The Matrix Transformation Technique for the Time‎- ‎Space Fractional Linear Schrödinger Equation

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Karamali Gholamreza; Mohammadi Firouzjaei Hadi

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Journal Paper

Paper Information

Journal: Iranian Journal of Mathematical Chemistry Year:2024 | Volume:15 | Issue:3 Page(s): 137-154

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Information Journal Paper

Title

The Matrix Transformation Technique for the Time‎- ‎Space Fractional Linear Schrödinger Equation

Author(s)

Karamali Gholamreza | Mohammadi Firouzjaei Hadi | Issue Writer Certificate

Keywords

Time-space fractional Schrödinger equation

Anomalous diffusion

Matrix transformation technique

Laplace transform

Abstract

‎This paper deals with a Time-space fractional Schrödinger equation with homogeneous Dirichlet boundary conditions‎. ‎A common strategy for discretizing time-fractional operators is finite difference schemes‎. ‎In these methods‎, ‎the time-step size should usually be chosen sufficiently small‎, ‎and subsequently‎, ‎too many iterations are required which may be time-consuming‎.‎To avoid this issue‎, ‎we utilize the Laplace transform method in the present work to discretize time-fractional operators‎. ‎By using the Laplace transform‎, ‎the equation is converted to some time-independent problems‎. ‎To solve these problems‎, ‎matrix transformation and improved Matrix transformation techniques are used to approximate the spatial derivative terms which are defined by the spectral fractional Laplacian operator‎. ‎After solving these stationary equations‎, ‎the numerical inversion of the Laplace transform is used to obtain the solution of the original equation‎. ‎The combination of finite difference schemes and the Laplace transform creates an efficient and easy-to-implement method for Time-space fractional Schrödinger equations‎. ‎Finally‎, ‎some numerical experiments are presented and show the applicability and accuracy of this approach‎.

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