A Sinc Method for an Eigenvalue Problem of a Differential Operator with Periodic Coefficients and Its Comparison with Hill's Method

2013 ◽  
Author(s):  
Kenichiro Tanaka
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Periodic Coefficients ◽  
Sinc Method ◽  
Hill’S Method

scholarly journals Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3005
Author(s):  
Roberto De Marchis ◽  
Arsen Palestini ◽  
Stefano Patrì
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Second Order ◽  
Rotational Invariance ◽  
Polar Coordinates ◽  
Elliptic Pde ◽  
Elliptic Differential Operator ◽  
Ladder Operators ◽  
Accidental Degeneracy

We consider the linear, second-order elliptic, Schrödinger-type differential operator L:=−12∇2+r22. Because of its rotational invariance, that is it does not change under SO(3) transformations, the eigenvalue problem −12∇2+r22f(x,y,z)=λf(x,y,z) can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called accidental degeneracy of L occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems.

scholarly journals Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Calogero Vetro
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Admissible Range ◽  
Indefinite Potential ◽  
Multiplicity Of Positive Solutions

AbstractWe consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation $$f(z,\cdot )$$ f ( z , · ) is $$(p-1)$$ ( p - 1 ) -sublinear and then the case where it is $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter $$\lambda \in {\mathbb {R}}$$ λ ∈ R which we specify exactly in terms of principal eigenvalue of the differential operator.

scholarly journals Numerical Analysis of the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Conditions

2009 ◽  
Vol 14 (1) ◽  
pp. 115-122 ◽  
Author(s):  
S. Sajavičius ◽  
M. Sapagovas
Keyword(s):  
Numerical Analysis ◽  
Differential Operator ◽  
Eigenvalue Problem ◽  
General Problem ◽  
Nonlocal Conditions ◽  
Spectral Structure ◽  
Integral Conditions ◽  
One Dimensional ◽  
Special Cases ◽  
Nonlocal Integral Conditions

In this paper the eigenvalue problem for one-dimensional differential operator with nonlocal integral conditions is investigated numerically. The special cases of general problem are analyzed and hypothesis about the dependence of the spectral structure of that problem on the coefficient of differential operator and the parameters of nonlocal conditions are formulated.

scholarly journals On a Parametric Nonlinear Dirichlet Problem with Subdiffusive and Equidiffusive Reaction

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Mean Curvature ◽  
Nonlinear Eigenvalue Problem ◽  
Nodal Solution ◽  
Smooth Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Nonhomogeneous Differential Operator

AbstractWe study a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator. Our setting incorporates equations driven by the p-Laplacian, the (p, q)-Laplacian, and the generalized p-mean curvature differential operator. Applying variational methods we show that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal). In the particular case of (p, 2)-equations, using Morse theory, we produce another nodal solution for a total of four nontrivial smooth solutions.

On the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Condition

2004 ◽  
Vol 9 (2) ◽  
pp. 109-116 ◽  
Author(s):  
R. Čiupaila ◽  
Ž. Jesevičiūtė ◽  
M. Sapagovas
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Integral Condition ◽  
Positive Eigenvalue ◽  
One Dimensional ◽  
Multiple Eigenvalues ◽  
Nonlocal Integral Condition

The article investigates the eigenvalue problem for ordinary onedimensional differential operator with nonlocal integral condition. Such a problem is met in the literature quite rarely and is considerably less investigated. Also the conditions for existence of non-positive eigenvalue or multiple eigenvalues are obtained.

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