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Jacobi Spectral Collocation Method for the Time Variable-Order Fractional Mobile-Immobile Advection-Dispersion Solute Transport Model

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Difference and functional equations Real functions Basic linear algebra Difference equations

Published online by Cambridge University Press:  20 July 2016

Heping Ma  and
Yubo Yang
Heping Ma*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, China
Yubo Yang*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, China Nanhu College, Jiaxing University, Jiaxing, Zhejiang 314001, China
*
*Corresponding author. Email addresses:[email protected] (H. Ma), [email protected] (Y. Yang)
*Corresponding author. Email addresses:[email protected] (H. Ma), [email protected] (Y. Yang)
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Abstract

An efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

Keywords

Coimbra variable-order fractional derivativeJacobi polynomialsspectral collocation methodMobile-immobile advection-dispersion model

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 65M70: Spectral, collocation and related methods 15A99: Miscellaneous topics 39A70: Difference operators
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 3 , August 2016 , pp. 337 - 352
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Miller, K.S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).Google Scholar
[2]Podlubny, I., Fractional Differential Equations, Academic Press (1999).Google Scholar
[3]Kilbas, A.A., Srivastava, H.M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier (2006).Google Scholar
[4]Podlubny, I., Skovranek, T. and Datsko, B., Recent advances in numerical methods for partial fractional differential equations, Proc. 15th ICC Conf., IEEE, pp. 454–457 (2014).Google Scholar
[5]Ichise, M., Nagayanagi, M. and Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode processes, J. Electroanal. Chem. Interfacial Electrochem. 33, 253–265 (1971).CrossRefGoogle Scholar
[6]Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M., Application of a fractional advection-dispersion equation, Water Resources Res. 36, 1403–1412 (2000).Google Scholar
[7]Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press (2010).Google Scholar
[8]West, B.J., Bologna, M. and Grigolini, P., Physics of Fractal Operators, Springer, NewYork (2003).Google Scholar
[9]Magin, R.L., Fractional Calculus in Bioengineering, Begell House Inc, Reading (2006).Google Scholar
[10]Cao, J.Y. and Xu, C.J., A high order schema for the numerical solution of the fractional ordinary differential equations, J. Comp. Phys. 238, 154–168 (2013).Google Scholar
[11]Li, C.P. and Ding, H.F., Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model. 38, 3802–3821 (2014).Google Scholar
[12]Ren, J.C. and Sun, Z.Z., Efficient numerical solution of the multi-term time fractional diffusion-wave equation, East Asian J. Appl. Math. 5, 1–28 (2015).Google Scholar
[13]Rui, H.X. and Huang, J., Uniformly stable explicitly solvable finite difference method for fractional diffusion equations, East Asian J. Appl. Math. 5, 29–47 (2015).Google Scholar
[14]Zeng, F.H., Li, C.P., Liu, F. and Turner, I., The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput. 35, A2976-A3000 (2013).Google Scholar
[15]Ervin, V.J., Heuer, N. and Roop, J.P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Num. Anal. 45, 572–591 (2007).CrossRefGoogle Scholar
[16]Zhang, H., Liu, F. and Anh, V., Garlerkin finite element approximations of symmetric space-fractional partial differential equations, Appl. Math. Comput. 217, 2534–2545 (2010).Google Scholar
[17]Li, X.J. and Xu, C.J., A space-time spectral method for the time fractional diffusion equation, SIAM J. Num. Anal. 47, 2108–2131 (2009).CrossRefGoogle Scholar
[18]Li, C.P., Zeng, F.H. and Liu, F., Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal. 15, 383–406 (2012).Google Scholar
[19]Zayernouri, M. and Karniadakis, G.E., Fractional spectral collocation method, SIAM J. Sci. Comput. 36, A40-A62 (2014).Google Scholar
[20]Lorenzo, C.F. and Hartley, T.T., Variable order and distributed order fractional operators, Nonlinear Dynam. 29, 57–98 (2009).CrossRefGoogle Scholar
[21]Samko, S.G. and Ross, B., Integration and differentiation to a variable fractional order, Integr. Trans. Spec. Funct. 1, 277–300 (1993).Google Scholar
[22]Coimbra, C.F.M., Mechanics with variable-order differential operators, Ann. Physik 12, 692–703 (2003).Google Scholar
[23]Sun, H.G., Chen, W. and Chen, Y.Q., Variable order fractional differential operators in anomalous diffusion modeling, Phys. A 388, 4586–4592 (2009).Google Scholar
[24]Ingman, D. and Suzdalnitsky, J., Control of damping oscillations by fractional differential operator with time-dependent order, Comp. Meth. Appl. Mech. Eng. 193, 5585–5595 (2004).Google Scholar
[25]Ramirez, L.E.S. and Coimbra, C.F.M., On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equ. 2010, Art. ID 846107: 1–16 (2010).Google Scholar
[26]Lin, R., Liu, F., Anh, V. and Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput. 212, 435–445 (2009).Google Scholar
[27]Zhuang, P., Liu, F., Anh, V. and Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Num. Anal. 47, 1760–1781 (2009).Google Scholar
[28]Zhang, H., Liu, F., Phanikumar, M.S. and Meerschaert, M.M., A novel numerical method for the time variable fractional ordermobile-immobile advection-dispersionmodel, Comput.Math. Appl. 66, 693–701 (2013).Google Scholar
[29]Sun, H.G., Chen, W., Li, C.P. and Chen, Y.Q., Finite difference schemes for variable-order time fractional diffusion equation, Int. J. Bifurcat. Chaos 22, 1250085: 1–16 (2012).Google Scholar
[30]Zhao, X., Sun, Z.Z. and Karniadakis, G.E., Second-order approximations for variable order fractional derivatives: Algorithms and applications, J. Comp. Phys. 293, 184–200 (2015).Google Scholar
[31]Chen, Y., Liu, L., Li, B. and Sun, Y., Numerical solution for the variable order linear cable equation with Bernstein polynomials, Appl. Math. Comput. 238, 329–341 (2014).Google Scholar
[32]Bhrawy, A.H. and Zaky, M.A., Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear. Dyn. 80, 101–116 (2015).Google Scholar
[33]Zayernouri, M. and Karniadakis, G.E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J. Comp. Phys. 293, 312–338 (2015).Google Scholar
[34]Abdelkawy, M.A., Zaky, M.A., Bhrawy, A.H. and Baleanu, D., Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys. 67, 1–19 (2015).Google Scholar
[35]Zhang, Y., Benson, D.A. and Reeves, D.M., Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water. Resources 32, 561–581 (2009).Google Scholar
[36]Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B., Fractal mobile/immobile solute transport, Water Resources Res. 39, 1296: 1–12 (2003).CrossRefGoogle Scholar
[37]Shen, J., Tang, T. and Wang, L.L., Spectral Methods: Algorithms, Analysis and Applications, Springer, New York (2011).Google Scholar
[38]Potts, D., Fast algorithms for discrete polynomial transforms on arbitrary grids, Linear Algebra Appl. 366, 353–370 (2003).Google Scholar
[39]Wang, H. and Wang, K.X., An O(Nlog2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comp. Phys. 230 7830–7839 (2011).Google Scholar
[40]Wang, H. and Wang, K.X., A Fast Finite ElementMethod for Space-Fractional Dispersion Equations on Bounded Domains in ℝ2, SIAM J. Sci. Comput. 37, A1614-A1635 (2015).Google Scholar
[41]Sun, H.G., Chen, Y.Q. and Chen, W., Random-order fractional differential equation models, Signal Process. 91, 525–530 (2011).CrossRefGoogle Scholar