scholarly journals Fractional Schrödinger Equation in the Presence of the Linear Potential

Mathematics ◽  
2016 ◽  
Vol 4 (2) ◽  
pp. 31 ◽  
Author(s):  
André Liemert ◽  
Alwin Kienle
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Linear Potential ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

scholarly journals Fractional schrödinger equation with zero and linear potentials

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Saleh Baqer ◽  
Lyubomir Boyadjiev
Keyword(s):  
Schrödinger Equation ◽  
Potential Field ◽  
Schrodinger Equation ◽  
Fractional Derivatives ◽  
Free Particle ◽  
Linear Potential ◽  
Fractional Schrödinger Equation ◽  
Zero Potential ◽  
Fractional Schrodinger Equation

AbstractThis paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox

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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