Abstract
We study the inhomogeneous nonlinear time-fractional Schrödinger equation for linear potential, where the order of fractional time derivative parameter α varies between 0 < alpha < 1. First, we begin from the original Schrödinger equation, and then by the Caputo fractional derivative method in natural units, we introduce the fractional time-derivative Schrödinger equation. Moreover, by applying a finite-difference formula to time discretization and cubic B-splines for the spatial variable, we approximate the inhomogeneous nonlinear time-fractional Schrödinger equation; the simplicity of implementation and less computational cost can be mentioned as the main advantages of this method. In addition, we prove the convergence of the method and compute the order of the mentioned equations by getting an upper bound and using some theorems. Finally, having solved some examples by using the cubic B-splines for the spatial variable, we show the plots of approximate and exact solutions with the noisy data in figures.
Highlights
The most famous equation in quantum mechanics that can explain the behavior of particles in Hilbert spaces is the Schrödinger equation
By applying a finite-difference formula to time discretization and cubic B-splines for the spatial variable, we approximate the inhomogeneous nonlinear time-fractional Schrödinger equation; the simplicity of implementation and less computational cost can be mentioned as the main advantages of this method
This equation is obtained in a different form in quantum physics, for example, in canonical quantization of the quantum mechanics, time evolution of the wave function leads to the Schrödinger equation
Summary
The most famous equation in quantum mechanics that can explain the behavior of particles in Hilbert spaces is the Schrödinger equation. In paper [14], according to Caputo’s fractional derivatives, Planck mass and constant were represented by fractal equations whose dimensions are fractal quantities By this technique, they have shown that the time-dependent fractal Schrödinger differential equation for a particle in the potential field exactly matches the standard form of the equation. In paper [3,4,5,6,7] presented by Laskin, a Schrödinger time-independent fractal equation was considered and applications of this equation such as determining the shape of the Schrödinger wave function and its exact solution, the wave function and the eigenvalues for infinite potential wells, and in particular the values for a linear potential field, were obtained for 0 < α < 1.
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