Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation

Affiliation(s)

^{1}
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

^{2}
Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic
University (IMSIU), Riyadh, KSA.

^{3}
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

ABSTRACT

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

KEYWORDS

Fractional Advection-Dispersion Equation, Caputo Fractional Derivative, Finite Difference Method, Chebyshev Pseudo-Spectral Method, Convergence Analysis

Fractional Advection-Dispersion Equation, Caputo Fractional Derivative, Finite Difference Method, Chebyshev Pseudo-Spectral Method, Convergence Analysis

Cite this paper

Sweilam, N. , Khader, M. and Adel, M. (2014) Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation.*Applied Mathematics*, **5**, 3240-3248. doi: 10.4236/am.2014.519301.

Sweilam, N. , Khader, M. and Adel, M. (2014) Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation.

References

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[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.

[3] Khader, M.M. (2012) Introducing an Efficient Modification of the Homotopy Perturbation Method by Using Chebyshev Polynomials. Arab Journal of Mathematical Sciences, 18, 61-71.

[4] Khader, M.M. and Hendy, A.S. (2012) The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudospectral Method. International Journal of Pure and Applied Mathematics, 74, 287-297.

[5] Khader, M.M. (2012) Introducing an Efficient Modification of the Variational Iteration Method by Using Chebyshev Polynomials. Application and Applied Mathematics: An International Journal, 7, 283-299.

[6] Sweilam, N.H., Khader, M.M. and Al-Bar, R.F. (2008) Homotopy Perturbation Method for Linear and Nonlinear System of Fractional Integro-Differential Equations. International Journal of Computational Mathematics and Numerical Simulation, 1, 73-87.

[7] Jafari, H. and Daftardar-Gejji, V. (2006) Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by ADM. Applied Mathematics and Computation, 180, 488-497.

[8] Yu, Q., Liu, F., Anh, V. and Turner, I. (2008) Solving Linear and Nonlinear Space-Time Fractional Reaction-Diffusion Equations by Adomian Decomposition Method. International Journal for Numerical Methods in Engineering, 74, 138-153.

http://dx.doi.org/10.1002/nme.2165

[9] Hashim, I., Abdulaziz, O. and Momani, S. (2009) Homotopy Analysis Method for Fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14, 674-684.

[10] Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Integro-Differential Equations. ANZIAM, 51, 464-475.

[11] Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Mathematics, 2012, Article ID: 764894.

[12] Smith, G.D. (1965) Numerical Solution of Partial Differential Equations. Oxford University Press, Oxford.

[13] Sweilam, N.H., Khader, M.M. and Nagy, A.M. (2011) Numerical Solution of Two-Sided Space-Fractional Wave Equation Using Finite Difference Method. Journal of Computational and Applied Mathematics, 235, 2832-2841.

[14] Sweilam, N.H., Khader, M.M. and Adel, M. (2012) On the Stability Analysis of Weighted Average Finite Difference Methods for Fractional Wave Equation. Fractional Differential Calculus, 2, 17-29.

[15] Yuste, S.B. and Acedo, L. (2005) An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Numerical Analysis, 42, 1862-1874.

http://dx.doi.org/10.1137/030602666

[16] Zhuang, P. and Liu, F. (2006) Implicit Difference Approximation for the Time Fractional Diffusion Equation. Journal of Applied Mathematics and Computing, 22, 87-99.

http://dx.doi.org/10.1007/BF02832039

[17] Zhuang, P., Liu, F., Anh, V. and Turner, I. (2008) New Solution and Analytical Techniques of the Implicit Numerical Methods for the Anomalous Sub-Diffusion Equation. SIAM Journal on Numerical Analysis, 46, 1079-1095.

http://dx.doi.org/10.1137/060673114

[18] Chechkin, A.V., Gonchar, V.Y., Klafter, J., Metzler, R. and Tanatarov, L.V. (2004) Lévy Flights in a Steep Potential Well. Journal of Statistical Physics, 115, 1505-1535.

[19] Langlands, T.A.M., Henry, B.I. and Wearne, S.L. (2009) Fractional Cable Equation Models for Anomalous Electrodiffusion in Nerve Cells: Infinite Domain Solutions. Journal of Mathematical Biology, 59, 761-808.

[20] Quintana-Murillo, J. and Yuste, S.B. (2011) An Explicit Numerical Method for the Fractional Cable Equation. International Journal of Differential Equations, 2011, Article ID: 231920.

[21] Rall, W. (1977) Core Conductor Theory and Cable Properties of Neurons. In: Poeter, R., Ed., Handbook of Physiology: The Nervous System, Vol. 1, Chapter 3, American Physiological Society, Bethesda, 39-97.

[22] Sokolov, I.M., Klafter, J. and Blumen, A. (2002) Fractional Kinetics. Physics Today, 55, 48-54.

