Abstract
In this paper we primarily consider the family of elliptic PDEs Δu + f(u) = 0 on the square region Ω = (0, 1) × (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an "action" functional, we consider Newton's method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functional in general.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 801-820 |
| Number of pages | 20 |
| Journal | International Journal of Bifurcation and Chaos |
| Volume | 11 |
| Issue number | 3 |
| DOIs | |
| State | Published - Mar 2001 |
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ASJC Scopus subject areas
- General
- Applied Mathematics
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Newton's method and Morse index for semilinear elliptic PDES. / Neuberger, John M; Swift, James W.
In: International Journal of Bifurcation and Chaos, Vol. 11, No. 3, 03.2001, p. 801-820.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Newton's method and Morse index for semilinear elliptic PDES
AU - Neuberger, John M
AU - Swift, James W
PY - 2001/3
Y1 - 2001/3
N2 - In this paper we primarily consider the family of elliptic PDEs Δu + f(u) = 0 on the square region Ω = (0, 1) × (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an "action" functional, we consider Newton's method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functional in general.
AB - In this paper we primarily consider the family of elliptic PDEs Δu + f(u) = 0 on the square region Ω = (0, 1) × (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an "action" functional, we consider Newton's method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functional in general.
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UR - http://www.scopus.com/inward/citedby.url?scp=0035606761&partnerID=8YFLogxK
U2 - 10.1142/S0218127401002444
DO - 10.1142/S0218127401002444
M3 - Article
VL - 11
SP - 801
EP - 820
JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
SN - 0218-1274
IS - 3
ER -