Numerical Solution of Space-Time Fractional Convection-Diffusion Equations with Variable Coefficients Using Haar Wavelets

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1 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations with Variable Coefficients Using Haar Wavelets Jinxia Wei, Yiming Chen, Baofeng Li 2 and Mingxu Yi Abstract: In this paper, we present a computational method for solving a class of space-time fractional convection-diffusion equations with variable coefficients which is based on the Haar wavelets operational matrix of fractional order differentiation. Haar wavelets method is used because its computation is sample as it converts the original problem into Sylvester equation. Error analysis is given that shows efficiency of the method. Finally, a numerical example shows the implementation and accuracy of the approach. Keywords: Haar wavelets, operational matrix, fractional convection-diffusion equation, Sylvester equation, error analysis, numerical solution. Introduction Fractional differential equations are generalized from classical integer order ones, which are obtained by replacing integer order derivatives by fractional ones. In the last few decades, fractional calculus and fractional differential equations have found a wide area of applications in several different fields [Hilfer (2); He (998); Chen, Yi, Chen, and Yu (22)]. For example, one could mention the problem of anomalous diffusion [EI-Sayed (996); Gafiychuk, Datsun, and Meleshko (28)], the nonlinear oscillation of earthquake can be modeled with fractional derivative [Delbosco and Rodino (996)] and many other [Ryabov and Puzenko (22)] recent developments in the description of anomalous transport by fractional dynamics. Though the concepts and the calculus of fractional derivative are few centuries old, it is realized only recently that these derivatives form an perfect framework for modeling real problems. Owing to the increasing applications, a considerable attention has been given to exact and numerical solutions of frac- College of Sciences, Yanshan University, Qinhuangdao, Hebei, China. 2 Department of Mathematics and Information Science, Tangshan Normal University, Tangshan, Hebei, China

2 482 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 tional differential equations. They have been solved by means of the numerical and analytical methods such as variational iteration method [Odibat (2)], Adomian decomposition method [EI-Sayed (998); EI-Kalla (2)], generalized differential transform method [Odibat and Momani (28); Momani and Odibat (27)], wavelet method [Yi and Chen (22)]. In this paper, our study focuses on the following space-time fractional convectiondiffusion equation with variable coefficients: α u(x,t) t α = b(x) u(x,t) < x <, < t subject to the initial conditions + a(x) β u(x,t) β c(x)u(x,t) + q(x,t) () u(x,) =, x (2) u(,t) = u(,t) =, < t (3) where α u(x,t)/ t α is fractional derivative of Caputo sense, β u(x,t)/ β is fractional derivative Riemann-Liouville sense [Podlubny (999)]. a(x), b(x), c(x), q(x, t) are the known continuous functions, u(x,t) is the unknown function, < α <, < β < 2. Many scholars have studied the time fractional diffusion equations in recent years. Lin and Xu [Lin and Xu (27)] used finite difference method to solve the time fractional diffusion equation. Meerschaert et al.[ Meerschaert, Scheffler, and Tadjeran (26)] applied the finite difference methods to solve two-dimensional fractional dispersion equation. Zhang [Zhang (29)] discussed a practical implicit method to solve a class of initial boundary value space-time fractional convection-diffusion equations with variable coefficients. Recently, the operational matrices of fractional order integration for the Legendre wavelets [Rehman and Khan (2)], Chebyshev wavelets [Li (2)], Haar wavelets [Li and Zhao (2)], CAS wavelets [Saeedi, Moghadam, Mollahasani, and Chuev (2)] and the second kind Chebyshev wavelets [Wang and Fan (22)] have been developed to solve the fractional differential equations. Our purpose is to proposed Haar wavelets operational matrix method to solve a class of space-time fractional convection-diffusion equations with variable coefficients. 2 Definitions of fractional derivatives and integrals In this section, we give some necessary definitions and preliminaries of the fractional calculus theory which will be used in this article [Podlubny (999)].

