Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method

Abstract

We construct the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences

“…The main difference compared to the non-fractional case is that, to evaluate L, the numerical solutions for all the n previous time values t 0 , t 1 , · · · , t n−1 are required, whereas for non-fractional equations, only the solution to the previous value t n−1 is used. The computational cost to obtain the solution at the time t n from the solution at the time t n−1 increases as n, that is, increases as the number of terms in the summation that compares in the second term of the (10). This implies that the computational effort to go from t 0 to t n grows as n 2 .…”

“…Therefore, new methods for obtaining approximate solutions of fractional nonlinear partial and ordinary differential equations have been developed. Recently, several methods have drawn special attention such as Adomian decomposition method [4][5][6], variational iteration method [7,8], homotopy analysis method [9][10][11][12], homotopy perturbation method [13][14][15][16], wavelet methods [17,18], finite difference methods [19,20], finite volume methods [21], finite element methods [22,23] and spectral methods [24]. Recently, a new approach has been developed [25][26][27][28][29] to find exact and numerical solutions of the fractional advection-diffusion-reaction equations involving Riemann-Liouville derivative by means the combination of the Lie symmetries theory and finite difference methods.…”

Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general‐shaped domains

“…(2) has been considered by many researchers in many theoretical papers. For a rather incomplete list, authors of [12] constructed the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method, Wei et al [13,14] considered an implicit fully discrete local discontinuous Galerkin method, Mohebbi et al [15] employed a meshless technique based on collocation and radial basis functions (RBFs), and Garrappa et al [16] applied Krylov projection methods to numerical simulation. In the space-fractional case, Amore et al [17] developed the collocation approach based on sinc functions which discretizes the Schrödinger equation on a uniform grid, Atangana [18] proposed a stable and convergent difference scheme for the space fractional variable-order Schrödinger equation with Caputo variable-order fractional derivative, Wang and Huang [19] studied an energy conservative Crank-Nicolson (CN) difference scheme for nonlinear Riesz space-fractional Schrödinger equations, and Zhao et al [20] combined the compact operator in space discretization so that a linearized difference scheme proposed for a two-dimensional nonlinear space fractional Schrödinger equation.…”

“…We consider the time fractional nonlinear Schrödinger equation as the following form[12,15] i ∂ α u(x, y, t) ∂t α + 1 2 u(x, y, t) + (1 − sin 2 xsin 2 y)u(x, y, t) + |u(x, y, t)| 2 u(x, y, t) = 0, (74) Graphs of absolute errors versus δt ∈ {1/500, 1/200, 1/100, 1/50, 1/20, 1/10}, for real and imaginary parts u(x, y, T ) with N = 121 and T = 1, on [0, 1] 2 for Example 1. [Color figure can be viewed at wileyonlinelibrary.com.…”

Galerkin finite element method for nonlinear fractional Schrödinger equations

“…In recent years, as the generalization of the standard nonlinear Schrödinger equation, there have been growing interests in the analysis and computing for the numerical solutions to nonlinear fractional Schrödinger equations (FSEs). For the time-fractional Schrödinger equations, Mohebbi et al [23] employed a meshless technique based on collocation methods and radial basis functions, Khan et al [16] derived approximating solutions by homotopy analysis methods, and Wei et al [35] gave discrete solution via a rigorous analysis of implicit fully discrete local discontinuous Galerkin method. For the space-fractional Schrödinger equations, some fully or linearly implicit difference methods were introduced and discrete conservation properties were analyzed in [30,31,33].…”