The Dirichlet problem for the time-fractional advection-diffusion equation in a line segment
- Yuriy Povstenko^{1}Email author and
- Joanna Klekot^{2}
https://doi.org/10.1186/s13661-016-0597-4
© Povstenko and Klekot 2016
Received: 11 June 2015
Accepted: 21 April 2016
Published: 28 April 2016
Abstract
The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered in a line segment. The fundamental solution to the Dirichlet problem and the solution of the problem with a constant boundary condition are obtained using the integral transform technique. The numerical results are illustrated graphically.
Keywords
fractional calculus Caputo fractional derivative non-Fickian diffusion fractional advection-diffusion equation Mittag-Leffler functionMSC
26A33 45K051 Introduction
A comprehensive survey of research on the fractional advection diffusion equation as well as of the numerical methods used for its solving can be found in [23]. In the literature there are only several papers in which the analytical solutions of fractional advections diffusion equation were considered [23–26]. In the present paper, we investigate the Dirichlet problem for equation (6) in a line segment \(0< x< L\). Two types of boundary conditions are considered: the Dirac delta boundary condition for the fundamental solution and the constant boundary condition for the sought-for function.
2 The fundamental solution to the Dirichlet problem
3 Constant boundary value of a function
4 Conclusions
We have considered the time-fractional advection-diffusion equation with the Caputo fractional derivative in a domain \(0< x< L\). The Laplace transform with respect to time t and the finite sin-Fourier transform with respect to the spatial coordinate x have been used. The fundamental solution to the Dirichlet problem and the solution to the problem with a constant boundary condition for the sought-for function have been obtained. The results of numerical calculations are displayed in the figures for different values of the nondimensional spatial variable x̄, the drift parameter v̄, the time parameter κ, and the order of the time-fractional derivative α. To evaluate the Mittag-Leffler functions \(E_{\alpha, \alpha}(-x)\) and \(E_{\alpha}(-x)\) we have used the algorithms suggested in [28] (the interested reader is also referred to the Matlab programs that implement these algorithms [29]). It should be emphasized that the first term in curly brackets in the solution (39) satisfies the boundary condition (31) and (32), whereas the second one equals zero at the ends of a line segment \(0< x< L\) due to (19).
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Kaviany, M: Principles of Heat Transfer in Porous Media, 2nd edn. Springer, New York (1995) View ArticleMATHGoogle Scholar
- Feller, W: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971) MATHGoogle Scholar
- Scheidegger, AE: The Physics of Flow Through Porous Media, 3rd edn. University of Toronto Press, Toronto (1974) MATHGoogle Scholar
- Van Kampen, NG: Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam (2007) MATHGoogle Scholar
- Risken, H: The Fokker-Planck Equation. Springer, Berlin (1989) View ArticleMATHGoogle Scholar
- Nield, DA, Bejan, A: Convection in Porous Media, 3rd edn. Springer, New York (2006) MATHGoogle Scholar
- Metzler, R, Klafter, J: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Rossikhin, Y, Shitikova, MV: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15-67 (1997) View ArticleGoogle Scholar
- West, BJ, Bologna, M, Grigolini, P: Physics of Fractal Operators. Springer, New York (2003) View ArticleGoogle Scholar
- Magin, RL: Fractional Calculus in Bioengineering. Begell House Publishers, Inc., Redding (2006) Google Scholar
- Gafiychuk, V, Datsko, B: Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl. 59, 1101-1107 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Mainardi, F: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010) View ArticleMATHGoogle Scholar
- Uchaikin, VV: Fractional Derivatives for Physicists and Engineers, Background and Theory. Springer, Berlin (2013) View ArticleMATHGoogle Scholar
- Povstenko, Y: Fractional Thermoelasticity. Springer, New York (2015) View ArticleMATHGoogle Scholar
- Povstenko, Y: Theory of diffusive stresses based on the fractional advection-diffusion equation. In: Abi Zeid Daou, R, Moreau, X (eds.) Fractional Calculus: Applications, pp. 227-241. Nova Science Publishers, New York (2015) Google Scholar
- Povstenko, Y: Fractional heat conduction equation and associated thermal stress. J. Therm. Stresses 28, 83-102 (2005) MathSciNetView ArticleGoogle Scholar
- Povstenko, Y: Thermoelasticity which uses fractional heat conduction equation. J. Math. Sci. 162, 296-305 (2009) MathSciNetView ArticleGoogle Scholar
- Povstenko, Y: Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys. Scr. T 136, 014017 (2009) View ArticleGoogle Scholar
- Povstenko, Y: Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418-435 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Gorenflo, R, Mainardi, F: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223-276. Springer, Wien (1997) View ArticleGoogle Scholar
- Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999) MATHGoogle Scholar
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
- Povstenko, Y: Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables. Math. Probl. Eng. 2014, 705364 (2014) MathSciNetView ArticleGoogle Scholar
- Liu, F, Anh, V, Turner, I, Zhuang, P: Time-fractional advection-dispersion equation. J. Appl. Math. Comput. 13, 233-245 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Huang, F, Liu, F: The time fractional diffusion equation and the advection-dispersion equation. ANZIAM J. 46, 317-330 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Huang, F, Liu, F: The fundamental solution of the space-time fractional advection-dispersion equation. J. Appl. Math. Comput. 18, 339-350 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Prudnikov, AP, Brychkov, YA, Marichev, OI: Integrals and Series. Vol. 1: Elementary Functions. Gordon & Breach, Amsterdam (1986) MATHGoogle Scholar
- Gorenflo, R, Loutchko, J, Luchko, Y: Computation of the Mittag-Leffler function and its derivatives. Fract. Calc. Appl. Anal. 5, 491-518 (2002) MathSciNetMATHGoogle Scholar
- Matlab File Exchange 2005, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy. Available for download at www.mathworks.com/matlabcentral/fileexchange/8738-Mittag-Leffler-function