NONLINEAR ELLIPTIC EIGENVALUE PROBLEM
The solutions (λ, u(x, y)) of ∇ 2u = -f(u) in the rectangleD(δ) ≡ {(x, y)|0 < x Y < t, 0 < y < 1 + δ}which have u=0 on the boundary of D(δ) are considered in detail for the cases f(u)=sinh u and f(u)=u-u 3. The trivial solut...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
1979
|
| _version_ | 1826278537217179648 |
|---|---|
| author | Budden, P Norbury, J |
| author_facet | Budden, P Norbury, J |
| author_sort | Budden, P |
| collection | OXFORD |
| description | The solutions (λ, u(x, y)) of ∇ 2u = -f(u) in the rectangleD(δ) ≡ {(x, y)|0 < x Y < t, 0 < y < 1 + δ}which have u=0 on the boundary of D(δ) are considered in detail for the cases f(u)=sinh u and f(u)=u-u 3. The trivial solution, where u=0, exists for all λ. Non-trivial solutions bifurcate from the trivial solution only when λ=λ n(δ) (where λ( n, n = 1, 2, are the eigenvalues of the linearized problem in which f(u) is replaced by u). If λ n(0) is not a simple eigenvalue (for exampkle ), λ 2 then, for δ and λ-λ n(δ) both small, secondary bifurcation occurs from the branches of non trivial solutions. Approximations to u(x, y) are found (i) near both the primary and secondary bifurcation points, where E ≡ ∫ D(δ)|∇u 2dx dy is small and (ii) when E ↑ ∞. Finite-difference solutions (computed using Newton's method), are found, and these match smoothly on to the appropriate analytical approximations as E ↑ 0 and as E ↑ ∞. Both the analystical and numerical techniques can be applied to wider classes of functions f(x, y; u) and boundary conditions. © 1979 Academic Press Inc. (London) Ltd. |
| first_indexed | 2024-03-06T23:45:24Z |
| format | Journal article |
| id | oxford-uuid:70baf8b3-f3c6-4a9f-aebc-ccbaf2d2e33b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:45:24Z |
| publishDate | 1979 |
| record_format | dspace |
| spelling | oxford-uuid:70baf8b3-f3c6-4a9f-aebc-ccbaf2d2e33b2022-03-26T19:39:03ZNONLINEAR ELLIPTIC EIGENVALUE PROBLEMJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:70baf8b3-f3c6-4a9f-aebc-ccbaf2d2e33bEnglishSymplectic Elements at Oxford1979Budden, PNorbury, JThe solutions (λ, u(x, y)) of ∇ 2u = -f(u) in the rectangleD(δ) ≡ {(x, y)|0 < x Y < t, 0 < y < 1 + δ}which have u=0 on the boundary of D(δ) are considered in detail for the cases f(u)=sinh u and f(u)=u-u 3. The trivial solution, where u=0, exists for all λ. Non-trivial solutions bifurcate from the trivial solution only when λ=λ n(δ) (where λ( n, n = 1, 2, are the eigenvalues of the linearized problem in which f(u) is replaced by u). If λ n(0) is not a simple eigenvalue (for exampkle ), λ 2 then, for δ and λ-λ n(δ) both small, secondary bifurcation occurs from the branches of non trivial solutions. Approximations to u(x, y) are found (i) near both the primary and secondary bifurcation points, where E ≡ ∫ D(δ)|∇u 2dx dy is small and (ii) when E ↑ ∞. Finite-difference solutions (computed using Newton's method), are found, and these match smoothly on to the appropriate analytical approximations as E ↑ 0 and as E ↑ ∞. Both the analystical and numerical techniques can be applied to wider classes of functions f(x, y; u) and boundary conditions. © 1979 Academic Press Inc. (London) Ltd. |
| spellingShingle | Budden, P Norbury, J NONLINEAR ELLIPTIC EIGENVALUE PROBLEM |
| title | NONLINEAR ELLIPTIC EIGENVALUE PROBLEM |
| title_full | NONLINEAR ELLIPTIC EIGENVALUE PROBLEM |
| title_fullStr | NONLINEAR ELLIPTIC EIGENVALUE PROBLEM |
| title_full_unstemmed | NONLINEAR ELLIPTIC EIGENVALUE PROBLEM |
| title_short | NONLINEAR ELLIPTIC EIGENVALUE PROBLEM |
| title_sort | nonlinear elliptic eigenvalue problem |
| work_keys_str_mv | AT buddenp nonlinearellipticeigenvalueproblem AT norburyj nonlinearellipticeigenvalueproblem |