Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational)
We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, \(f(u)=u^{3}\), the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of t...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
2002-02-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Subjects: | |
| Online Access: | https://www.ictp.acad.ro/jnaat/journal/article/view/709 |
| _version_ | 1818521083172618240 |
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| author | Călin-Ioan Gheorghiu Damian Trif |
| author_facet | Călin-Ioan Gheorghiu Damian Trif |
| author_sort | Călin-Ioan Gheorghiu |
| collection | DOAJ |
| description | We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, \(f(u)=u^{3}\), the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct variational methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method. |
| first_indexed | 2024-12-11T01:46:15Z |
| format | Article |
| id | doaj.art-c5a8d1b2d3514c5c89ace9fc13b6f8db |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-12-11T01:46:15Z |
| publishDate | 2002-02-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-c5a8d1b2d3514c5c89ace9fc13b6f8db2022-12-22T01:24:53ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2002-02-01311Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational)Călin-Ioan Gheorghiu0Damian Trif1Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy“Babes-Bolyai” University, Cluj-Napoca, RomaniaWe consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, \(f(u)=u^{3}\), the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct variational methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method.https://www.ictp.acad.ro/jnaat/journal/article/view/709nonlinear reaction-diffusionpositive solutionconserved integralbifurcationvariational formulationLagrange multiplier |
| spellingShingle | Călin-Ioan Gheorghiu Damian Trif Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational) Journal of Numerical Analysis and Approximation Theory nonlinear reaction-diffusion positive solution conserved integral bifurcation variational formulation Lagrange multiplier |
| title | Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational) |
| title_full | Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational) |
| title_fullStr | Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational) |
| title_full_unstemmed | Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational) |
| title_short | Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational) |
| title_sort | direct and indirect approximations to positive solution for a nonlinear reaction diffusion problem i direct variational |
| topic | nonlinear reaction-diffusion positive solution conserved integral bifurcation variational formulation Lagrange multiplier |
| url | https://www.ictp.acad.ro/jnaat/journal/article/view/709 |
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