Abstract
The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established. Error estimates for the approximate eigenvalue and the corresponding approximate positive eigenfunction are proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.
Highlights
We study the nonlinear eigenvalue problem of finding the minimal eigenvalue λ ∈ Λ, Λ= [0, ∞), corresponding to a positive eigenfunction u(x), x ∈ Ω, Ω =(0,π ), Ω =[0,π ], satisfying the following equations
We assume that p(μ), r(μ), μ ∈ Λ, and s(x), x ∈ Ω, are smooth positive functions
A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established
Summary
We study the nonlinear eigenvalue problem of finding the minimal eigenvalue λ ∈ Λ, Λ= [0, ∞), corresponding to a positive eigenfunction u(x), x ∈ Ω, Ω =(0,π ), Ω =[0,π ], satisfying the following equations. A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established. Nonlinear eigenvalue problems of the form (1), (2) arise in modeling the plasma of radio-frequency discharge at reduced pressures. Numerical methods for solving matrix eigenvalue problems with nonlinear dependence on the parameter were constructed and investigated in the papers [5-13]. Mesh methods for solving differential nonlinear eigenvalue problems were studied in [14-16]. In the papers [2430], numerical methods for solving applied nonlinear boundary value problems and variational inequalities have been studied
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