scholarly journals Time independent fractional Schrödinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative

2018 ◽  
Vol 59 (2) ◽  
pp. 022111 ◽  
Author(s):  
Tapas Das ◽  
Uttam Ghosh ◽  
Susmita Sarkar ◽  
Shantanu Das
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Schrodinger Equation ◽  
Higher Dimension ◽  
Fractional Schrödinger Equation ◽  
Type Potential ◽  
Fractional Schrodinger Equation

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.

Space-Time Fractional Schrödinger Equation With Composite Time Fractional Derivative

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Johan L.A. Dubbeldam ◽  
Zivorad Tomovski ◽  
Trifce Sandev
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Schrodinger Equation ◽  
Time Derivative ◽  
Adomian Decomposition Method ◽  
Space Time ◽  
Infinite Domain ◽  
Fractional Schrödinger Equation ◽  
Adomian Decomposition ◽  
Fractional Schrodinger Equation

AbstractThe fractional Schrödinger equation has recently received substantial attention. We generalize the fractional Schrödinger equation to the Hilfer time derivative and the Caputo space derivative, and solve this equation for an infinite potential by using the Adomian decomposition method. The infinite domain solution of the space-time fractional Schrödinger equation in the case of Riesz space fractional derivative is obtained in terms of the Fox H-functions. We interpret our results for the fractional Schrödinger equation by introducing a complex effective potential in the standard Schrödinger equation, which can be used to describe quantum transport in quantum dots.

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.

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