The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems
We study the Dirichlet boundary value problem u″t+λfut=0,−1<t<1,u−1=u1=0, generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases when fu=σ−uexp−K/1+u and fu=∏i=...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2022-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2022/2941463 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832562375964753920 |
|---|---|
| author | Huizeng Qin Youmin Lu |
| author_facet | Huizeng Qin Youmin Lu |
| author_sort | Huizeng Qin |
| collection | DOAJ |
| description | We study the Dirichlet boundary value problem u″t+λfut=0,−1<t<1,u−1=u1=0, generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases when fu=σ−uexp−K/1+u and fu=∏i=1mai—u, respectively. We prove first that all positive solutions of the first problem are less than or equal to σ, obtain more specific lower and upper bounds for these solutions, and compute a curve in the σK -plane with accuracy up to 10−6, below which the first problem has a unique positive solution and above which it has exactly three positive solutions. For the second problem, we determine its number of positive solutions and find a formula for the value of λ that separates the regions of λ, in which the problem has different numbers of solutions. We also computed the graphs for some special cases of the second problem, and the results are consistent with the existing results. Our code in Mathematica is available upon request. |
| format | Article |
| id | doaj-art-036028902fa9460b845e94fd0a918762 |
| institution | Kabale University |
| issn | 1687-0425 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-036028902fa9460b845e94fd0a9187622025-02-03T01:22:42ZengWileyInternational Journal of Mathematics and Mathematical Sciences1687-04252022-01-01202210.1155/2022/2941463The Bifurcation Curves of a Category of Dirichlet Boundary Value ProblemsHuizeng Qin0Youmin Lu1School of Mathematics and StatisticsDepartment of Mathematical and Digital SciencesWe study the Dirichlet boundary value problem u″t+λfut=0,−1<t<1,u−1=u1=0, generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases when fu=σ−uexp−K/1+u and fu=∏i=1mai—u, respectively. We prove first that all positive solutions of the first problem are less than or equal to σ, obtain more specific lower and upper bounds for these solutions, and compute a curve in the σK -plane with accuracy up to 10−6, below which the first problem has a unique positive solution and above which it has exactly three positive solutions. For the second problem, we determine its number of positive solutions and find a formula for the value of λ that separates the regions of λ, in which the problem has different numbers of solutions. We also computed the graphs for some special cases of the second problem, and the results are consistent with the existing results. Our code in Mathematica is available upon request.http://dx.doi.org/10.1155/2022/2941463 |
| spellingShingle | Huizeng Qin Youmin Lu The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems International Journal of Mathematics and Mathematical Sciences |
| title | The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems |
| title_full | The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems |
| title_fullStr | The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems |
| title_full_unstemmed | The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems |
| title_short | The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems |
| title_sort | bifurcation curves of a category of dirichlet boundary value problems |
| url | http://dx.doi.org/10.1155/2022/2941463 |
| work_keys_str_mv | AT huizengqin thebifurcationcurvesofacategoryofdirichletboundaryvalueproblems AT youminlu thebifurcationcurvesofacategoryofdirichletboundaryvalueproblems AT huizengqin bifurcationcurvesofacategoryofdirichletboundaryvalueproblems AT youminlu bifurcationcurvesofacategoryofdirichletboundaryvalueproblems |