Analytical Solutions of Nonlinear Time-Space Fractional Schrödinger Equation

2017 ◽  
Vol 6 (2) ◽  
pp. 297-302 ◽  
Author(s):  
S. Azimi-Kavar ◽  
M. S. Hashemi
Keyword(s):  
Schrödinger Equation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Fractional Schrödinger Equation ◽  
Time Space ◽  
Fractional Schrodinger Equation

scholarly journals Modified Fractional Variational Iteration Method for Solving the Generalized Time-Space Fractional Schrödinger Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Baojian Hong ◽  
Dianchen Lu
Keyword(s):  
Schrödinger Equation ◽  
Iteration Method ◽  
Schrodinger Equation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Fractional Schrödinger Equation ◽  
Variational Iteration ◽  
Time Space ◽  
Generalized Time ◽  
Fractional Schrodinger Equation

Based on He’s variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations.

The Numerical Computation of the Time Fractional Schrödinger Equation on an Unbounded Domain

2018 ◽  
Vol 18 (1) ◽  
pp. 77-94
Author(s):  
Dan Li ◽  
Jiwei Zhang ◽  
Zhimin Zhang
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Unbounded Domain ◽  
Caputo Fractional Derivative ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Main Idea ◽  
Fractional Schrödinger Equation ◽  
Direct Scheme ◽  
Fractional Schrodinger Equation

AbstractA fast and accurate numerical scheme is presented for the computation of the time fractional Schrödinger equation on an unbounded domain. The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.

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