[23] Snyder, M.A. (1966) Chebyshev Methods in Numerical Approximation. Prentice-Hall, Inc., Englewood Cliffs.

[24] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 16, 2535-2542.

[25] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2011) Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations. Applied Mathematical Modelling, 35, 5662-5672.

http://dx.doi.org/10.1016/j.apm.2011.05.011

[1] Liu, F., Yang, Q. and Turner, I. (2011) Stability and Convergence of Two New Implicit Numerical Methods for the Fractional Cable Equation. Journal of Computational and Nonlinear Dynamics, 6, Article ID: 01109.

[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.

[3] Khader, M.M. (2012) Introducing an Efficient Modification of the Homotopy Perturbation Method by Using Chebyshev Polynomials. Arab Journal of Mathematical Sciences, 18, 61-71.

[4] Khader, M.M. and Hendy, A.S. (2012) The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudospectral Method. International Journal of Pure and Applied Mathematics, 74, 287-297.

[5] Khader, M.M. (2012) Introducing an Efficient Modification of the Variational Iteration Method by Using Chebyshev Polynomials. Application and Applied Mathematics: An International Journal, 7, 283-299.

[6] Sweilam, N.H., Khader, M.M. and Al-Bar, R.F. (2008) Homotopy Perturbation Method for Linear and Nonlinear System of Fractional Integro-Differential Equations. International Journal of Computational Mathematics and Numerical Simulation, 1, 73-87.

[7] Jafari, H. and Daftardar-Gejji, V. (2006) Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by ADM. Applied Mathematics and Computation, 180, 488-497.

[8] Yu, Q., Liu, F., Anh, V. and Turner, I. (2008) Solving Linear and Nonlinear Space-Time Fractional Reaction-Diffusion Equations by Adomian Decomposition Method. International Journal for Numerical Methods in Engineering, 74, 138-153.

http://dx.doi.org/10.1002/nme.2165

[9] Hashim, I., Abdulaziz, O. and Momani, S. (2009) Homotopy Analysis Method for Fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14, 674-684.

[10] Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Integro-Differential Equations. ANZIAM, 51, 464-475.

[11] Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Mathematics, 2012, Article ID: 764894.

[12] Smith, G.D. (1965) Numerical Solution of Partial Differential Equations. Oxford University Press, Oxford.

[13] Sweilam, N.H., Khader, M.M. and Nagy, A.M. (2011) Numerical Solution of Two-Sided Space-Fractional Wave Equation Using Finite Difference Method. Journal of Computational and Applied Mathematics, 235, 2832-2841.

[14] Sweilam, N.H., Khader, M.M. and Adel, M. (2012) On the Stability Analysis of Weighted Average Finite Difference Methods for Fractional Wave Equation. Fractional Differential Calculus, 2, 17-29.

[15] Yuste, S.B. and Acedo, L. (2005) An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Numerical Analysis, 42, 1862-1874.

http://dx.doi.org/10.1137/030602666

[16] Zhuang, P. and Liu, F. (2006) Implicit Difference Approximation for the Time Fractional Diffusion Equation. Journal of Applied Mathematics and Computing, 22, 87-99.

http://dx.doi.org/10.1007/BF02832039

[17] Zhuang, P., Liu, F., Anh, V. and Turner, I. (2008) New Solution and Analytical Techniques of the Implicit Numerical Methods for the Anomalous Sub-Diffusion Equation. SIAM Journal on Numerical Analysis, 46, 1079-1095.

http://dx.doi.org/10.1137/060673114

[18] Chechkin, A.V., Gonchar, V.Y., Klafter, J., Metzler, R. and Tanatarov, L.V. (2004) Lévy Flights in a Steep Potential Well. Journal of Statistical Physics, 115, 1505-1535.

[19] Langlands, T.A.M., Henry, B.I. and Wearne, S.L. (2009) Fractional Cable Equation Models for Anomalous Electrodiffusion in Nerve Cells: Infinite Domain Solutions. Journal of Mathematical Biology, 59, 761-808.

[20] Quintana-Murillo, J. and Yuste, S.B. (2011) An Explicit Numerical Method for the Fractional Cable Equation. International Journal of Differential Equations, 2011, Article ID: 231920.

[21] Rall, W. (1977) Core Conductor Theory and Cable Properties of Neurons. In: Poeter, R., Ed., Handbook of Physiology: The Nervous System, Vol. 1, Chapter 3, American Physiological Society, Bethesda, 39-97.

[22] Sokolov, I.M., Klafter, J. and Blumen, A. (2002) Fractional Kinetics. Physics Today, 55, 48-54.

[23] Snyder, M.A. (1966) Chebyshev Methods in Numerical Approximation. Prentice-Hall, Inc., Englewood Cliffs.

[24] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 16, 2535-2542.

[25] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2011) Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations. Applied Mathematical Modelling, 35, 5662-5672.

http://dx.doi.org/10.1016/j.apm.2011.05.011