Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection–diffusion equation
- M. A. Zaky^{1}Email authorView ORCID ID profile,
- D. Baleanu^{2, 3},
- J. F. Alzaidy^{4} and
- E. Hashemizadeh^{5}
https://doi.org/10.1186/s13662-018-1561-7
© The Author(s) 2018
Received: 30 November 2017
Accepted: 15 March 2018
Published: 22 March 2018
Abstract
In this paper, we investigate numerical solution of the variable-order fractional Galilei advection–diffusion equation with a nonlinear source term. The suggested method is based on the shifted Legendre collocation procedure and a matrix form representation of variable-order Caputo fractional derivative. The main advantage of the proposed method is investigating a global approximation for the spatial and temporal discretizations. This method reduces the problem to a system of algebraic equations, which is easier to solve. The validity and effectiveness of the method are illustrated by an easy-to-follow example.
Keywords
1 Introduction
Recently, kinetic equations with fractional derivatives were recognized as a useful tool for description of anomalous diffusion phenomena. Examples include systems exhibiting underground water pollution, Hamiltonian chaos, disordered medium, dynamics of protein molecules, reactions in complex systems, motions under the influence of optical tweezers, and more; see reviews on fractional kinetics [1–4]. The kinetic equations with time-fractional derivative are used for description of subdiffusion processes, that is, those for which the mean-squared displacement grows in time slower than linearly [5]. Also, it describes slow relaxation processes that are characterized by stretched exponential or power-law response function [6]. It became clear that further theoretical investigations are required to incorporate adequate tools for description of more realistic random processes, which are described by a set of characteristic exponents and are therefore of multifractional type. An adequate kinetic description of these processes requires the use of generalized fractional kinetics based on the concept of variable-order fractional (V-OF) operators. This calculus was proposed in [7, 8] and very recently was introduced in physics [9, 10].
The V-OF operators are nonlocal with singular kernels, which makes the V-OF models complicated. Hence, solving V-OF models is also more complicated. Numerical computation of the V-OF operators is the key to understand the behavior and physical meaning of the V-OF models. Lin et al. [11] investigated the stability and convergence of an explicit finite-difference approximation for a nonlinear V-OF diffusion equation. Chen et al. [12] proposed two numerical schemes for a V-OF anomalous subdiffusion equation, one with first-order temporal accuracy and fourth-order spatial accuracy and the other with second-order temporal accuracy and fourth-order spacial accuracy. Yang et al. [13] proposed a finite difference scheme for solving V-OF reaction–diffusion equation. Abdelkawy et al. [14] proposed a new spectral method to achieve high accurate solution for the V-OF mobile–immobile advection–dispersion model. Chen et al. [15] proposed a numerical method to estimate the V-OF derivatives of an unknown signal in noisy environment. Tavares et al. [16] presented a numerical tool to solve partial differential equations involving V-OF Caputo derivatives. Bhrawy and Zaky [17] proposed a numerical method for solving the V-OF nonlinear cable equation based on shifted Jacobi collocation procedure together with the shifted Jacobi operational matrix for V-OF derivatives. They also proposed an accurate and robust approach to approximate solutions of V-OF functional boundary value problems[18]. Zaky et al. [19] proposed the Jacobi wavelets collocation approach based on the Jacobi wavelets operational matrix of V-OF derivative for solving a general class of V-OF differential equations arising in turbulent fluid dynamics. Doha et al. [20, 21] used polynomial collocation techniques to solve V-OF integro–differential equations. Moghaddam and Tenreiro Machado [22–24] proposed algorithms based on finite difference approximations and B-spline interpolation for different definitions of V-OF derivatives.
Spectral methods are of fundamental importance in computational physics because of their ability in achieving desired solution with a small number of degrees of freedom, which often allows gains in accuracy with considerable reduction in computational cost [25]. Collocation method is one of more applicable types of the spectral methods and is frequently used to solve various types of differential equations, such as the Schrödinger equation [26, 27], Rayleigh–Stokes equation [28], diffusion equation [29], mobile–immobile advection–dispersion equation [14], and cable equation [17]. It is well known that the majority of the fractional differential equations have no exact solutions. Therefore, numerical methods to obtain an approximate solution have become the preferred approach for such equations [30–37]. Approaches for numerically approximating the solution of fractional differential equations have been extensively studied; see, e.g., [38–42].
This paper is organized as follows. In Sect. 2, we first present some preliminaries from fractional calculus and introduce some properties of the shifted Legendre polynomials. In Sect. 3, we derive the operational matrix of the V-OF derivative for the shifted Legendre polynomials. In Sect. 4, the V-OF Galilei invariant advection–diffusion equation with a nonlinear source term is numerically investigated. In Sect. 5, numerical results are discussed. Finally, In Sect. 6, we outline the main conclusions.
2 Preliminaries
In this section, we recall some mathematical preliminaries of the V-OF operators (see [44]) and relevant properties of Legendre polynomials (see [25, 45–48]).
