New Iterative Method for Time-Fractional Schrödinger Equations

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1 ISSN , England, UK World Journal of Modelling and Simulation Vol ) No. 2, pp New Iterative Method for Time-Fractional Schrödinger Equations Ambreen Bibi 1, Abid Kamran 2, Umer Hayat 3, Syed Tauseef Mohyud-Din 4 Department of Mathematics, HITEC University, Taxila Cantt. Pakistan Received March , Revised July , Accepted April ) Abstract. New Iterative Method NIM) is applied to tackle time- fractional Schrödinger equations. The proposed technique is fully compatible with the complexity of these problems and obtained results are highly encouraging. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm. Keywords: new iterative method, fractional Schrödinger partial differential equations, nonlinear problems 1 Introduction Nonlinear partial differential equations [1 12, 14 19, 24, 25 are of extreme importance in applied and engineering sciences. The through study of literature reveals that most of the physical phenomena are nonlinear in nature and hence there is a dire need to fine their appropriate solutions, see [1 25 and the references therein. Recently, scientists have observed that number of real time problems is modeled by fractional nonlinear differential equations [1 11, 14 17, 19, 24, 25 which are very hard to tackle. In 2006, Dafardar-Gejji and Jafari [8 have proposed a new technique for solving linear and nonlinear functional equations, namely the New Iterative Method. The Method has proven useful in solving equations such as algebraic equations, integral equations, ordinary and partial differential equations of integer and fractional order see [5 8 and the references therein. In the similar context, we apply New Iterative Method NIM) to solve time- fractional Schrödinger partial differential equations [15, 17, 25 : or D t ux, t) + ιux, t) = 0; ux, 0) = fx), i 2 = 1, 1) id t ux, t) + ux, t) y ux, t) 2 ux, t) = 0; ux, 0) = fx), ι 2 = 1, 2) where 0 1. The fractional derivatives are considered in the Caputo sense. It is to be highlighted that such equations arise frequently in applied, physical and engineering sciences. The basic motivation of this paper is the extension of a very reliable and efficient technique which is called New Iterative Method NIM) to find approximate solutions of time-fractional Schrödinger partial differential equations. It is observed that the proposed algorithms is fully synchronized with the complexity of fractional differential equations. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm. 2 Definitions Definition 1. A real function fx), x > 0, is said to be in the space C µ, µ R if there exists a real number p> µ), such that fx) = x p f 1 x), where f 1 x) C [0, ), and it is said to be in the space Cµ if f m C µ, µ 1, m N. Corresponding author. address: syedtauseefs@hotmail.com. Published by World Academic Press, World Academic Union

2 90 A. Bibi & A. Kamran & et al.: New Iterative Method for Time-Fractional Definition 2. The Riemann-Liouville fractional integral operator of order 0, of a function f C, 1, is defined as J fx) = 1 x Γ ) 0 x t) 1 ft)dt. > 0, x > 0, J 0 fx) = fx). Properties of the operator j can be found in [1 3, 8 11, 14, 16, we mention only the following For f C, 1,, β 0 and γ > 1: 1. J J β ft) = J +β ft), 2. J J β ft) = J β J ft), 3. J x γ = Γ γ+1) Γ +γ+1) x+γ. Definition 3. The fractional derivative of fx) in the Caputo sense is defined as D fx) = J m D m fx) = 1 x Γ m ) 0 x t)m 1 f m t)dt, for m 1 < m, m Z, x > 0, f C 1 m. Also, we need here two of its basic properties. Lemma 1. if m 1 < m, m N and f C m, 1, then D J fx) = fx), and J D fx) = fx) m 1 f k) 0 + )x k k=0 k!, x > 0. 3 Analysis of New Iterative Method Consider the following general functional equation u x) = f x) + Nu x)), 3) where N is a nonlinear operator from a Banach space B B and f is a known function. x = x 1, x 2, x 3,, x n ). we are looking for a solution u of Eq. 3) having the series form u x) = The nonlinear operator N can b decomposed as ) N u = Nu 0 ) + From Eqs. 4) and 5), Eq. 3) is equivalent to u i x). 4) { i ) i 1 N u j N u j )}. 5) j=1 ) { u i = f + Nu 0 ) + N We define the recurrence relation u 0 = f, u 1 = Nu 0 ), u m+1 = Nu u m ) Nu u m 1 ), m = 1, 2, Then And j=1 j=0 i i 1 u j ) N u j )}. 6) u u m+1 ) = Nu u m ), m = 1, 2, 8) j=0 u i = f + N u i ). 9) The k-term approximate solution of Eq. 3) and Eq. 4) is given by u = u 0 + u u k 1 [4, 18. 7) WJMS for contribution: submit@wjms.org.uk

