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Spline Method for Solving the Linear Time Fractional Diffusion Equation Talaat S. El-Danaf Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El-Koom, Egypt. Email: [email protected] Abstract In this paper, we are concerned with the problem of applying cubic parametric spline functions to develop a numerical method for obtaining approximation for the solution for the linear time fractional diffusion equation. The special parametric spline used in this paper is in fact trigonometric-polynomial splines. The truncation error of the method is theoretically analyzed. Using Von Neumann method, the proposed method is shown to be conditionally stable. A numerical example is included to illustrate the practical implementation of the proposed method. Keywords: Parametric spline; Time fractional diffusion; Von Neumann stability. 1 Introduction In this article, we propose a parametric spline based method to obtain numerical solutions for the time fractional diffusion equation of the form u 2u (1) g ( x, t ), a x b, 0 1, t 0 , t x 2 subject to the conditions (2) u(a, t ) 1 t , u(b, t ) 2 t , t 0 . and ux, t 0 f x , a x b. (3) The spline functions proposed have the form [2,6,7], T3 span1, x, sin x, cos x where is the frequency of the trigonometric part of the spline functions which will be used to raise the accuracy of the method. Numerical solution for the linear time fractional diffusion equation of the form u 2u (4) 0 0 x 1, t 0 t x 2 subject to the conditions and u (0, t ) 0, u (1, t ) 0, ux,0 x. (5) . (6) based on implicit finite difference method [3], have been proposed. This paper is organized as follows: In section 2, a new method depends on the use of the parametric spline is derived. In section 3 the stability analysis is theoretically discussed. Using Von Neumann method, the proposed method is shown to be conditionally stable. Finally, in section 4 one numerical example is included to illustrate the practical implementation of the proposed method. ________________________________________________________________________________ Corresponding Author: Talaat S. El- Danaf: talaat11@ yahoo.com 2 Numerical Method We set up a grid in the x , t plane with grid spacing h and k and grid points xi , t j , where xi a ih, for each i 0,1,..., N 1, and t j jk , for each j 0,1,.... j Let Z i be an approximation to u ( xi , t j ), obtained by the segment Pi ( x, t j ) of the mixed spline function passing through the points ( xi , Z i j ) and ( xi 1 , Z i j 1 ) . Each segment has the form [2,6,7] Pi ( x, t j ) ai (t j ) cos ( x xi ) bi (t j ) sin ( x xi ) ci (t j ) ( x xi ) di (t j ) , (7) for each i 0,1,, N . To obtain expressions for the coefficients of Eq. (7) in terms of Z i , Z i j1 , j S i , and S i j 1 , we first define j Pi ( xi , t j ) Z i j , Pi ( xi 1 , t j ) Z i j 1 , Pi ( 2) ( xi , t j ) Si j , and Pi ( 2) ( xi 1 , t j ) Si j 1 By using Eqs. (7) and (8), we have ai d i Z i j , ai cos bi sin ci h d i Z i j 1 , (8) (9) ai S i , 2 j ai 2 cos bi 2 sin S i j 1 , ai ai (t j ), bi bi (t j ), ci ci (t j ), d i d i (t j ) , and h. By solving the last four where equations, we obtain the following expressions: h 2 cos S i j S i j 1 Z i j 1 Z i j h S i j 1 S i j h2 j h2 , (10) , c d Si Z ij . ai 2 S i j , bi i i 2 2 2 h sin 2.1 Spline Relations Using the continuity condition of the first derivative at x xi , that is Pi (1) ( xi , t j ) Pi (11) ( xi , t j ) , we obtain bi ci ai 1 sin bi 1 cos ci 1 . (11) Using expressions in Eq. (10), Eq. (11) becomes h 2 cos S i j S i j 1 Z i j 1 Z i j hS i j 1 S i j h 2 sin 2 Z i j Z i j 1 hS i j S i j 1 h 2 cos S i j 1 S i j h 2 j S sin cos i 1 h 2 2 sin 2 After slight rearrangements, the last equation becomes Z i j 1 2Z i j Z i j 1 S i j 1 S i j S i j 1 , i 1,2,, N . (12) h2 h2 2h 2 cos 2h 2 , 2 , and h . where sin 2 sin Remark 1. 