Definition 2.1
Definition 2.2
Definition 2.3
3 Operational matrices based on Legendre polynomials
Theorem 3.1
Proof
This theorem is a generalization of that in [48], where \(0 < \gamma _{\min } < \gamma (t) < \gamma _{\max } < 1\) and \(\tau=1\).
4 The collocation method
After the construction of the V-OF differentiation matrices of Caputo type, we now use the Legendre–Gauss–Lobatto collocation technique in combination with the shifted Legendre operational matrix of V-OF fractional differentiation.
5 Numerical example
The maximum absolute errors
γ(x,t) | Method [43] | Proposed method | ||
---|---|---|---|---|
\(\tau ^{2} = h^{2} = \frac{1}{{256}}\) | N = M = 4 | N = M = 8 | N = M = 12 | |
\(\frac{10-xt}{300} \) | 1.1311 × 10^{−4} | 41806 × 10^{−5} | 4.8553 × 10^{−11} | 6.5749 × 10^{−16} |
\(\frac{20-e^{xt}}{600}\) | 9.2323 × 10^{−5} | 4.1813 × 10^{−5} | 4.8554 × 10^{−11} | 9.9033 × 10^{−16} |
\(\frac{12+x^{3}-t^{5}}{300}\) | 3.7142 × 10^{−4} | 4.1810 × 10^{−5} | 3.9710 × 10^{−11} | 6.0284 × 10^{−16} |
\(\frac{15+\cos (xt)}{450}\) | 3.7155 × 10^{−5} | 4.1810 × 10^{−5} | 3.9711 × 10^{−11} | 5.4710 × 10^{−16} |
\(\frac{10-\sin (xt)}{310}\) | 9.6551 × 10^{−5} | 4.1808 × 10^{−5} | 4.8554 × 10^{−11} | 7.0597 × 10^{−16} |
\(\frac{10+(xt)^{2}-(xt)^{3}}{300}\) | 1.2258 × 10^{−5} | 4.1808 × 10^{−5} | 4.8552 × 10^{−11} | 6.9103 × 10^{−16} |
\(\frac{13-xt+\cos (xt)}{400}\) | 1.6057 × 10^{−4} | 4.1804 × 10^{−5} | 4.8551 × 10^{−11} | 6.0096 × 10^{−16} |
\(\frac{11+(xt)^{2}-\sin (xt)}{330}\) | 2.0982 × 10^{−5} | 4.1815 × 10^{−5} | 4.8553 × 10^{−11} | 5.3889 × 10^{−16} |
\(\frac{18-sin^{2}(xt)+\cos ^{3}(xt)}{630}\) | 9.0181 × 10^{−5} | 4.1785 × 10^{−5} | 4.8618 × 10^{−11} | 4.9293 × 10^{−16} |
6 Conclusions
In this paper, we proposed an efficient method for solving the V-OF Galilei invariant advection–diffusion equation with a nonlinear source term. The proposed method based on the Legendre–Gauss–Lobatto collocation technique combined with the associated operational matrices of V-OF derivatives. This algorithm was employed for solving a class of variable-order fractional differential equations. The method has the advantage of transforming the problem into the solution of a system of algebraic equations, which greatly simplifies it. Finally, we presented a numerical example to demonstrate the efficiency of the proposed method.
Declarations
Acknowledgements
The authors would like to express their gratitude to the anonymous reviewers for their constructive comments, which gave the paper its final form.
