Search extension method for multiple solutions of a nonlinear problem

“…In [3,4], the authors of this paper proposed a new Search-Extension Method (SEM) independent of the minimax methods above. But, in the one-dimensional case under consideration, these multiple solutions can be simply obtained by nonlinear orthogonal expansion.…”

Section: Introductionmentioning

confidence: 99%

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

Xie

Chen

2004

Acta Mathematicae Applicatae Sinica, English Series

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“…A new search-extension method (SEM) [13] was proposed by the authors of this paper, which is independent of the mountain pass and minimax theorems above. SEM includes two basic parts: search and extension, i.e.…”

Section: Introductionmentioning

confidence: 99%

Analysis of search-extension method for finding multiple solutions of nonlinear problem

Chen

Xie

2008

Sci. China Ser. A-Math.

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For numerical computations of multiple solutions of the nonlinear elliptic problem Δu + f (u) = 0 in Ω, u = 0 on Γ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u ∈ H 1+α , α > 0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.

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“…The most closely related numerical results are done for the Henon equation with a zero Dirichlet boundary condition, see, e.g., [10,11,24,3]. In this paper we try to modify the local minimax method (LMM) developed in [10,11] to solve for multiple positive solutions to (1.1).…”

mentioning

confidence: 99%

“…Comparing to the work done in [10,11,24,3], in addition to the usual difficulties due to high nonlinearity, solution multiplicity and instability in the problem, the current numerical work has three major difficulties to overcome (1) the Henon problem has only one trivial solution (a local minimum) u ≡ 0 with MI(0) = 0. So it is always put in the support L of LMM.…”

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confidence: 99%

On Finding Multiple Solutions to a Singularly Perturbed Neumann Problem

Xie¹,

Yongjun²,

Zhou³

2012

SIAM J. Sci. Comput.

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Abstract. In this paper, in order to numerically solve for multiple positive solutions to a singularly perturbed Neumann boundary value problem in mathematical biology and other applications, a local minimax method is modified with new local mesh refinement and other strategies. Algorithm convergence and other related properties are verified. Motivated by the numerical algorithm and convinced by the numerical results, a Morse index approach is used to identify the Morse index of the root solution u 1 ε = 1 at any perturbation value, its bifurcation points and then the critical perturbation value. Many interesting numerical solutions are computed for the first time and displayed with their contours and mesh profiles to illustrate the theory and method.

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Search extension method for multiple solutions of a nonlinear problem

Cited by 44 publications

References 15 publications

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

Analysis of search-extension method for finding multiple solutions of nonlinear problem

On Finding Multiple Solutions to a Singularly Perturbed Neumann Problem

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