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Numerical solutions of nonlinear systems derived from semilinear elliptic equations

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dc.contributor.authorCruceanu, Stefan-Gicu, author
dc.contributor.authorAllgower, Eugene, advisor
dc.contributor.authorTavener, Simon, advisor
dc.date.accessioned2024-03-13T19:26:10Z
dc.date.available2024-03-13T19:26:10Z
dc.date.issued2007
dc.description.abstractThe existence and the number of solutions for N-dimensional nonlinear boundary value problems has been studied from a theoretical point of view, but there is no general result that states how many solutions such a problem has or even to determine the existence of a solution. Numerical approximation of all solutions (complex and real) of systems of polynomials can be performed using numerical continuation methods. In this thesis, we adapt numerical continuation methods to compute all solutions of finite difference discretizations of boundary value problems in 2-dimensions involving the Laplacian. Using a homotopy deformation, new solutions on finer meshes are obtained from solutions on coarser meshes. The issue that we have to deal with is that the number of the solutions of the complex polynomial systems grows with the number of mesh points of the discretization. Hence, the need of some filters becomes necessary in this process. We remark that in May 2005, E. Allgower, D. Bates, A. Sommese, and C. Wampler used in [1] a similar strategy for finding all the solutions of two-point boundary value problems in 1-dimension with polynomial nonlinearities on the right hand side. Using exclusion algorithms, we were able to handle general nonlinearities. When tracking solutions sets of complex polynomial systems an issue of bifurcation or near bifurcation of paths arises. One remedy for this is to use the gamma-trick introduced by Sommese and Wampler in [2]. In this thesis we show that bifurcations necessarily occur at turning points of paths and we use this fact to numerically handle the bifurcation, when mappings are analytic.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierETDF_Cruceanu_2007_3266385.pdf
dc.identifier.urihttps://hdl.handle.net/10217/237666
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.rights.licensePer the terms of a contractual agreement, all use of this item is limited to the non-commercial use of Colorado State University and its authorized users.
dc.subjectbifurcations
dc.subjectsemilinear elliptic equations
dc.subjectmathematics
dc.titleNumerical solutions of nonlinear systems derived from semilinear elliptic equations
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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