The global analysis on the spectral collocation method for time fractional Schrödinger equation

2020 ◽  
Vol 365 ◽  
pp. 124689 ◽  
Author(s):  
Minling Zheng ◽  
Fawang Liu ◽  
Zhengmeng Jin
Keyword(s):  
Schrödinger Equation ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Global Analysis ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

Solving Time Fractional Schrodinger Equation in the Sense of Local Fractional Derivative

2021 ◽  
Author(s):  
Mine Aylin Bayrak ◽  
Ali Demir
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Chebyshev Collocation Method ◽  
Fractional Schrödinger Equation ◽  
Local Fractional Derivative ◽  
Chebyshev Collocation ◽  
Mathematical Problems ◽  
Fractional Schrodinger Equation

Abstract The motivation of the studies solving the mathematical problems including time fractional Schrodinger equation by means of a method which is a combination of Chebyshev collocation method and Residual power series method (RPSM). The time fractional derivative in local fractional derivative sense is discretized with the help of Chebyshev collocation method to reduce time fractional Schrodinger equation into a system including two fractional ordinary differential equations. At this stage applying RPSM produce the truncated solution of the mathematical problem. Given examples illustrated that this method is applicable and compatible for solving mathematical problems with fractional derivative.

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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