Homotopy Analysis Transform Method for Time-fractional Schrödinger Equations

Transcription

1 International Journal of Modern Mathematical Sciences, 2013, 7(1): International Journal of Modern Mathematical Sciences Journal homepage:wwwmodernscientificpresscom/journals/ijmmsaspx ISSN: X Florida, USA Article Homotopy Analysis Transform Method for Time-fractional Schrödinger Equations Fitnat Saba, Saudia Jabeen and Syed Tauseef Mohyud-Din* Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan * Author to whom correspondence should be addressed; syedtauseefs@hotmailcom Article history: Received 22 March 2013, Received in revised form 30 May 2013, Accepted 5 June 2013, Published 8 June 2013 Abstract: Homotopy Analysis Transform Method (HATM) is applied to tackle timefractional Schrödinger equations The proposed HATM is an elegant coupling of Homotopy Analysis Method (HAM) and Laplace transform Numerical results coupled with graphical representation explicitly reveal the complete reliability of the proposed algorithm Keywords: Homotopy Analysis Transform Method (HATM), Homotopy Analysis Method (HAM), Fractional equations, nonlinear problems Mathematics Subject classification: 0230Jr, 0545Yv, 0230IK 1 Introduction This paper witnesses application of a newly developed algorithm namely Homotopy Transform Analysis Method (HATM) which is formulated by combing the traditional Homotopy Analysis Method (HAM) [1-14] and Laplace transformation [15] for solving linear and nonlinear time-fractional Schrödinger equations [16-18] The linear time-fractional Schrödinger equations are of the form: (1) and the nonlinear time- fractional Schrödinger partial differential equations are (2)

2 27 and (3) ( )with a cubic and power law nonlinearities respectively, is a constant and is a complex function The fractional derivatives are considered in the Caputo sense It is observed that the proposed technique (HATM) is fully compatible with the versatility of such problems and obtained results are of very high accuracy 2 Homotopy Analysis Transform Method To illustrate the basic idea of this method, let us consider a general fractional nonlinear nonhomogeneous partial differential equation of the form (4) subject to the initial conditions (5) is the source term, R is the linear differential operator and N represents the general nonlinear differential operator, is the Caputo fractional derivative of function which is defined as (6) denotes the Gamma function The properties of fractional derivative can be found in [17, 18] Laplace transform (denoted throughout this paper by L) of the Caputo operator is an important property will be used in this paper [ ] Now taking the Laplace transform of both sides of eq (4) and (5), we have [ ] [ ] [ ] [ ] (7) Using the property of the Laplace transform, we have [ ] [ ] [ ] (8) ( ) We define the nonlinear operator

3 28 [ ] [ ] [ ] [ ] then the so-called zero-order deformation equation of the Laplace equation (8) has the form [ ] [ ] (9) [ ] is an embedding parameter, ћ is a non-zero auxiliary parameter, is an auxiliary function When and, we have and respectively Thus, as increasing from 0 to 1, varies from to Expanding in Taylor series with respect to q, we have (10) (11) Define the vector { } Differentiating Equation (10) m times with respect to the embedding parameter q, and then setting and and finally dividing them by m!, we have the so-called mthorder deformation equation ( ) (12) and ( ) { [ ] (13) solution, Applying the inverse Laplace transform of both sides of (12), then we have a power series

4 29 In what follows, we will apply the HATM to four physical models to illustrate the strength of the method and to establish exact solutions for these models 3 Application In this section we are applying Homotopy Analysis Transform Method (HATM) to find the exact solution of linear and nonlinear Schrodinger equations Example 1: Consider the following linear time-fractional Schrödinger equation (14) with initial condition (15) Taking the Laplace transform on both sides of Eqs (14-15) [ ] [ ] (16) using the property of the Laplace transform, we have, [ ] (17) so then, ( [ ]) (18) The zero-order deformation equation of the Laplace equation (18) has the form [ ] [ ] [ ] ( [ ]) Now the m th -order deformation equation is ( ) (19) ( ) (20) and { Now from Eq (19) and (20), we have so then,

5 30 Now (21) Now taking the inverse Laplace transform of both sides of Eq (21), we have [ Thus, we have the solution of (22) in series form for, ] (22) [ ] which is the exact solution of the Schrödinger equation (23) (a) (b) (c) (d) Figure 1 The surface shows solution for the Eq (22), when (a), (b), (c), (d), (e) exact solution Eq (14) (e)

6 31 Example 2: Consider the following linear time-fractional Schrödinger equation (24) with initial condition (25) Taking the Laplace transform on both sides of Eq (24-25) [ ] [ ] using the property of the Laplace transform, we have, ( ) [ ] (26) we have, ( [ ]) (27) the zero-order deformation equation of the Laplace equation (27) has the form [ ] [ ] [ ] ( [ ]) Now the m th -order deformation equation is ( ) (28) ( ) ( ) (29) and { Now from Eq (28) and (29), we have ( ) so then, ( ) Now (30)

