# American Institute of Mathematical Sciences

## Solving the time-fractional Schrodinger equation with the group preserving scheme

 1 Department of Mathematics, University of Bonab, Bonab, Iran 2 Department of Mathematics, Firat University, Science Faculty, Elazig, 23119, Turkey 3 Department of Computer Engineering, Biruni University, Istanbul, Turkey 4 Department of Mathematics, Cankaya University, Ankara, Turkey

* Corresponding author: Mustafa Bayram

Received  May 2019 Revised  June 2019 Published  November 2019

In this paper a powerful numerical scheme is proposed to gain the numerical solutions of the time-fractional Schrodinger equation: $i^{C}\mathcal{D}_{0^+, t}^{\alpha}w(x, t) + \vartheta\frac{\partial^2 w(x, t)}{\partial x^2}+ \delta \vert w(x, t)\vert^2 w(x, t)+P(x, t) w(x, t) = \mathcal{F}(x, t), 0<\alpha \leq 1$, $\vartheta$ and $\delta$ are real constants, $p(x, t)$ is trapping potential. The time fractional $^{C}\mathcal{D}_{0^+, t}^{\alpha} w$ is defined in Caputo definition. By using a fictitious $\iota$ we can convert the variable $w(x, t)$ into a new variable by: $(1+\iota)^k w(x, t) = \Xi(x, t, \iota)$, where $0< k \leq 1$. In new space with a semi-discretization of $w(x, t, \iota)$ the group-preserving scheme is used to solve the problem.

Citation: Mohammad Parto-haghighi, Mustafa Inc, Mustafa Bayram, Dumitru Baleanu. Solving the time-fractional Schrodinger equation with the group preserving scheme. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020146
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##### References:
Plot of exact and approximate solutions of real part for example 1
Plot of exact and approximate solutions of imaginary part for example 1
Contour plot of error of real part for example 1
Contour plot of error of imaginary part for example 1
Plot of exact and approximate solutions of real part for example 2
Plot of exact and approximate solutions of imaginary part for example 2
Contour plot of error of real part for example 2
Contour plot of error of imaginary part for example 2
Plot of exact and approximate solutions of real part for example 3
Plot of exact and approximate solutions of imaginary part for example 3
Contour plot of error of real part for example 3
Contour plot of error of imaginary part for example 3
Plot of exact and approximate solutions of real part for example 4
Plot of exact and approximate solutions of imaginary part for example 4
Contour plot of error of real part for example 4
Contour plot of error of imaginary part for example 4
Plot of exact and approximate solutions of real part for example 5
Plot of exact and approximate solutions of imaginary part for example 5
Contour plot of error of real part for example 5
Contour plot of error of imaginary part for example 5
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