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

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1 Applied Matheatics, 4, 5, Published Online Noveber 4 in SciRes. Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation N. H. Sweila, M. M. Khader,3, M. Adel Departent of Matheatics, Faculty of Science, Cairo University, Giza, Egypt Departent of Matheatics and Statistics, College of Science, Al-Ia Mohaad Ibn Saud Islaic University (IMSIU), Riyadh, KSA 3 Departent of Matheatics, Faculty of Science, Benha University, Benha, Egypt Eail: nsweila@sci.cu.edu.eg, ohaedbd@yahoo.co Received 3 August 4; revised Septeber 4; accepted 6 October 4 Copyright 4 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Coons Attribution International License (CC BY). Abstract Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied atheatics, bio-engineering and others. However, any researchers reain unaware of this field. In this paper, an efficient nuerical ethod for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The ethod is based on Chebyshev approxiations. The properties of Chebyshev polynoials are used to reduce ADE to a syste of ordinary differential equations, which are solved using the finite difference ethod (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived forula are given. Nuerical solutions of ADE are presented and the results are copared with the exact solution. Keywords Fractional Advection-Dispersion Equation, Caputo Fractional Derivative, Finite Difference Method, Chebyshev Pseudo-Spectral Method, Convergence Analysis. Introduction Ordinary and partial fractional differential equations (FDEs) have been the focus of any studies due to their frequent appearance in various applications in fluid echanics, viscoelasticity, biology, physics and engineering [] []. Consequently, considerable attention has been given to the solutions of FDEs of physical interest. Most How to cite this paper: Sweila, N.H., Khader, M.M. and Adel, M. (4) Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation. Applied Matheatics, 5,

2 N. H. Sweila et al. FDEs do not have exact solutions, so approxiate and nuerical techniques [3] [4], ust be used. Recently, several nuerical ethods to solve FDEs have been given such as variational iteration ethod [5], hootopy perturbation ethod [3] [6], Adoian decoposition ethod [7] [8], hootopy analysis ethod [9], collocation ethod [] [] and finite difference ethod []-[7]. We introduce soe necessary definitions and atheatical preliinaries of the fractional calculus theory that will be required in the present paper. Definition The Caputo fractional derivative operator D of order is defined in the following for ( f ) ( t) x D f ( x) = d t, >, x >, + Γ( ) x t where Γ. is the gaa function. Siilar to integer-order differentiation, Caputo fractional derivative operator is a linear operation <,, ( λ + µ ) = λ + µ, D f x g x D f x D g x where λ and µ are constants. For the Caputo s derivative we have [] D x n D C =, C is a constant, (), for n and n< ; = Γ ( n + ) n x, for n and n. Γ ( n + ) We use the ceiling function to denote the sallest integer greater than or equal to and = {,,, }. Recall that for, the Caputo differential operator coincides with the usual differential operator of integer order. For ore details on fractional derivatives definitions and its properties see []. Anoalous, or non-fickian, dispersion has been an active area of research in the physics counity since the introduction of continuous tie rando walks (CTRW) by Montroll and Weiss [965]. These rando walks extended the predictive capability of odels built on the stochastic process of Brownian otion, which is the basis for the classical advectiondispersion equation (ADE). A fractional ADE. is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE, the fractional ADE has solutions that reseble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies. When a fractional Fick s law replaces the classical Fick s law in an Eulerian evaluation of solute transport in a porous ediu, the result is a fractional ADE. It describes the spread of solute ass over large distances via a convolutional fractional derivative. We consider the initial-boundary value proble of the fractional Advection-dispersion equation which is usually written in the following for β t µ < β, (, ) u xt, = D u xt, D u xt, + s xt,, < x<, t T, (3) where <, s xt is the source ter, µ is a constant and D γ is the Caputo fractional derivative with respect to x and of order γ, where γ =, β. Under the zero boundary conditions u, t = u, t =, (4) and the following initial condition g( x) u x, =. (5) In the last few years appeared any papers to study this odel (3)-(5) [] [8]-[], the ost of these papers study the ordinary case of such proble but in this paper we study the fractional case. Our idea is to apply the Chebyshev collocation ethod to discretize (3) to get a linear syste of ODEs thus greatly siplifying the proble, and use FDM [] to solve the resulting syste. () 34

