On the Solution of Modified Fractional Diffusion Equations
Transcription
On the Solution of Modified Fractional Diffusion Equations
On the Solution of Modified Fractional Diffusion Equations BADR S. ALKAHTANI Mathematics Department, College of Science, King Saud University, P.O.Box 1142, Riyadh 11989, Saudi Arabia Email: [email protected] ADEM KILICMAN Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia, Email: [email protected] Abstract In this paper, we applied the Homotopy Analysis method (HAM) to obtain the analytical solutions of the general space-time fractional diffusion equation. The explicit solutions of the equations have been presented in the closed form by using initial conditions. Further examples are also discussed to explain this method. MSC 2010: 26A33, 34A08, 34K37, 35R11 Keywords: Homotopy Analysis Method , Analytical solution, Fractional diffusion equation, Jumarie‟s fractional derivative. 1. Introduction In recent years, fractional derivatives are fruitfully applied in many mathematical problems occur in the area of mathematical physics, chemistry, engineering etc. The solutions of fractional differential equations have great importance in the field of science and engineering. Many nonlinear and linear equations are appeared in various fields. There is no analytical technique or method to solve all differential equations. Only approximate solutions can be derived using the linearization or perturbation methods (see [1]-[6]). The Homotopy analysis method provides a powerful approach to find analytical approximate solution to linear and nonlinear problems, and it is powerful tool for scientists, engineers, and applied mathematicians, because it provides immediate analytic and numerical solution for both linear and nonlinear fractional differential equations without linearization or discretization. 2. Jumarie’s Fractional Derivative Recently, a new modified Riemann-Liouville left derivative is proposed by G. Jumarie [78] (see also [9] and [10]). Comparing with the classical Caputo derivative, the definition of the fractional derivative is not required to satisfy higher integer-order derivative than .v Secondly th derivative of a constant is zero. For these merits, Jumarie modified derivative successfully applied in the probability calculus [9], fractional Laplace problem [10]. According to the definition of Jumarie‟s Fractional Derivative, define f : R R , x f ( x) denotes a continuous (but not necessarily differentiable) function and let the partition h 0 in the interval [0, 1]. Through the fractional Riemann Liouville integral 0 I x f ( x) x 1 ( x ) 1 f ( )d , 0 1 0 )1) The Modified Fractional Derivative is defined as 0 D x f ( x) x 1 dn ( x ) n ( f ( ) f (0)) d , n (n ) dx 0 (2) where x [0,1], n 1 n and n 1. G. Jumarie [8] (seel also in [9] and [10]) has defined the fractional difference in following ways ( FW 1) f ( x) (1) k f [ x ( k )h], 0 k (3) where FWf ( x) f ( x h) . Then the fractional derivative is defined as the following limit, ( ) f lim h0 f ( x) . h (4) The proposed modified Riemann–Liouville derivative as shown in Eq. (2) is strictly equivalent to Eq. (4). Further, Jumarie has introduced various properties of the modified Riemann –Liouville derivative as given below (see [9] for further properties). (a) Fractional Leibniz product law 0 Dx (uv) u ( ) v uv ( ) , (5) (b) Fractional Leibniz formulation 0 I x 0 Dx f ( x) f ( x) f (0), 0 1, (6) Therefore, the integration by part can be used during the fractional calculus a I b u ( ) v (uv) / ba a I b uv ( ) , (c) Integration with respect to (d ) . (7) Assume f (x) denotes a continuous R R function, we use the following equality for the integral with respect to (d ) x 1 ( x ) 1 f ( ) d , 0 1 0 I x f ( x) 0 (8) x 1 f ( )(d ) . (1 ) 0 3. Space-Time Fractional Differential Equation If we consider the following modified one-dimensional non-homogenous space-time fractional diffusion equation considered which is a modification of Fractional Diffusion Equations used in [14] u ( x, t ) d ( x) D2 u ( x, t ) q( x, t ), t (9) on a finite domain x L x x R with 0 1 . Note that D 2 is Jumarie‟s Fractional Derivatives of order 2 . It is assume that the diffusion coefficient (or diffusivity) d ( x) 0 and initial condition u( x, t 0) s( x) for x L x x R along with Dirichlet boundary conditions of the form u( x L , t ) 0 and u( x R , t ) bR (t ). Next, we consider two-dimensional space-time fractional diffusion equation u ( x, y, t ) d ( x, y) D2 u ( x, y, t ) e( x, t ) D2 u( x, y, t ) q( x, y, t ), t (10) on finite rectangular domain x L x x H and y L y y H , where fractional orders 0 1 and 0 1 , and the diffusion coefficients d ( x) 0 and e( x, y) 0 . The „forcing‟ function q( x, y, t ) can be used to represent sources and sinks. We will assume that the fractional diffusion equation has a unique and sufficiently smooth solution under the following initial and boundary conditions. Assume the initial condition u( x, y, t 0) f ( x, y) for x L x x H , y L y y H and Dirichlet boundary condition u( x, y, t ) B( x, y, t ) on the boundary (perimeter) of the rectangular region x L x x H , y L y y H , with the additional restriction that B( x L , y, t ) B( x, y L , t ) 0 . The classical dispersion equation in twodimensions is given by and 1 . The values of 0 1 , or 0 1 model a super diffusive process in that coordinate Eq. (10) also uses Jumarie‟s fractional derivatives of order and . 