3 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 483 Definition. Riemann-Liouville definition of fractional differential operator is given by D α t u(t) = { d r u(t) dt, α = r N r d r t u(t ) dt r dt, r < α < r (t T ) α r+ Γ(r α) (4) The Riemann-Liouville fractional integral operator J α of order α is defined as J α u(t) = t (t T ) α u(t )dt, t >, J u(t) = u(t) (5) Γ(α) Definition 2. The Caputo definition of fractional differential operator is given by D α u(t) = { d r u(t) dt r, α = r N; Γ(r α) t u (r) (T ) (t T ) α r+ dt, r < α < r. (6) The Caputo fractional derivatives of order α is also defined as D α u(t) = J r α D r u(t), where D r is the usual integer differential operator of order r. The relation between the Riemann- Liouville operator and Caputo operator is given by the following expressions: D α J α u(t) = u(t) (7) J α D α r u(t) = u(t) u (k) ( + ) tk k!, t > (8) k= 3 Haar wavelets and function approximation For t [,], Haar wavelets functions are defined as follows [Ray (22)]: h (t) = m h i (t) = 2 j/2 k, t < k /2 2 j 2 j 2 m j/2 k /2, t < k 2 j 2 j, otherwise (9) () where i =,,2,...,m, m = 2 p+ and p is a positive integer which is called the maximum level of resolution. j and k represent integer decomposition of the index i, i.e. i = 2 j + k.

4 484 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 For arbitrary function u(x,t) L 2 ([,) [,)), it can be expanded into Haar series by u(x,t) = m- m- i= j= u i j h i (x)h j (t) () where u i j = h i (x), u(x,t),h j (t) are wavelets coefficients, h i (x),h j (x) = h i(x)h j (x)dx. Let H m (x) = [h (x),h (x),...,h m (x)] T, H m (t) = [h (t),h (t),...,h m (t)] T, then Eq.() will be written as u(x,t) = H T m(x) U H m (t). In this paper, we use wavelet collocation method to determine the coefficients u i j. These collocation points are shown in the following: x l =t l = (l /2)/m, l =,2,...,m. (2) Discreting Eq.() by the step (2), we can obtain the matrix form of Eq.() C = H T U H (3) where U = [u i j ] m m and C = [u(x i,t j )] m m. H is called Haar wavelets matrix of order m, i.e. h (t ) h (t ) h (t m ) h (t ) h (t ) h (t m ) H = h m (t ) h m (t ) h m (t m ) From the definition of Haar wavelets functions, we may know easily that H is a orthogonal matrix, then we have U = H C H T (4) 4 Haar wavelets operational matrix of fractional order integration and differentiation The integration of the H m (t)can be approximated by Chen and Hsiao [Chen and Hsiao (997)]: t H m (s)ds = PH m (t) (5) where P is called the Haar wavelets operational matrix of integration.

5 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 485 Now, we are able to derive the Haar wavelets operational matrix of fractional order integration. For this purpose, we may make full use of the definition of Riemann- Liouville fractional integral operator J α which is given by Definition. The Haar wavelets operational matrix of fractional order integration P α will be deduced by P α H m (t) = J α H m (t) where = [J α h (t),j α h (t),...,j α h m (t)] T [ t = (t T ) α h (T )dt, Γ(α) Γ(α) Γ(α) t (t T ) α h m (T )dt = [Ph (t),ph (t),...,ph m (t)] T t α Ph (t) = m Γ(α + ) Ph i (t) = m where λ (t) = ] T t (t T ) α h (T )dt,..., t [,) (6), t < k 2 j 2 j/2 k λ (t), t < k /2 2 j 2 j 2 j/2 k /2 λ 2 (t), t < k 2 j 2 j 2 j/2 k λ 3 (t), t < 2 j ( t k ) α Γ(α + ) 2 j ; λ 2 (t) = Γ(α + ) λ 3 (t) = Γ(α + ) ( t k 2 j ( t k 2 j ) α 2 Γ(α + ) ) α 2 Γ(α + ) ( t k /2 2 j ( t k /2 2 j ) α ; ) α + Γ(α + ) (7) ( t k 2 j ) α. The derived Haar wavelets operational matrix of fractional integration is P α = (P α H) H T. Let D α is the Haar wavelets operational matrix of fractional differentiation. According to the property of fractional calculus D α P α = I, we can obtain the matrix D α by inverting the matrix P α.