3 World Journal of Modelling and Simulation, Vol ) No. 2, pp Solving PDEs using NIM Consider the partial differential equation of arbitrary order D t ux, t) = Au, u) + Bx, t), m 1 < m, m N, 10) k u t k x, 0) = h kx), k = 0, 1,, m 1, 11) where A is a nonlinear function of u and u partial derivatives of u with respect to x and t), and B is the source function. In view of 2.4), the initial value problems considered in section 4 are equivalent to the following integro-partial differential equation: ux, t) = m 1 k=0 h k x) tk k! + I t B + I t A = f + Nu), 12) where f = Σ m 1 k=0 h kx) tk k! + I t BIt and Nu) = It A. We get the solution of Eq. 12) by employing the algorithm 7). 4 Numerical examples In this section, we apply New Iterative Method NIM) to solve time- fractional Schrödinger equations. Numerical results are very encouraging. Example 1. We first consider the linear time-fractional Schrödinger equations. D t u + ιu xx = 0, 0 < 1, 13) with initial conditions ux, 0) = 1 + cosh2x). The problem is equivalent to the following integro-partial differential equation: where f = 1 + cosh2x) It iu xx ) and Nu) = 0. In view of algorithm 7), we get u 0 x, t) = 1 + cosh2x), t u 1 x, t) = 4ι cosh2x) u 2 x, t) = 4ι) 2 ι cosh2x) Γ +1), t 2 Γ 3+1), u = 1 + cosh2x) I t iu xx ) = f + Nu), u 3 x, t) = 4ι) 3 ι cosh2x) t ux, t) =1 + cosh2x) 1 4ι cosh2x) Γ + 1) + 4ι) 2 t 2 ι cosh2x) Γ 2 + 1) 4ι)3 ι cosh2x) Γ 3 + 1) + For the special case = 1, we obtain from Eq. 14) ). 14) ux, t) = 1 + cosh2x)e 4ιt, 15) which is the exact solution of the Schrödinger equation [15. The results for the exact solution Eq. 15) and the approximate solution Eq. 14) considering the first four term series solution using New Iterative Method, for = 0.25, 0.50, 0.75 and 1, are shown in Fig. 1. WJMS for subscription: info@wjms.org.uk