4. Homotopy Analysis Method (HAM) For the following nonlinear differential equation FD u x, y, t 0, (11) where FD is a nonlinear operator for this problem, x,y and t denote an independent variables, u x, y, t is unknown function. To apply Homotopy Analysis Method (HAM), we need to construct the following deformation: 1 q L U x, y, t; q u0 x, y, t q H x, y, t FD U x, y, t ; q , (12) where q 0, 1 is the embedding parameter, 0 is an auxiliary parameter, H x, y, t 0 is an auxiliary function, L is an auxiliary linear operator, u0 x, y, t is an initial guess of u x, y, t and U x, y, t; q is an unknown function of the independent variables x, t and q. Obviously, when q 0 and q 1, it holds U x, y, t;0 u0 x, y, t , U x, y, t;1 u x, y, t , (13) Respectively. Using the parameter q, we expand U x, t; q in Taylor series as follows: U x, y, t; q u0 x, y, t um x, y, t q m , (14) m 1 where m 1 U x, y , t ; q um q 0, m! mq Assume that the auxiliary linear operator, the initial guess, the auxiliary parameter and the auxiliary function H x, t are selected such that the series (12) is convergent at q 1 , then due to (12) we have u x, t u 0 x, t u m x, t . (15) m 1 Let us define the vector u n x, t u0 x, t , u1 x, t ,..., u n x, t , Differentiating equations (10) m times with respect to the embedding parameter q, then setting q 0 and finally dividing them by m!, we have the so-called mth-order deformation equation Lu m x, t m u m1 x, t H x, t Rm u m1 , (16) where 1 Rm um1 m 1! and m 1 FD U x, t ; q m1q q0 , 0 m 1, 1 m 1. m Finally, for the purpose of computation, we will approximate the HAM solution of Eq. (9) by the following truncated series: m 1 m x, t uk x, t . k 0 The Homotopy Analysis Method (HAM) contains the auxiliary parameter . which provides us with a simple way to adjust and control the convergence region of solution series for large value of t ([11, 12], see also [12]) . In this paper, we will solve space-time diffusion equation by Homotopy Analysis Method . The derivatives are understood in the Jumarie‟s Fractional Derivative sense. By the present method, numerical results can be obtained with using a few iterations. Now we will obtain the solution of Space-Time (One and Two Dimensional) Fractional Differential Equations. 5. Numerical Applications In this section, we apply the proposed algorithm of Homotopy Analysis Method (HAM) using Jumarie‟s approach for fractional order diffusion equation: Example 5.1 We consider a one-dimensional fractional diffusion equation for the Eq. (1), as taken [14,16] u ( x, t ) 1.8u ( x, t ) d ( x) q( x, t ), t x1.8 (17) on a finite domain 0 x 1 , with the diffusion coefficient d ( x) (2.2) x 2.8 6 0.183634 x 2.8 , (18) the source/sink function q( x, t ) (1 x)e t x 3 , 0 x 1, (19) with the initial conditions u( x, 0) x 3 , and the boundary conditions u(0, t ) 0, u(1, t ) e t , t 0. (20) According to Eq. (12), the zeroth-order deformation can be given by 1 q LU x, t; q u0 x, t qH x, t u( x, t ) d ( x) We choose the auxiliary linear operator t u ( x, t ) q( x, t ) . 1.8 x 1.8 (21) LUx, t; q D t Ux, t; q , with the property LC 0, where C is an integral constant. We also choose the auxiliary function to be Hx, t 1. Hence, the mth-order deformation can be given by Lum x, t mum 1 x, t H x, t Rm um 1 , where u m 1 ( x, t ) 1.8 u m 1 ( x, t ) Rm u m1 d ( x) q( x, t ). t x1.8 Now the solution of the mth-order deformation equation (14) for m 1 become u m x, t m u m1 x, t L1 Rm u m1 . (22) (23) Consequently, the first few terms of the HAM series solution for 1 are as follows: u0 e t x 3 e t x 4 x 4 , 4( e t 1 t ) x 5 u1 x, t ( e 1) x , 2.2 t u 2 x, t t 4 4(e 1 t ) x 2.2 5 t 1 t)x5 2! , 3.2 2.2 2 80(e t It obvious that the “noise” terms appear between the components u0 and u1 , and it is cancelled. The closed form solution is u( x, t ) et x3. Fig. 5.1 The surface shows the solution u ( x, t ) for Eq (17). . Example 5.2 Now, we consider a two-dimensional fractional diffusion equation for the Eq. (2), considered in [15,16] u ( x, y, t ) 1.8u ( x, y, t ) 1.6u ( x, y, t ) d ( x, y) e ( x , y) q( x, y, t ), t x1.8 y1.6 (24) on a finite rectangular domain 0 x 1 , 0 y 1 , for 0 t Tend with the diffusion coefficients d ( x, y) (2.2) x 2.8 y 6, (25) e( x, y) 2 xy 2.6 (4.6) , (26) and the forcing function q( x, y, t ) (1 2 xy )e t x 3 y 3.6 , (27) with the initial condition u( x, y,0) x 3 y 3.6 , (28) and Dirichlet boundary conditions on the rectangle in the form u( x,0, t ) u(0, y, t ) 0, u( x,1, t ) e t x 3 , (29) u(1, y, t ) e t y 3.6 , (30) and for all t 0. According to Eq. (12), the zeroth-order deformation can be given by 1 q L U x, y, t; q u0 x, y, t u ( x, y, t ) 1.8u ( x, y, t ) 1.6u ( x, y, t ) q H x, y , t d ( x, y ) e ( x , y ) q ( x, y , t ) , 1.8 1.6 t x y (31) We choose the auxiliary linear operator L U x, y, t; q Dt U x, y, t; q , with the property LC 0, where C is an integral constant. We also choose the auxiliary function to be Hx, t 1. Hence, the mth-order deformation can be given by L um x, y, t mum1 x, y, t H x, y, t Rm um1 , where Rm um1 um 1 ( x, y, t ) t d ( x, y) 1.8um 1 ( x, y, t ) x1.8 e( x, t ) 1.6um 1 ( x, y, t ) y1.6 q( x, y, t ) (32) Now the solution of the mth-order deformation Eq. (14) for m 1 become um x, y, t mum1 x, y, t L1 Rm um1 . (33) Consequently, the first few terms of the HAM series solution for 1 are as follows u0 x 3 y 3.6 e t 2 x 4 y 4.6 e t 2 x 4 y 4.6 , 2 4.6 5 5.6 8 t u1 x, t x 4 y 4.6 ( e t 1) x y ( e 1 t ), 2 . 2 3 u 2 x, t 1106 5 5.6 t 9101827 6 6.6 t t2 x y (e 1 t ) x y (e 1 t , 165 272250 2! , It obvious that the “noise” terms appear between the components u0 and u1 , and it is cancelled. The closed form solution is u( x, y, t ) x 3 y 3.6 e t . Fig. 5.2 The surface shows the solution u ( x, t ) for Eq. (24). 6. Conclusions In this paper, the application of Homotopy Analysis Method (HAM) was extended to obtain explicit and numerical solutions of linear and non-homogeneous space-time fraction diffusion equations. The obtained results demonstrate the reliability of the algorithm and its wider applicability to fractional differential equations. The advantage of HAM is the auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. Also it is clear that the solutions agree with the exact solutions appear in [14,16]. 7. Acknowledgements This project was supported by Deanship of Scientific Research, College of Science Research Center, King Saud University. Authors are sincerely thankful to referee for his constructive comments to improve this paper. References [1] S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comp. Simul. 70 (2), (2005), 110-118. [2] S. Momani and R. Ibrahim, Analytical solutions of a fractional oscillator by the decomposition method, Int. J. Pure Appl. Math. 37 (1), (2007), 119-132. [3] S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput. 170 (2), (2005), 1126-1134. [4] S. Das, Solution of Fractional Vibration Equation by the Variational Iteration Method and Modified Decomposition Method, Int. J. Nonl. Sci. Numer. Simul. 9 (2008) 361365. [5] S. Momani and Z. Odibat, A. Alawneh, Variational iteration method for solving the space-fractional and time- fractional KdV equation, Num. Meth. For Part. Diff. Equ. 2007. [6] Z. Odibat, S. Momani and A. Alawneh, Analytic study on time-fractional Schrödinger equations: Exact solutions by GDTM, J. Phy.: Conference Series 96 (2008), 012066. [7] G. Jumarie, Stochastic differential equations with fractional Brownian motion input, Int. J. Syst. Sci. 6 (1993), 1113-1132. [8] G. Jumarie, New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Math. Comput. Model. 44 (2006), 231-254. [9] G.Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett. 22 (2009). [10] G. Jumarie, Laplace transform of fractional order via the Mittag–Leffler function and modified Riemann- Liouville derivative, Appl. Math. Lett. 22 (2009), 1659-1664. [11] S. J. Liao, Beyond perturbation: Introduction to the Homotopy Analysis Method , Chapman and Hall/CRC Press, Boca Raton, 2003. [12] S. J. Liao, on the analytic solution of magneto hydro dynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488 (2003), 189–212. [13] S.J. Liao, Notes on the Homotopy Analysis Method : some definitions and theorems Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 983–997. [14] M. M. Meerschaert, H. Scheffler and C. Tadjeran, Finite difference methods for twodimensional fractional dispersion equation, J.Comput. Phys. 211 (2006), 249–261. [15] C. Tadjeran and M. M. Meerschaert and H. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), 205-213. [16] S. S. Ray, K.S. Chaudhuri and R. K. Bera, Application of modified decomposition method for the analytical solution of space-time diffusion equation, Appl. Math. Comput. 196 (2008), 294–302.