THE EXISTENCE OF THE GLOBAL SOLUTION OF THE SEMI-LINEAR SCHRÖDINGER EQUATION WITH NUMERICAL COMPUTATION ON THE DIRICHLET BOUNDARY

Pius W. M. Chin

doi:10.11948/20240161

2025 Volume 15 Issue 2

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Pius W. M. Chin. THE EXISTENCE OF THE GLOBAL SOLUTION OF THE SEMI-LINEAR SCHRÖDINGER EQUATION WITH NUMERICAL COMPUTATION ON THE DIRICHLET BOUNDARY[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 786-809. doi: 10.11948/20240161

Citation:

Pius W. M. Chin. THE EXISTENCE OF THE GLOBAL SOLUTION OF THE SEMI-LINEAR SCHRÖDINGER EQUATION WITH NUMERICAL COMPUTATION ON THE DIRICHLET BOUNDARY[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 786-809. doi: 10.11948/20240161

THE EXISTENCE OF THE GLOBAL SOLUTION OF THE SEMI-LINEAR SCHRÖDINGER EQUATION WITH NUMERICAL COMPUTATION ON THE DIRICHLET BOUNDARY

Pius W. M. Chin^1, ,

1.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, Ga-rankuwa, Pretoria, South Africa

Corresponding author: Email: pius.chin@smu.ac.za(P. W. M. Chin)

Abstract

Abstract

This paper concerns the analysis of the initial boundary value problem for the semi-linear Schrödinger equation. In the paper, we design a reliable scheme coupling the nonstandard finite difference method in time with the Galerkin combined with the compactness method in the space variables to analyze the problem. The analysis begins by showing that, given initial solutions in specified space, the global solution of the Schrödinger equation exists uniquely. We further show using the a priori estimates obtained from the existence process, that the numerical solution from the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme replicates or preserves the qualitative properties of the exact solution. Numerical experiments are conducted using a carefully chosen example to justify our theoretical proposition.
- Schrödinger equation /
- semi-linear equation /
- nonstandard finite difference method /
- Galerkin method /
- optimal rate of convergence
MSC: 35J47, 35J70, 65N15, 65N30

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References

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