4 For the special case 1, we obtain from 4.2), 1 cosh ) which is the exact solution of the Schrödinger equation [17. The results for the exact solution Eq. 4.6) and the approximate solution Eq. 4.7) considering the which is the exact solution of the Schrödinger equation [17. The results for the exact solution Eq. first four term series solution using New Iterative Method, for 0.25, 0.50, , are 4.3) and the approximate solution Eq. 4.2) considering the first four term series solution using shown in Figure New Iterative Method, for 0.25, 0.50, , are shown in Figure A. 1. Bibi & A. Kamran & et al.: New Iterative Method for Time-Fractional Fig. 1. The surface shows solution ux, t) for the Eq. 14) when a) = 0.25, b) = 0.50, c) = 0.75, d) = 1, e) exact solution 15) 5 Example 2. Consider the following linear time-fractional Schrödinger equation Fig. 2. The surface shows solution ux, t) for the Eq. 18) when a) = 0.25, b) = 0.50, c) = 0.75, d) = 1, e) exact solution 17). Figure 2. The surface shows solution, for the Eq. 4.6) when a) 0.25, b) 0.50, c) 0.75, d) 1, e) exact solution Eq.4.5). Example 4.3 Consider the following nonlinear time-fractional Schrödinger equation 2 0, where 0 1, 4.7) with initial conditions, 0, 2, D t u + ιu xx = 0, 0 < 1, 16) with initial conditions ux, 0) = e 3ιx. The given equation is equivalent to the integro-partial differential equation: 7 where f = e 3ix It iu xx ) and Nu) = 0. In view of the algorithm 7), we get u 0 x, t) = e 3ιx, u 1 x, t) = 9ιe 3ιx t u 2 x, t) = 9ι) 2 e 3ιx Γ +1), t2 t3 u = e 3ix I t iu xx ) = f + Nu), u 3 x, t) = 9ι) 3 e 3ιx Γ 3+1), ux, t) = e 1 3ιx t + 9ι Γ + 1) + t 2 9ι)2 Γ 2 + 1) + t 3 ) 9ι)3 Γ 3 + 1) +, 17) For the special case = 1, we obtain the form Eq. 17) ux, t) = e 3ιx+3t), 18) which is the exact solution of the Schrödinger equation [15. The results for the exact solution Eq. 18) and the approximate solution Eq. 19) considering the first four term series solution using New Iterative Method, for = 0.25, 0.50, 0.75 and 1, are shown in Fig. 2. Example 3. Consider the following nonlinear time-fractional Schrödinger equation with initial conditions ux, 0) = e ιx. where f = e ιx + I t iu xx ) and Nu) = I t 2iu 2 ū), ιd t u + u xx + 2 u 2 ū = 0, 0 < 1, 19) u = e ιx + I t iu xx + 2iu 2 ū) = f + N, WJMS for contribution: submit@wjms.org.uk

5 World Journal of Modelling and Simulation, Vol ) No. 2, pp In view of the algorithm 7), we get u 0 x, t) = e ιx, u 1 x, t) = ιe ιx t Γ +1), u 2 x, t) = ι 2 e ιx [ u 3 x, t) = ι 3 e ιx 5 t2 2Γ 1+2) Γ +1)) 2 Γ 3+1), ux, t) = e 1 ιx t + ι Γ + 1) + t 2 [ ι2 Γ 2 + 1) + ι3 5 For the special case = 1, we obtain the form Eq. 20) 2Γ 1 + 2) Γ + 1)) 2 Γ 3 + 1) + ). 20) ux, t) = e ιx+t), 21) which is the exact solution of the Schrödinger equation [15. The results for the exact solution Eq. 20) and the approximate solution Eq. 21) considering the first four term series solution using New Iterative Method, for = 0.25, 0.50, 0.75 and 1, are shown in Fig. 3. Fig. 3. The surface shows solution ux, t) for the Eq. 21) when a) = 0.25, b) = 0.50, c) = 0.75, d) = 1, e) exact solution 20) Figure 3 The surface shows solution, for the Eq.4.9 ) when a) 0.25, b) 0.50, c) 0.75, d) 1, e) exact solution Eq. 4.8). Example 4.4 Consider the following nonlinear time-fractional Schrödinger equation 2 0, where 0 1, 4.10) with initial conditions, 0. 2, Fig. 4. The surface shows solution ux, t) for the Eq. 24) when a) = 0.25, b) = 0.50, c) = 0.75, d) = 1, e) exact solution 23) Example 4. Consider the following nonlinear time-fractional Schrödinger equation with initial conditions ux, 0) = e ιx. 9 where f = e ιx + It iu xx ) and Nu) = It 2iu 2 ū), In view of the algorithm 7), we get u 0 x, t) = e ιx, u 1 x, t) = 3ιe ιx t u 2 x, t) = 3ι) 2 e ιx Γ +1), t2 [ u 3 x, t) = ι) 3 e ιx 18Γ 1+2) 63 Figure 4 The surface shows solution, for the Eq. 4.11) when a) 0.25, b) 0.50, c) 0.75, d) 1, e) exact solution Eq. 4.12). 5. Conclusions: New Iterative Method NIM) has been implemented to find appropriate solutions of time-fractional nonlinear Schrödinger equations. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm. ιd t u + u xx 2 u 2 ū = 0, 0 < 1, 22) u = e ιx + I t iu xx 2iu 2 ū) = f + N, Γ +1)) 2 Γ 3+1), 11 WJMS for subscription: info@wjms.org.uk

