An Approximate Analytical Solution of the Nonlinear Schrödinger Equation with Harmonic Oscillator Using Homotopy Perturbation Method and Laplace-Adomian Decomposition Method

Hindawi Advances in Mathematical Physics Volume 2018, Article ID 6765021, 11 pages https://doi.org/10.1155/2018/6765021 Research Article An Approximate Analytical Solution of the Nonlinear Schrödinger Equation with Harmonic Oscillator Using Homotopy Perturbation Method and Laplace-Adomian Decomposition Method Emad K. Jaradat,1 Omar Alomari ,2 Mohammad Abudayah ,2 and Ala’a M. Al-Faqih1 1 2 Department of physics, Mutah University, Jordan German Jordanian University, Amman, Jordan Correspondence should be addressed to Omar Alomari; Received 2 October 2018; Revised 22 November 2018; Accepted 26 November 2018; Published 5 December 2018 Academic Editor: Antonio Scarfone Copyright © 2018 Emad K. Jaradat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Laplace-Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) are both utilized in this research in order to obtain an approximate analytical solution to the nonlinear Schrödinger equation with harmonic oscillator. Accordingly, nonlinear Schrödinger equation in both one and two dimensions is provided to illustrate the effects of harmonic oscillator on the behavior of the wave function. The available literature does not provide an exact solution to the problem presented in this paper. Nevertheless, approximate analytical solutions are provided in this paper using LADM and HPM methods, in addition to comparing and analyzing both solutions. 1. Introduction The Schrödinger equation is often encountered in many branches of science and engineering, including quantum mechanics, nonlinear optics, plasma physics, hydrodynamics, and superconductivity. It is a mathematical partial differential equation used to describe the motion and behavior change of the physical system over time. In classical mechanics, it plays the role of Newton’s law and conservation of energy. In quantum mechanics, we describe systems using wave function. The Schrödinger equation has two “forms”; one is the time-dependent wave equation that describes how the wave function of a particle will evolve in time. The other is the time independent wave equation in which the time dependence has been “removed”; it describes what the allowed energies are of the particle [1, 2]. In recent years, a considerable amount of research focused on finding analytical solution to the Schrödinger equations using various methods, among which are Adomian Decomposition Method [3–8], Elzaki decomposition method [9], Variation Iteration method [10], Nikiforod–Uvarov (NV) method [11], and Homotopy Perturbation Method [3, 4, 12–16]. Additionally, Borhanifar [17] solved the nonlinear Schrödinger and coupled Schrödinger equations with a differential transformation method. Shidfar and Molabahrami [18, 19] investigated the d-dimensional Schrödinger equation with a power-law nonlinearity, Zhenga et al. [20] solved the time-dependent Schrödinger equation using homotopy analysis method (HAM) and the Adomian decomposition technique (ADM), and Amador et al. [21] solved nonlinear Schrödinger equations with variable coefficients using Riccati equations and similarity transformations. Finally, Khan and Wu [22] applied Homotopy perturbation transform method (HPTM) to solve nonlinear equations; HPTM uses the Homotopy Perturbation Method together with the Laplace transformation to solve the nonlinear equations. Also, Hosseini et al. [23–27] investigated various forms of the nonlinear Schrödinger equation (NLSE). This paper is organized in several sections. The HPM method is briefly explained in “Homotopy Perturbation Method”. Then the LADM model is described in the “Laplace-Adomian Decomposition Method”. Then in the 2 Advances in Mathematical Physics “One-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator”, the solution to the One-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator equation in its nonlinear version is provided with a numerical example. Similarly, the solution of the Two-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator is presented in the “Two-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator”. Finally, in the “Conclusion”, we summarize our findings and present our final remarks. Since the exact solution to this problem is not available, we compare our numerical results with the results obtained using Mathematica function NDsolve. From Laplace transform of first derivative and substituting the initial condition, we get 𝑓 (𝑥) 𝑖 (4) + L {𝑅Ψ + 𝑁Ψ} . 𝑠 𝑠 Next step is replacing the wave function by an infinite series of terms to be determined later as per the Adomian Decomposition Method (ADM): L {Ψ (𝑥, 𝑡)} = ∞ Ψ (𝑥, 𝑡) = ∑ Ψ𝑛 (𝑥, 𝑡) , (5) 𝑛=0 and the nonlinear terms are replaced by the series: ∞ 𝑁Ψ = ∑ 𝐴 𝑛 (Ψ0 , Ψ1 , . . . , Ψ𝑛 ) , 2. Laplace-Adomian Decomposition Method The Adomian Decomposition Method (ADM) is a method to solve differential equations by expressing the analytic solution in terms of a series. The method separates the linear and nonlinear parts of a differential equation. The nonlinear part can be expressed in terms of what is called Adomian Polynomials [28–30]. The initial condition and the terms that contain the independent variables will be used as the initial approximation. Then by means of a recurrence relation, it is possible to find the terms of the series that give the approximate solution of the differential equation. The Laplace transform is an integral transform that is powerful and useful technique to solve differential equations, which transforms the original differential equation into an algebraic equation. Below are the definitions of Laplace transform and inverse Laplace transform. Definition 1. Given a function 𝑓(𝑡) defined for all 𝑡 ≥ 0, the Laplace transform of 𝑓 is the function 𝐹 defined by ∞ −𝑠𝑡 𝐹 (𝑠) = L {𝑓 (𝑡)} = ∫ 𝑓 (𝑡) 𝑒 𝑑𝑡, 0 (1) and the inverse Laplace transform is defined as follows. Definition 2. Given a continuous function 𝑓(𝑡), if 𝐹(𝑠) = L{𝑓(𝑡)}, then 𝑓(𝑡) is called the inverse Laplace transform of 𝐹(𝑠) denoted 𝑓(𝑡) = L−1 {𝐹(𝑠)}. The Laplace-Adomian Decomposition Method (LADM) was first introduced by Suheil A. Khuri [31] and has been effectively used to find solutions to general nonlinear equations. The added value of this method utilizes the two methods (Laplace Transform and ADM) to obtain the solution for nonlinear equations. Consider the following equation: 𝐿Ψ − 𝑖𝑅Ψ − 𝑖𝑁Ψ = 0, Ψ (𝑥, 0) = 𝑓 (𝑥) , (2) where 𝐿 = 𝜕/𝜕𝑡 and 𝑅 = 𝜕2 /𝜕𝑥2 , 𝐿 and 𝑅 are Linear operators, and 𝑁 is a nonlinear operator. Laplace-Adomian Decomposition Method consists of applying Laplace transform to both sides of (2) and yields L {𝐿Ψ} − L {𝑖𝑅Ψ} − L {𝑖𝑁Ψ} = 0. (6) 𝑛=0 (3) where 𝐴 𝑛 (Ψ0 , Ψ1 , . . .)’s are the Adomian Polynomials, defined by 󵄨󵄨 ∞ 󵄨󵄨 𝑑𝑛 1 𝐴𝑛 = [ 𝑛 𝑁 ( ∑ 𝜆𝑖 Ψ𝑖 (...truncated)