Superconvergence of high order FEMs for eigenvalue problems with periodic boundary conditions

2009 ◽  
Vol 198 (30-32) ◽  
pp. 2246-2259 ◽  
Author(s):  
H.-T. Huang ◽  
S.-L. Chang ◽  
C.-S. Chien ◽  
Z.-C. Li
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions ◽  
High Order

scholarly journals Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications

2021 ◽  
Vol 41 (4) ◽  
pp. 489-507
Author(s):  
Abdelrachid El Amrouss ◽  
Omar Hammouti
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Critical Point Theory ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Difference Operator ◽  
Periodic Boundary Conditions ◽  
Point Theory ◽  
Existence Of A Solution ◽  
Order Difference

Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory.

The interlacing of eigenvalues for periodic multi-parameter problems

1978 ◽  
Vol 80 (3-4) ◽  
pp. 357-362 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Image Position ◽  
Periodic Equation

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

scholarly journals Solvability of a fourth-order boundary value problem with periodic boundary conditions II

1991 ◽  
Vol 14 (1) ◽  
pp. 127-137 ◽  
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Special Case

Letf:[0,1]×R4→Rbe a function satisfying Caratheodory's conditions ande(x)∈L1[0,1]. This paper is concerned with the solvability of the fourth-order fully quasilinear boundary value problemd4udx4+f(x,u(x),u′(x),u″(x),u‴(x))=e(x),   0<x<1, withu(0)−u(1)=u′(0)−u′(1)=u″(0)-u″(1)=u‴(0)-u‴(1)=0. This problem was studied earlier by the author in the special case whenfwas of the formf(x,u(x)), i.e., independent ofu′(x),u″(x),u‴(x). It turns out that the earlier methods do not apply in this general case. The conditions need to be related to both of the linear eigenvalue problemsd4udx4=λ4uandd4udx4=−λ2d2udx2with periodic boundary conditions.

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