6 94 A. Bibi & A. Kamran & et al.: New Iterative Method for Time-Fractional ux, t) = e 1 ιx t + 3ι Γ + 1) + t 2 [ 3ι)2 Γ 2 + 1) ι)3 63 For the special case = 1, we obtain the from Eq. 23) 18Γ 1 + 2) Γ + 1)) 2 Γ 3 + 1) + ). 23) ux, t) = e ιx+3t), 24) which is the exact solution of the Schrödinger equation [15. The results for the exact solution Eq. 24) and the approximate solution Eq. 23) considering the first four term series solution using New Iterative Method, for = 0.25, 0.50, 0.75 and 1, are shown in Fig Conclusion New Iterative Method NIM) has been implemented to find appropriate solutions of time-fractional nonlinear Schrödinger equations. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm. References [1 S. Abbasbandy. Numerical solutions of nonlinear klein-gordon equation by variational iteration method. International Journal for Numerical Methods in Engineering, 2007, 70: [2 M. Abdou, A. Soliman. New applications of variational iteration method. Physica D: Nonlinear Phenomena, 2005, 211: 1 8. [3 M. Abdou, A. Soliman. Variational iteration method for solving burger s and coupled burger s equations. Journal of Computational and Applied Mathematics, 2005, [4 A.Yildirim, H. Kocak. Homotopy perturbation method for solving the space-time fractional advection-dispersion equation. Advances in Water Resources, 2009, 3212): [5 V. Daftardar-Gejji, S. Bhalekar. An iterative method for solving fractional differential equations. Applied Mathematics and Mchanics, [6 V. Daftardar-Gejji, S. Bhalekar. Solving fractional diffusion-wave equations using a new iterative method. Fractional Calculus and Applied Analysis, 2008, 112): [7 V. Daftardar-Gejji, S. Bhalekar. Solving fractional boundary value problems with dirichlet boundary conditions using a new iterative method. Computers & Mathematics with Applications, 2010, [8 V. Daftardar-Gejji, H. Jafari. An iterative method for solving nonlinear functional equations. Journal of Mathematical Analysis and Applications, 2006, [9 J. He. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 1998, 167: [10 J. He. Some applications of nonlinear fractional differential equations and their approximations. Bull Science Technology, 1999, 152): [11 J. He. Some asymptotic methods for strongly nonlinear equation. International Journal of Modern Physics B, 2006, 2010): [12 S. Mohyud-Din. On the conversion of partial differential equations. Zeitschrift für Naturforschung A- A Journal of Physical Sciences, 2010, 65a: [13 S. Mohyud-Din. Variational iteration techniques for boundary value problems. VDM Verlag, [14 S. Momani. An algorithm for solving the fractional convection-diffusion equation with nonlinear source term. Communications in Nonlinear Science and Numerical Simulation, 2007, 12: [15 N. Khan, M. Jamil, A. Ara. Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method. Mathematical Physics, 2012, ): 11. [16 I. Podlubny. Fractional differential equations. Academic Press, New York, [17 P. Rozmej, B. Bandrowski. On fractional schr?dinger equation. Computational Methods in Science and Technology, 2010, 162): [18 S. Mohyud-Din, M. Noor, et al. Solution of singular equations by he s variational iteration method. International Journal of Nonlinear Sciences and Numerical Simulation, 2010, 112): WJMS for contribution: submit@wjms.org.uk

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