3 N. H. Sweila et al. The organization of this paper is as follows. In the next section, we obtain the approxiation of fractional derivative D y( x). In Section 3, we prove the error analysis of the proposed forula. In Section 4, we ipleent chebyshev collocation ethod to the solution of (3). As a result a syste of ordinary differential equations is fored and the solution of the considered proble is introduced. In Section 5, we give soe nuerical results to clarify the proposed ethod. Also a conclusion is given in Section 6.. Derivation of the Approxiate Forula The well known Chebyshev polynoials are defined on the interval [,] of the following recurrence forula [3] T z = zt z T z, T z =, T z = z, n =,,. n+ n n The analytic for of the Chebyshev polynoials T where [ ] n = ( ) n and can be deterined with the aid z of degree n is given by n i n i i=!! n i, (6) ( n i! ) ( i) ( n i) T z n z n denotes the integer part of n. The orthogonality condition is π, for i = j = ; Ti( zt ) j ( z) π d z =, for i = j ; z, for i j., we define the so called shifted Chebyshev polynoials by introducing the change of variable z = x. So, the shifted Chebyshev polynoials are defined as Tn ( x) = Tn( x ) = T n( x). T x of degree n is given by In order to use these polynoials on the interval [ ] The analytic for of the shifted Chebyshev polynoials n n k n k ( n+ k! ) n k = ( k)!( n k)! k T x = n x, n =,,. (7) The function u( x ), which belongs to the space of square integrable in [ ] shifted Chebyshev polynoials as where the coefficients c i are given by u x ct x ( x) i i i=,, ay be expressed in ters of =, (8) u xt ckj = x h = h = j = hjπ x x j d,, j,,,. (9) In practice, only the first ( + ) -ters of shifted Chebyshev polynoials are considered. Then we have u x = ct x. () i i i= Khader [4] introduced a new approxiate forula of the fractional derivative and used it to solve nuerically the fractional diffusion equation. u x is given in the following theore. The ain approxiate forula of the fractional derivative of Theore [4] Let u( x ) be approxiated by Chebyshev polynoials as in () and also suppose > then where w ik, is given by i = k () i= k= ( ) ( ) i ik, D u x cw x 34

4 N. H. Sweila et al. w ( ) ik, = i k k i( i k! ) ( k ) ( i k)! ( k)! Γ ( k+ ) + Γ + Also, in this section, special attention is given to study the convergence analysis and evaluate an upper bound of the error of the proposed approxiate forula. Theore (Chebyshev truncation theore) [3] The error in approxiating u( x ) by the su of its first ters is bounded by the su of the absolute values of all the neglected coefficients. If then for all u( x ), all, and all [,] k k k = () u x = ct x, (3) E u x u x c, (4) T k k= + t. Theore 3 [5] The Caputo fractional derivative of order for the shifted Chebyshev polynoials can be expressed in ters of the shifted Chebyshev polynoials theselves in the following for where k i ( i ) = Θi, jk, j, (5) D T x T x k= j= i k ( ) i( i+ k! ) Γ k + Θ i, jk, =, j =,,. hjγ k+ ( i k)! Γ( k j+ ) Γ ( k+ j + ) Theore 4 [] E Dux Du x The error = in approxiating Dux T E c i k by Du ( x) is bounded by Θ,,. (6) T i i jk i= + k= j= 3. Procedure Solution of the Fractional Advection-Dispersion Equation Consider the fractional Advection-dispersion equation of type given in Equation (3). In order to use Chebyshev u xt, as collocation ethod, we first approxiate Fro Equations (3), (7) and Theore we have (, ) i i i= u xt = u t T x. (7) du i i i ( t) k ( β) k β Ti ( x) = ui( t) wik, x µ ui( t) wik, x + s( x, t). (8) dt i= i= k= i= β k= β We now collocate Equation (8) at ( + ) points x, p =,,, p as i i i i p = i ik, p i ik, p + p i= i= k= i= β k= β k µ ( β) k β (, ) u t T x u t w x u t w x s x t. (9) For suitable collocation points we use roots of shifted chebyshev polynoial T ( x)