6 486 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 For instance, if α =.5,m = 8, we have D /2 = The fractional order differentiation of the function t was selected to verify the correctness of matrix D α. The fractional order differentiation of the function u(t) = t is obtained in the following: D α u(t) = Γ(2) Γ(2 α) t α (8) When α =.5,m = 32, the comparison result for fractional differentiation is shown in Fig Our result Exact solution Figure :.5-order differentiation of the function u(t) = t. 5 Numerical solution of the fractional partial differential equations Consider the space-time fractional convection-diffusion equations with variable coefficients Eq.(). If we approximate the function u(x,t)by using Haar wavelets, we have u(x,t) = H T m(x) U H m (t) (9)

7 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 487 Then we can get α u(x,t) t α u(x,t) β u(x,t) β = α (H T m(x)uh m (t)) t α = (HT m(x)uh m (t)) = Hm(x)U T α (H m (t)) t α = Hm(x)UD T α H m (t) (2) [ ] Hm (x) T = UH m (t) = H T m(x)[d ] T UH m (t) (2) [ ] T β (H T = m(x)uh m (t)) β H m (x) β = β UH m (t) = Hm(x)[D T β ] T UH m (t) The function q(x,t) of Eq.() can be also expressed as (22) q(x,t) = H T m(x) Q H m (t) (23) where Q = [q i j ] m m. Substituting Eq.(9), Eq.(2), Eq.(2), Eq.(22) and Eq.(23) into Eq.(), we have H T (x)ud α H(t) = b(x)h T (x)[d ] T UH(t) + a(x)h T (x)[d β ] T UH(t) c(x)h T (x)uh(t) + q(x,t) (24) Dispersing Eq.(24) by the points (x i,t j ), i =,2,...,m and j =,2,...,m, we can obtain H T UD α H = A H T [D ] T UH + A 2 H T [D β ] T UH A 3 H T UH + H T QH (25) namely { HA H T [D ] T HA 2 H T [D β ] T + HA 3 H T } U +UD α = Q (26) where b(x ) a(x ) b(x ) A =......, A a(x ) 2 =......, b(x m ) a(x m ) c(x ) c(x ) A 3 = c(x m ) Eq.(26) is a Sylvester equation. The Sylvester equation can be solved easily by using Matlab software.

8 488 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 6 Error analysis In this section, we assume that u(x,t) is continuous and bounded on (,) (,), there is M >, x,t (,) (,), u(x,t) Suppose u m (x,t) is the following approximation of u(x,t) u m (x,t) = m m n= l= M (27) u nl h n (x)h l (t) (28) where m = 2 p+, p =,,2,... Then u(x,t) u m (x,t) = n=m l=m u nl h n (x)h l (t) = n=2 p+ l=2 p+ u nl h n (x)h l (t) (29) Theorem 6. Assume u(x,t) L 2 ([,) [,)) and u m (x,t) be defined by Eq.(28), ( then we have u(x,t) u m (x,t) E M 3, where u(x,t) m 3 E = u2 (x,t)dxdt Proof. The orthonormality of the sequence {h i (t)} on [,) implies that { /m, n = n h n (x)h n (x)dx =, n n (3) Then we have ) /2. u(x,t) u m (x,t) 2 E = [u(x,t) u m (x,t)] 2 dxdt ( )( ) = u nl u n l h n (x)h n (x)dx h n (t)h n (t)dt l=2 p+ n =2 p+ l =2 p+ n=2 p+ u 2 nl n=2 p+ l=2 p+ = m 2 (3) where u nl = h n (x), u(x,t),h l (t). According to Eq.(9) and Eq.(), we obtain u(x,t),h l (t) = u(x,t)h l (t)dt = 2 j/2 m ( (k 2 )2 j (k )2 j ) k2 j u(x,t)dt u(x,t)dt (k 2 )2 j (32)