1- The truncation error for Eq. (12), that is T * i u ij1 u ij1 2u ij D x2 u ij1 D x2 u ij1 D x2 u ij j can be obtained by expanding this equation in Taylor series in terms of derivatives as follows 2 u ( xi , t j ) and its h2 h2 j T * i h 2 2 Dx2 uij h 2 Dx4 uij h 4 Dx6 uij . 12 360 12 From this expression of the local truncation error, for 2 h 2 our scheme is of Oh 2 , but h2 our scheme is of O h 4 . 12 h 2 4h 2 , 2 h 2 , and system (12) reduces to 2- As 0, that is 0, then , , 6 6 ordinary cubic spline, that is h2 Z i j 1 2Z i j Z i j 1 ( S i j 1 4S i j S i j 1 ), i 1,2,, N . 6 Using Eq. (1) , we can write S i j in the form for 2 h 2 and 2 Z ij Z ij j g i t x 2 Using the Caputo partial fractional derivative[3,5], we have Sij Z xi , t j (13) t j Z x , s 1 i t j s ds, t j jk , 0 1. 1 0 t t Using a piecewise technique, Eq. (14) becomes j 1 Z xi , t j ( q 1) k Z x , s 1 i t j s ds, 0 1. qk 1 q 0 t t (14) (15) Since t j s does not change sign on qk , q 1k , the Weighted Mean Value Theorem for Integrals[1], can be applied to each integration in the last summation as follows[3,4], q 1k Z xi , s Z xi , s * q 1k t s ds t s ds, qk s * q 1k . j j qk qk t t This implies that ( q 1) k qk Z xi , s Z iq 1 Z iq t j s ds t k ( q 1) k qk t s ds j 1 1 Z iq 1 Z iq t j qk t j qk k k 1 Z iq 1 Z iq jk qk 1 jk qk k 1 k 1 1 1 1 Z iq 1 Z iq j q j q 1 . k 1 then the discrete approximation for the partial fractional derivative (15) can be written in the form j 1 Z xi , t j (16) j ,q Z iq 1 Z iq , 0 1. t q 0 1 where j , q j q 1 j q 11 , and . Formula (16) allows us to express S i j in 1 1 k the form 3 j 1 S i j j ,q Z iq 1 Z iq g ij (17) q 0 which gives us the following useful formulas Si1 Z i1 Z i0 g i0 , 1,0 1, j 1. and j 2 (18) S i j Z i j Z i j 1 j ,q Z iq 1 Z iq g ij , j , j 1 1, j 2 (19) q 0 The use of Eqs. (18) and (19) in Eq. (12) respectively gives us the following systems 1 Z i11 2 Z i1 1 Z i11 Z i01 Z i0 Z i01 i1 , j 1, i 1,2,, N . 1 where i g i11 g i1 g i11 , j 1, i 1,2,, N . and Ai Z i j 1 Bi Z i j Ai Z i j 1 Ai* Z i j 11 Bi* Z i j 1 Ai* Z i j 11 i j , (20) (21) i 1,2,..., N and j 2 where Ai , Ai 1 , * Bi , Bi 2 , * and i j ,q Z j 2 j q 0 q 1 i 1 Z Z j 2 q i 1 q 0 j ,q q 1 i Z Z j 2 q i q 0 j ,q q 1 i 1 Z iq1 i j , j 2 . or j 2 j 2 q 0 q 0 i j j ,q Z iq11 Z iq 1 Z iq11 j ,q Z iq1 Z iq Z iq1 i j , j 2 (22) System (21) consists of N equations in the unknowns Z i , i 0,..., N 1. To get a solution to this system we need 2-additional equations. These equations are obtained from conditions in (2). Remark 2. For , 0, h 2 , System (21) reduces to the finite difference method[3], that is 1 j 2 1 Z i 1 2 Z i j 2 Z i j 1 Z i j 1 i j , 2 h h h j 1, i 1,2,, N . i1 g i1 , and i j j ,q Z iq 1 Z iq g ij , j 2. j 2 where q 0 3 Stability Analysis The Von Neumann technique will be carried out to investigate the stability of systems (20) and (21). The key part of Von Neumann analysis is to assume a solution of the form (23) Z i j j exp Iih , I 1. Inserting the latter expression for Z i j in Eq. (20), we obtain the characteristic equations in the form 2 1 exp Iih 1 1 exp I i 1h exp I i 1h 0 exp Iih 0 exp I i 1h exp I i 1h , 4 (24) After simple calculations, Eq. (24) becomes 1 0 2 cos , h . The quantity 2 cos is surely 21 cos 2 cos positive if we choose 0, and 0 such that 2 but 1 cos is positive or equal to (25) where zero. Then we obtain 1 0 , and 0 1. which implies 1 0 (26) Substituting Eq.(23) into Eqs.