5 N. H. Sweila et al. Also, by substituting Equations (7) in the boundary conditions (4) we can obtain equations as follows i ( ) ui( t) =, ui( t) =. () i= i= Equation (9), together with + of ordinary differential equations which can be solved, for the unknowns u i, i =,,,, using the finite difference ethod, as described in the following section. 4. Nuerical Results equations of the boundary conditions (), give In this section, we present a nuerical exaple to illustrate the efficiency and the validation of the proposed nuerical ethod when applied to solve nuerically the fractional Advection-dispersion equation. Consider the ADE (3) with µ = and the following source ter t β Γ ( + ) β s( xt, ) = e ( x x )! + x β! Γ( β + ) and the boundary conditions u(, t) = u(, t) =, with the initial condition (,) The exact solution of Equation (3) in this case is (, ) e t β, () β u x = x x. u xt = x x. () We apply the proposed ethod with = 3, and approxiate the solution as follows Using Equation (9) we have where p 3 3, i i i= k u xt = u t T x. (3) 3 3 i 3 i i i p i ik, p i ik, p p i= i= k= i= k= ( β) k β u t T x = u t w x u t w x + s x, t, p =,, (4) x are roots of shifted Chebyshev polynoial T ( x) x, i.e. =.46447, x = By using Equations () and (4) we can obtain the following syste of ODEs u t + k u t + k u t = R u t + R u t + R u t + s t, (5) where + + = u( t) u( t) u( t) u3( t) u ( t) u ( t) u ( t) u ( t) u t l u t l u t Qu t Q u t Q u t s t, (6) =, (7) =, (8) 3 k = T x, k = T x, = L x, = L x, 3 3 ( β) β ( β) β ( β) β =,, =,, +,, ( ) β ( ( β) ( β) ( β) ), β, β ( β ) β ( x ) ( ) β ( ( β) ( β) ( β) ). R w x R w x w x w x R = x w + w x x w + w x + w x 3 3, 3,3 3, 3, 3,3 ( β) ( β) =, =,, +, Q w x Q w x w x w Q = x w + w x x w + w x + w x 3 3, 3,3 3, 3, 3,3 Now, to use FDM for solving the syste (5)-(8), we will use the following notations: t i = iτ to be the integration tie ti T τ = T N for i =,,, N. Define u n i = ui( tn), s n i = si( tn). Then the syste (5)-(8), is discretized in tie and takes the following for, 344

6 N. H. Sweila et al. n+ n n+ n n+ n u u u u u3 u3 n+ n+ n+ n+ + k + k = Ru + Ru + R3u3 + s, (9) τ τ τ n+ n n+ n n+ n u u u u u3 u3 n+ n+ n+ n+ + + = Qu + Qu + Q3u3 + s, (3) τ τ τ u u + u u =, (3) n+ n+ n+ n+ 3 u + u + u + u =. (3) n+ n+ n+ n+ 3 We can write the above syste (9)-(3) in the following atrix for as follows n+ n n+ k τr τr k τr3 u k k u s τq τq τq3 u u s = + τ u u u3 u3 We use the notation for the above syste. (33) n+ n n+ n+ n n+ AU = BU + S, or U = A BU + A S, (34) n n n n n where = T n n n and T U u, u, u, u 3 S = τs, τs,, The obtained nuerical results by eans of the proposed ethod are shown in Table and Figures -4. In Table, the absolute error between the exact solution u ex and the approxiate solution u at approx = 3 and = 5 with the final tie T = are given. Also, in Figure and Figure, coparison between the exact solution and the approxiate solution at T =.5 with tie step τ =.5, = 3 and = 5, are presented, respectively. Also, in Figure 3 and Figure 4, the behavior of the approxiate solution at T =.5 and = 5 with different values of and β are presented, respectively. Fro, these figures, we can see that the behavior of the approxiate solution depends on the order of the fractional derivative. 5. Conclusion and Rearks The properties of the Chebyshev polynoials are used to reduce the fractional Advection-dispersion equation to the solution of syste of ODEs which solved by using FDM. The fractional derivative is considered in the Table. The absolute error between the exact solution and the approxiate solution at = 3, = 5 and T =. x uex u at = 3 u u at = 5 ex approx approx e e e e e e e e e e e e e 3.387e e e e 3.98e e 3.575e e e 5 345

7 N. H. Sweila et al. Figure. Coparison between the exact solution and the approxiate solution at T =.5 with =.5; = 3. Figure. Coparison between the exact solution and the approxiate solution at T =.5 with =.5; = 5. Figure 3. The behavior of the approxiate solution at different values of at β =