9 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 489 Using mean value theorem of integrals: t,t 2 (k ) 2 j t < (k 2 ) 2 j, (k 2 ) 2 j t 2 < k 2 j such that u(x,t),h l (t) = 2 j/2 {[(k m 2 )2 j (k )2 j ]u(x,t ) [k2 j (k } 2 )2 j ]u(x,t 2 ) = 2 j/2 m (u(x,t ) u(x,t 2 )) (33) hence u nl = = = h n (x), 2 j/2 2 j/2 (u(x,t ) u(x,t 2 )) m h n (x)(u(x,t ) u(x,t 2 ))dx m 2 j/2 = 2m m ( ( (k 2 )2 j (k )2 j k2 j + u(x,t 2 )dx (k 2 )2 j ) h n (x)u(x,t )dx h n (x)u(x,t 2 )dx k2 j u(x,t )dx Using mean value theorem of integrals again: ) (k 2 )2 j u(x,t )dx (k 2 )2 j (k )2 j u(x,t 2 )dx x,x 2,x 3,x 4 (k ) 2 j x,x 3 < (k 2 ) 2 j, (k 2 ) 2 j x 2,x 4 < k 2 j such that u nl = { [(k 2m 2 )2 j (k )2 j ]u(x,t ) [k2 j (k 2 )2 j ]u(x 2,t ) [(k 2 )2 j (k )2 j ]u(x 3,t 2 ) + [k2 j (k } 2 )2 j ]u(x 4,t 2 ) (34) = 2 j+2 m [(u(x,t ) u(x 2,t )) (u(x 3,t 2 ) u(x 4,t 2 ))] therefore u 2 nl = 2 2 j+4 m 2 [(u(x,t ) u(x 2,t )) (u(x 3,t 2 ) u(x 4,t 2 ))] 2 (35)

10 49 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 Using mean value theorem of derivatives: ξ,ξ 2 x ξ < x 2, x 3 ξ 2 < x 4 such that [ u 2 nl = 2 2 j+4 m 2 (x 2 x ) u(ξ,t ) (x 4 x 3 ) u(ξ ] 2,t 2 ) 2 { [ ] 2 2 j+4 m 2 (x 2 x ) 2 u(ξ,t ) 2 [ + (x 4 x 3 ) 2 u(ξ2,t 2 ) } +2(x 2 x )(x 4 x 3 ) u(ξ,t ) u(ξ 2,t 2 ) Putting Eq.(27) and Eq.(36) together, we get ] 2 (36) u 2 nl 4M2 2 4 j+4 m 2 = M2 2 4 j+2 m 2 (37) Substituting Eq.(37) into Eq.(3), then we have u(x,t) u m (x,t) 2 E = m 2 Therefore u(x,t) u m (x,t) E = m 2 m 2 = M2 m 4 u 2 nl n=2 p+ l=2 ( p+ 2 j+ 2 j+ l=2 j u 2 nl ) j=p+ n=2 ( j 2 j+ 2 j+ M2 j=p+ n=2 j l=2 j j=p+ = M2 3m 4 = M2 3 This theorem is complete. 2 2(p+) m 6 ( 2 j+ 2 j+ n=2 j 2 4 j+2 m 2 ) 2 l=2 4 j+2 j ) (38) M 3 m 3 (39) From the Eq.(39), we can see that u(x,t) u m (x,t) E when m. A conclusion is drawn that Haar wavelets method is convergent when it is used to solve the numerical solution of fractional differential equations.

11 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 49 7 A numerical example In this section, to demonstrate the validity and applicability of the approach, we consider the space-time fractional convection-diffusion equation with variable coefficients Eq.(). Let a(x) = Γ(2.8)x/2, [ b(x) = x.8, c(x) = x.5, when α =.8,β =.5, and q(x,t) = Γ(3)x 2 ( x)t.2 Γ(2.2) + 2x.8 3x 2.8 Γ(2.8)Γ(3) 2Γ(.5) x.5 + Γ(2.8)Γ(4) 2Γ(2.5) x x 3.5 x ]t the exact solution is u(x,t) = x 2 ( x)t 2. Figs. 2-5 show the numerical solutions for various m. The absolute error for different m is shown in Table..2.5 u(x,t)..5.5 t.2.4 x.6.8 Figure 2: Numerical solution of m = u(x,t)..5.5 t.2.4 x.6.8 Figure 3: Numerical solution of m = 32

12 492 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , u(x,t)..5.5 t.2.4 x.6.8 Figure 4: Numerical solution of m = u(x,t)..5.5 t.2.4 x.6.8 Figure 5: Exact solution

