Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

Adel R Hadhoud,Abdulqawi A M Rageh,Taha Radwan

doi:10.3390/fractalfract6030127

Adel R Hadhoud, Abdulqawi A M Rageh + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract6030127

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Abstract

This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The L1−approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms L2 and L∞ are also calculated at different values of N and t to evaluate the performance of the suggested method.

Highlights

The nonlinear Schrödinger equation is one of the most fundamental equations of quantum physics, and can be used to describe many nonlinear phenomena such as fluid dynamics, waves in water, plasma, and self-focusing in laser pulses
The order of derivatives in fractional calculus can be any real number, which distinguishes it from ordinary calculus, where the order of derivatives can only be natural numbers
The conventional Schrödinger equation has been generalized to a fractional order partial differential equation that takes into consideration the Riemann–Liouville, Caputo, and Riesz derivatives instead of the classical Laplacian [4,5,6,7]

Summary

Introduction

The nonlinear Schrödinger equation is one of the most fundamental equations of quantum physics, and can be used to describe many nonlinear phenomena such as fluid dynamics, waves in water, plasma, and self-focusing in laser pulses. Crank–Nicolson difference algorithm for solving the time-space FSEs. Space fractional variable-order Schrödinger equation solved numerically via the Crank-Nicolson scheme by Atangana and Cloot [19]. Yaseen et al [22] discussed the solution of the sub-diffusion equation of fractional order using a cubic trigonometric B-spline method. Bhrawya and Abdelkawy [23] developed the collocation method to solve one-and two-dimensional fractional Schrödinger equations subject to initial-boundary and non-local conditions. Et al in [26] applied cubic B-splines based on the finite-difference formula for solving the TFSEs. the MFVIM is used for finding approximate and exact solutions of the TFSEs by Hong [10]. To obtain a finite element scheme for solving TFSE, the first-order approximation of time fractional Caputo derivative will be discretized utilizing the so-called L1−approximation [3,38]:. Λ Re (t) and Λ Im (t) are the real and imaginary parts of the Λ(t), respectively

Derivation of the Numerical Method

Stability Analysis

Numerical Results

Computed

Conclusions