8 N. H. Sweila et al. Figure 4. The behavior of the approxiate solution at different values of β at =.8. Caputo sense. In this article, special attention is given to studying the convergence analysis and estiating an upper bound of the error for the proposed approxiate forula of the fractional derivative. The solution obtained using the suggested ethod is in excellent agreeent with the already existing ones and shows that this approach can be solved the proble effectively. Fro the resulted nuerical solution, we can conclude that the used techniques in this work can be applied to any other probles. It is evident that the overall errors can be ade saller by adding new ters fro the series (3). Coparisons are ade between the approxiate solution and the exact solution to illustrate the validity and the great potential of the technique. All coputations in this paper are done using Matlab 8. References [] Liu, F., Yang, Q. and Turner, I. () Stability and Convergence of Two New Iplicit Nuerical Methods for the Fractional Cable Equation. Journal of Coputational and Nonlinear Dynaics, 6, Article ID: 9. [] Podlubny, I. (999) Fractional Differential Equations. Acadeic Press, New York. [3] Khader, M.M. () Introducing an Efficient Modification of the Hootopy Perturbation Method by Using Chebyshev Polynoials. Arab Journal of Matheatical Sciences, 8, 6-7. [4] Khader, M.M. and Hendy, A.S. () The Approxiate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudospectral Method. International Journal of Pure and Applied Matheatics, 74, [5] Khader, M.M. () Introducing an Efficient Modification of the Variational Iteration Method by Using Chebyshev Polynoials. Application and Applied Matheatics: An International Journal, 7, [6] Sweila, N.H., Khader, M.M. and Al-Bar, R.F. (8) Hootopy Perturbation Method for Linear and Nonlinear Syste of Fractional Integro-Differential Equations. International Journal of Coputational Matheatics and Nuerical Siulation,, [7] Jafari, H. and Daftardar-Gejji, V. (6) Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by ADM. Applied Matheatics and Coputation, 8, [8] Yu, Q., Liu, F., Anh, V. and Turner, I. (8) Solving Linear and Nonlinear Space-Tie Fractional Reaction-Diffusion Equations by Adoian Decoposition Method. International Journal for Nuerical Methods in Engineering, 74, [9] Hashi, I., Abdulaziz, O. and Moani, S. (9) Hootopy Analysis Method for Fractional IVPs. Counications in Nonlinear Science and Nuerical Siulation, 4, [] Sweila, N.H. and Khader, M.M. () A Chebyshev Pseudo-Spectral Method for Solving Fractional Integro- Differential Equations. ANZIAM, 5, [] Sweila, N.H., Khader, M.M. and Mahdy, A.M.S. () Nuerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Matheatics,, Article ID:

9 N. H. Sweila et al. [] Sith, G.D. (965) Nuerical Solution of Partial Differential Equations. Oxford University Press, Oxford. [3] Sweila, N.H., Khader, M.M. and Nagy, A.M. () Nuerical Solution of Two-Sided Space-Fractional Wave Equation Using Finite Difference Method. Journal of Coputational and Applied Matheatics, 35, [4] Sweila, N.H., Khader, M.M. and Adel, M. () On the Stability Analysis of Weighted Average Finite Difference Methods for Fractional Wave Equation. Fractional Differential Calculus,, 7-9. [5] Yuste, S.B. and Acedo, L. (5) An Explicit Finite Difference Method and a New von Neuann-Type Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Nuerical Analysis, 4, [6] Zhuang, P. and Liu, F. (6) Iplicit Difference Approxiation for the Tie Fractional Diffusion Equation. Journal of Applied Matheatics and Coputing,, [7] Zhuang, P., Liu, F., Anh, V. and Turner, I. (8) New Solution and Analytical Techniques of the Iplicit Nuerical Methods for the Anoalous Sub-Diffusion Equation. SIAM Journal on Nuerical Analysis, 46, [8] Chechkin, A.V., Gonchar, V.Y., Klafter, J., Metzler, R. and Tanatarov, L.V. (4) Lévy Flights in a Steep Potential Well. Journal of Statistical Physics, 5, [9] Langlands, T.A.M., Henry, B.I. and Wearne, S.L. (9) Fractional Cable Equation Models for Anoalous Electrodiffusion in Nerve Cells: Infinite Doain Solutions. Journal of Matheatical Biology, 59, [] Quintana-Murillo, J. and Yuste, S.B. () An Explicit Nuerical Method for the Fractional Cable Equation. International Journal of Differential Equations,, Article ID: 39. [] Rall, W. (977) Core Conductor Theory and Cable Properties of Neurons. In: Poeter, R., Ed., Handbook of Physiology: The Nervous Syste, Vol., Chapter 3, Aerican Physiological Society, Bethesda, [] Sokolov, I.M., Klafter, J. and Bluen, A. () Fractional Kinetics. Physics Today, 55, [3] Snyder, M.A. (966) Chebyshev Methods in Nuerical Approxiation. Prentice-Hall, Inc., Englewood Cliffs. [4] Khader, M.M. () On the Nuerical Solutions for the Fractional Diffusion Equation. Counications in Nonlinear Science and Nuerical Siulation, 6, [5] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. () Efficient Chebyshev Spectral Methods for Solving Multi-Ter Fractional Orders Differential Equations. Applied Matheatical Modelling, 35,

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