13 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 493 Table : The absolute error of different m (x,t) m = 8 m = 6 m = 32 m = 64 (,) e e e e-9 (/8,/8) e e e e-7 (2/8,2/8).45388e e e e-6 (3/8,3/8) e e e e-6 (4/8,4/8) e e e e-6 (5/8,5/8) e e e-5.75e-5 (6/8,6/8) e e e e-5 (7/8,7/8) e e e e-5 From the Figs. 2-5 and Table, we can conclude that the numerical solutions are more and more close to the exact solution when m increases. Compared with the finite difference method in Ref. [Zhang (29)], taking advantage of above method can greatly reduce the computation. Moreover, the method in this paper is easy implementation. 8 Conclusion A operational matrix for the Haar wavelets operational matrix of fractional differentiation has been derived. This matrix is used to solve the numerical solutions of a class of space-time fractional convection-diffusion equations with variable coefficients effectively. We transform the fractional partial differential equation into a Sylvester equation which is easily to solve. Numerical example illustrates the powerful of the proposed method. The solutions obtained using the suggested method show that numerical solutions are in very good coincidence with the exact solution. From Theorem 6., we have illustrated the convergence of this algorithm. Acknowledgement: This work is supported by the Natural Foundation of Hebei Province (A222347). References Chen, Y. M.; Yi, M. X.; Chen, C.; Yu, C. X. (22): Bernstein polynomials method for fractional convection-diffusion equation with variable coefficients. Computer Modeling in Engineering & Sciences, vol. 83, no. 6, pp Chen, C. F.; Hsiao, C. H. (997): Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl., vol.44, no., pp

14 494 Copyright 22 Tech Science Press CMES, vol.89, no.6, pp , 22 Delbosco, D.; Rodino, L. (996): Existence and uniqueness for a nonlinear fractional differential equation. Journal of Mathmaticals Analysis and Applications, vol. 24, pp EI-Sayed, A. M. A. (996): Fractional order diffusion wave equation. International Journal of Theoretical Physics, vol. 35, pp EI-Sayed, A. M. A. (998): Nonlinear functional differential equations of arbitrary orders. Nonliear Analysis, vol. 33, pp EI-Kalla, I. L. (2): Error estimate of the series solution to a class of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat., vol. 6, pp Gafiychuk, V.; Datsun, B.; Meleshko, V. (28): Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics, vol. 22, pp Hilfer, R. (2): Application of Fraction Calculus in Physics. World Scientific Publishing Company. He, J. H. (998): Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Method in Applied Mechanics and Engineering, vol. 67, pp Lin, Y. M.; Xu, C. J. (27): Finite difference/spectral approximations for the time- fractional diffusion equation. Journal of Computational Physics, vol. 225, pp Li, Y. L. (2): Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear Sci. Numer. Simul., vol. 5, pp Li, Y. L.; Zhao, W. W. (2): Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematics and Computation, vol. 26, no. 8, pp Momani, S.; Odibat, Z. (27): Generalized differential transform method for solving a space and time-fractional diffusion-wave equation. Physics Letters A, vol. 37, pp Meerschaert, M. M.; Scheffler, H. P.; Tadjeran, C. (26): Finite difference methods for two- dimensional fractional dispersion equation. Journal of Computational Physics, vol. 2, pp Odibat, Z. M. (2): A study on the convergence of variational iteration method. Mathematical and Computer Modelling, vol. 5, pp Odibat, Z.; Momani, S. (28): Generalized differential transform method: Application to differential equations of fractional order. Applied Mathematics and Computation, vol. 97, pp

15 Numerical Solution of Space-Time Fractional Convection-Diffusion Equations 495 Podlubny, I. (999): Fractional Differential Equations, Academic Press, San Diego. Ryabov, Y. ; Puzenko, A. (22): A damped oscillations in view of the fraction oscillator equation. Physics Review B, vol. 66, pp Rehman, M. ; Khan, R. A. (2): The Legendre wavelet method for solving fractional differential equations. Commun. Nonlinear Sci. Numer. Simul., vol. 6. pp Ray, S. S. (22): On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation Applied Mathematics and Computation, vol. 28, pp Saeedi, H.; Moghadam, M. M.; Mollahasani, N.; Chuev, G. N. (2): A CAS wavelet method for solving nonlinear Fredholm integro-differential equation of fractional order. Commun. Nonlinear Sci. Numer. Simulat., vol. 6, pp Wang, Y. X.; Fan, Q. B. (22): The second kind Chebyshev wavelet method for solving fractional differential equations. Applied Mathematics and Computation, vol. 28, pp Yi, M. X.; Chen, Y. M. (22): Haar wavelet operational matrix method for solving fractional partial differential equations. Computer Modeling in Engineering & Sciences, vol. 88, no. 3, pp Zhang, Y. (29): A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, vol. 25, pp

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