(21) and (22) gives us the characteristic equation Bi j exp Iih Ai j exp I i 1h exp I i 1h Bi* j 1 exp Iih Ai* j 1 exp I i 1h exp I i 1h i j , j2 which can be simplified as 21 cos 2 cos j 2 cos j 1 1 i j exp Iih (27) and j 2 i j j ,q q 1 exp Iih exp I exp I q 0 j 2 j ,q q exp Iih exp I exp I , j 2 (28) q 0 which simplifies to j 2 1 i j 2 cos j ,q q 1 q , j 2 exp Iih q 0 Using Eqs. (29) and (27), we obtain (29) j 2 j j 1 j ,q q 1 q , j 2 (30) q 0 For j 2 , we obtain 2 1 2,0 1 0 2 {(1 2,0 ) 1 2,0 0 } The quantities , 1 2,0 ,and 2,0 are positive. Then 2 1 2,0 1 2,0 0 Inequality (26) implies 2 1 2,0 1 2,0 0 1 2,0 0 2,0 0 0 This gives 2 0 where 0 1 for 0, and 0 such that 2 . For j 3 , Eq. (30) implies 3 2 3,0 1 0 3,1 2 1 3 {1 3,1 2 3,1 3,0 1 3,0 0 } then 3 1 3,1 2 3,1 3,0 1 3,0 0 5 (31) Since , 1 3,1 , 3,1 3,0 ,and 3, 0 are positive. Using inequalities (26) and (31), we obtain 3 1 3,1 2 3,1 3,0 1 3,0 0 1 3,1 0 3,1 3, 0 0 3, 0 0 0 (32) 3 0 where 0 1 for 0, and 0 such that 2 . By the same method we can prove that j 0 Z i0 f i , j 1 and we have stability for 0, and 0 such that 2 . 4 Numerical Illustrations We now obtain the numerical solutions of the linear time fractional diffusion equation (1). Example Consider the linear time fractional diffusion equation [3] u 2u 0, 0 1, 0 x 1. t x 2 with boundary conditions u (0, t ) u (1, t ) 0, t 0, and initial conditions ux,0 x1 x, 0 x 1. (33) (34) (35) The solution will be approximated using the proposed method, with k 1 / 256, h 0.01, h 2 / 12 and h 2 2 . The results are presented in Tables 1 – 2. Table 1 : The numerical approximation by using the proposed method at t 0.5, h 2 / 12, and h 2 2 Numerical Solution Numerical Solution x 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.5 0.75 0.0122731 0.0168020 0.0196778 0.0206627 0.0196778 0.0168020 0.0122731 0.00860642 0.01179210 0.01381840 0.01451310 0.01381840 0.01179210 0.00860642 Table 2 : The numerical approximation by using the proposed method at t 1.5, h 2 / 12, and h 2 2 Numerical Solution Numerical Solution x 0.75 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00711150 0.00973602 0.01140260 0.01197350 0.01140260 0.00973602 0.00711150 6 0.00340607 0.00466484 0.00546479 0.00573890 0.00546479 0.00466484 0.00340607 h2 12 2 and h 2 at some different times t = 0.5 and 1.5 respectively. In Each figure the top curve is at 0.25 , the middle curve is at 0.5 and the bottom curve is at 0.75 . Figs.1 and 2 illustrate the behavior of the numerical solution for k 1 / 256, h 0.01, Fig. 2 Fig. 1 Z x, 0.5 . Z x,1.5 0.020 . 0.015 0.015 0.010 0.010 0.005 0.005 x values 0.2 0.4 0.6 0.8 x values 1.0 0.2 Figs.1: The behavior of the numerical solution for k 1 / 256, h 0.01, where t = 0.5 0.4 0.6 0.8 1.0 Figs.2: The behavior of the numerical solution 2 2 for k 1 / 256, h 0.01, h and h 2 2 12 where t = 1.5 and 0.25, 0.5 and 0.75 h and h 2 2 12 and 0.25, 0.5 and 0.75 The figures show that as -increases the curve of the numerical solution decays. 5 Conclusion In this paper, we considered a numerical treatment for the linear time fractional diffusion equation.. The method is shown to be conditionally stable using Von-Neumann method. References [1] Burden R.L. and Faires J. D, ”Numerical Analysis(eighth edition), Thomson Brooks/Cole” 2005. [2] El Danaf T. S. and Abd Alaal F.E.I. ”Non-polynomial spline method for the solution of the dissipative wave equation”, International Journal of Numerical methods for Heat and Fluid Flow, 19,950-959,(2009). [3] Diego A. Murio ”Implicit finite difference approximation for time fractional diffusion equations”, Computers and Mathematics with applications 56,1138-1145, 2008. [4] Jerome S. and Keith B. Oldham ” The fractional Calculus” Academic Press, Inc. 1974. 7 [5] Podlubny I,. ” Fractional differential equations” Academic Press, 1999. [6] Ramadan M. A., El Danaf T. S. and Abd Alaal F.E.I, ” Application of non-polynomial spline approach to the solution of Burgers’ equation”, The Open Applied Mathematics Journal, Vol.1,15-20,2007. [7] Rashidinia, J. and Mohammadi R, ” Non-polynomial cubic spline methods for the solution of parabolic equations” International Journal of Computer Mathematics , 85: 5, 843-850, 2008. 8