Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary
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In this paper, we investigate some mathematical and numerical aspects of a one-dimensional nonlinear Schrödinger problem defined in a noncylindrical domain. By a change of variable, we transform the original problem into an equivalent one defined in a cylindrical domain. To obtain the existence and uniqueness of the solution, we apply the Faedo-Galerkin method and results of compactness. The numerical simulation is performed by means of the finite element method in the associated space and the finite difference method in the temporal part, to get an approximate numerical solution. In addition, we will make an analysis of the rate of convergence of the applied methods. Finally, we will show that the results of the numerical simulation are in agreement with the theoretical analysis.
KeywordsNonlinear Schrödinger problem Noncylindrical domain Existence and uniqueness Numerical simulation Newton’s method
Mathematics Subject Classification (2010)MSC 35K55 MSC 65M06 MSC 65M60 MSC 35R37
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The author Rincon, M.A acknowledge the partial support from research fellowship of CNPq, Brazil. We would like to thank Prof. L. A. Medeiros, for the suggestions and collaboration in the development of this work.
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