Authors’ contributions
The authors have equal contributions to each part of this paper. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
- Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986) MathSciNetView ArticleMATHGoogle Scholar
- Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) View ArticleMATHGoogle Scholar
- Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284(1), 521–541 (2002) View ArticleMATHGoogle Scholar
- Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Fractional diffusion in inhomogeneous media. J. Phys. A, Math. Gen. 38(42), L679–L684 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today 55, 48–54 (2000) View ArticleGoogle Scholar
- Samko, S., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transforms Spec. Funct. 1, 277–300 (1993) MathSciNetView ArticleMATHGoogle Scholar
- Samko, S.: Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn. 71, 653–662 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12, 692–703 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Lin, R., Liu, F., Anh, V., Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212(2), 435–445 (2009) MathSciNetMATHGoogle Scholar
- Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32(4), 1740–1760 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Yang, Q., Moroney, T., Liu, F., Turner, I.: Computationally efficient methods for solving time-variable-order time-space fractional reaction–diffusion equation. In: Proceedings of the 5th IFAC Symposium on Fractional Differentiation and its Applications (2012) Google Scholar
- Abdelkawy, M.A., Zaky, M.A., Bhrawy, A.H., Baleanu, D.: Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model. Rom. Rep. Phys. 67, 773–791 (2015) Google Scholar
- Chen, Y., Weia, Y., Liu, D., Boutat, D., Chen, X.: Variable-order fractional numerical differentiation for noisy signals by wavelet denoising. J. Comput. Phys. 311, 338–347 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Tavares, D., Almeida, R., Torres, D.F.M.: Caputo derivatives of fractional variable order: numerical approximations. Commun. Nonlinear Sci. Numer. Simul. 35, 69–87 (2016) MathSciNetView ArticleGoogle Scholar
- Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80, 101–116 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A.: Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn. 85, 1815–1823 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zaky, M.A., Ameen, I.G., Abdelkawy, M.A.: A new operational matrix based on Jacobi wavelets for a class of variable-order fractional differential equations. Proc. Rom. Acad., Ser. A 18, 315–322 (2017) Google Scholar
- Doha, E.H., Abdelkawy, M.A., Amin, A.Z.M., Lopes, A.M.: On spectral methods for solving variable-order fractional integro-differential equations. Comput. Appl. Math. (2017). https://doi.org/10.1007/s40314-017-0551-9 Google Scholar
- Doha, E.H., Abdelkawy, M.A., Amin, A.Z.M., Baleanu, D.: Spectral technique for solving variable-order fractional Volterra integro-differential equations. Numer. Methods Partial Differ. Equ. (2017). https://doi.org/10.1002/num.22233 Google Scholar
- Moghaddam, B.P., Tenreiro Machado, J.A.: A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fract. Calc. Appl. Anal. 20, 1023–1042 (2017) MathSciNetMATHGoogle Scholar
- Moghaddam, B.P., Tenreiro Machado, J.A.: A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Comput. Math. Appl. 73, 1262–1269 (2017) MathSciNetView ArticleGoogle Scholar
- Moghaddam, B.P., Tenreiro Machado, J.A.: SM-algorithms for approximating the variable-order fractional derivative of high order. Fundam. Inform. 151, 293–311 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988) View ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A.: Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput. Math. Appl. 73, 1100–1117 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A.: An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A., Alzaidy, J.F.: Two shifted Jacobi–Gauss collocation schemes for solving two-dimensional variable-order fractional Rayleigh–Stokes problem. Adv. Differ. Equ. 2016, 272 (2016) MathSciNetView ArticleGoogle Scholar
- Zaky, M.A., Ezz-Eldien, S.S., Doha, E.H., Machado, J.T., Bhrawy, A.H.: An efficient operational matrix technique for multi-dimensional variable-order time fractional diffusion equations. J. Comput. Nonlinear Dyn. 11, 061002 (2016) View ArticleGoogle Scholar
- Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205(2), 719–736 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Sub-diffusion equations of fractional order and their fundamental solutions. In: Mathematical Methods in Engenering, pp. 20–48 (2006) Google Scholar
- Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection–diffusion equation. Appl. Math. Comput. 191, 12–20 (2007) MathSciNetMATHGoogle Scholar
- Chen, C., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation. Appl. Math. Comput. 198(2), 754–769 (2008) MathSciNetMATHGoogle Scholar
- Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y.Q., Vinagre Jara, B.M.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228(8), 3137–3153 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Yuste, S.B., Acedo, L.: On an explicit finite difference method for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862–1874 (2005) MathSciNetView ArticleMATHGoogle Scholar
- MacDonald, C.L., Bhattacharya, N., Sprouse, B.P., Silva, G.A.: Efficient computation of the Grünwald–Letnikov fractional diffusion derivative using adaptive time step memory. J. Comput. Phys. 297, 221–236 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A., Baleanu, D., Abdelkawy, M.A.: A novel spectral approximation for the two-dimensional fractional sub-diffusion problems. Rom. J. Phys. 60, 344–359 (2015) Google Scholar
- Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281(15), 876–895 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A.: Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl. Math. Model. 40, 832–845 (2016) MathSciNetView ArticleGoogle Scholar
- Bhrawy, A.H., Zaky, M.A.: A fractional-order Jacobi tau method for a class of time-fractional PDEs with variable coefficients. Math. Methods Appl. Sci. 39, 1765–1779 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zaky, M.A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math. (2017). https://doi.org/10.1007/s40314-017-0530-1 Google Scholar
- Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term. Appl. Math. Comput. 217(12), 5729–5742 (2011) MathSciNetMATHGoogle Scholar
- Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical method for the variable-order fractional advection–diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47(3), 1760–1781 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, A.H., Zaky, M.A., Van Gorder, R.A.: A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation. Numer. Algorithms 71(1), 151–180 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zaky, M.A.: An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. Comput. Math. Appl. (2017). https://doi.org/10.1016/j.camwa.2017.12.004 MathSciNetGoogle Scholar
- Bhrawy, A.H., Zaky, M.A., Baleanu, D.: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67(2), 340–349 (2015) Google Scholar
- Wang, L., Ma, Y., Yang, Y.: Legendre polynomials method for solving a class of variable order fractional differential equation. Comput. Model. Eng. Sci. 101(2), 97–111 (2014) MathSciNetMATHGoogle Scholar