7 32 now taking the inverse Laplace transform of both sides of eq (30), we have [ Thus, we have the solution of (31) in series form for, ] (31) [ ] (32) which is the exact solution of the Schrödinger equation (a) (b) (c) (d) Figure 2 The surface shows solution for the Eq (32) when (a), (b), (c), (d), (e) exact solution Eq (24) (e) Example 3: Consider the following linear time-fractional Schrödinger equation (33) with initial conditions (34) Taking the Laplace transform on both sides of Eqs (33-34)

8 33 [ ] [ ] [ ] using the property of the Laplace transform, we have [ ] (35) we have [ ] (36) The zero th -order deformation equation of the Laplace equation (36) has the form [ ] [ ] [ ] ( [ ]) Now the m th -order deformation equation is ( ) (37) ( ) ( ) (38) and { Now from Eq (37) and (38), we have ( ) so then, Now (39) Now taking the inverse Laplace transform of both sides of eq (39), we have [ ] (40)

9 34 Thus, we have the solution of (40) in series form for, [ ] which is the exact solution of the Schrödinger equation (41) (a) (b) (c) (d) (e) Figure 3 The surface shows solution for the Eq (39), when (a), (b), (c), (d), (e) exact solution Eq (33) Example 4: Consider the following nonlinear time-fractional Schrödinger equation (42) with initial condition (43) Taking the Laplace transform on both sides of Eq (42-43) [ ] [ ( )]

10 35 using the property of the Laplace transform, we have [ ] (44) we have, [ ] (45) The zero-order deformation equation of the Laplace equation (45) has the form [ ] [ ] [ ] ( [ ]) Now the m th -order deformation equation is ( ) (46) ( ) ( ) (47) and { Now from Eq (46) and (47), we have ( ) So then, Now (48) Now taking the inverse Laplace transform of both sides of eq (48), we have [ ] (49)

11 36 Thus, we have the solution of (49) in series form for, [ ] which is the exact solution of the Schrödinger equation (50) (a) (b) (c) (d) (e) Figure 4 The surface shows solution for the Eq (49), when (a), (b), (c), (d), (e) exact solution Eq (42) Example 5: We finally consider the nonlinear Schrodinger equation (51) with initial condition ( ) (52) Taking the Laplace transform on both sides of Eq (51-52) [ ] [ ( )] using the property of the Laplace transform, we have ( ) [ ] (53)

12 37 we have, ( ) [ ] (54) The zero-order deformation equation of the Laplace equation (54) has the form [ ] [ ] [ ] ( ) ( [ ]) Now the m th -order deformation equation is ( ) (55) ( ) ( ) ( ) (56) and { Now from Eq (55) and (56), we have ( ) ( ) so then, ( ) ( ) ( )

13 38 Now (57) Now taking the inverse Laplace transform of both sides of eq (57), we have ( ) [ ] (58) Thus, we have the solution of (58) in series form for, ( ) [ ] ( ) (59) which is the exact solution of the Schrödinger equation (a) (b) (c) (d) (e) Figure 5 The surface shows solution for the Eq (58), when (a), (b), (c), (d), (e) exact solution Eq (51)

14 39 4 Conclusion Homotopy Analysis Transform Method (HATM) proved to be extremely effective for finding solutions of time-fractional Schrödinger equations Numerical results re-confirm the accuracy of suggested scheme References [1] S Abbasbandy, Homotopy analysis method for generalized Benjamin-Bona-Mahony equation, Z Angew Math Phys, 59(2008): [2] S Abbasbandy and F S Zakaria, Soliton solutions for the fifth-order K-dV equation with the homotopy analysis method, Nonlinear Dyn, 51(2008): [3] S Abbasbandy, Homotopy analysis method for the Kawahara equation Nonlinear Anal (B), 11(2010): [4] S Abbasbandy, E Shivanian and K Vajravelu, Mathematical properties of curve in the framework of the homotopy analysis method, Commun Nonlin Sci Numer Simul 16(2011): [5] S Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys Lett A, 360(2006): [6] S Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota- Satsuma coupled KdV equation, Phys Lett A, 361(2007): [7] S J Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton, Chapman and Hall, 2003 [8] S J Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput 147(2004): [9] S J Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Appl Math Comput 169(2005): [10] Y Tan, S Abbasbandy, Homotopy analysis method for quadratic Riccati differential Equation, Commun Nonlin Sci Numer Simul, 13(2008): [11] T Hayat, M Khan, S Asghar, Homotopy analysis of MHD flows of an Oldroyd 8 constant fluid, Acta Mech, 168(2004): [12] T Hayat, M Khan, Homotopy solutions for a generalized second-grade fluid past a porous Plate, Nonlinear Dyn 42(2005): [13] S T Mohyud-Din and A Yildirim, The numerical solution of three dimensional Helmholtz equation, Chinese Physics Letters, 27(6) (2010):

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