms,2000,25(1): 213-222.