scholarly journals Asymptotic eigenvalue estimates for a Robin problem with a large parameter

2014 ◽  
Vol 71 (2) ◽  
pp. 141-156 ◽  
Author(s):  
Pavel Exner ◽  
Alexander Minakov ◽  
Leonid Parnovski
Keyword(s):  
Robin Problem ◽  
Large Parameter ◽  
Eigenvalue Estimates ◽  
Asymptotic Eigenvalue

Asymptotic expansions of the oblate spheroidal eigenvalues and wave functions for large parameter c

2001 ◽  
Vol 79 (5) ◽  
pp. 813-831 ◽  
Author(s):  
Tam Do-Nhat
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Expansions ◽  
Wave Functions ◽  
Laguerre Functions ◽  
Large Parameter ◽  
Oblate Spheroidal ◽  
Asymptotic Eigenvalue ◽  
Expansion Coefficients ◽  
Analytical Expressions

The asymptotic expansion of the oblate spheroidal eigenfunctions can be expanded in terms of the Laguerre functions of the first and second kinds, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating frequency. Analytical expressions of the eigenvalue coefficients, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results. PACS Nos.: 02.30Gp, 03.65ge

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS OF THE ROBIN PROBLEM WITH LARGE PARAMETER

2017 ◽  
Vol 22 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
Large Parameter ◽  
Eigenvalues And Eigenfunctions ◽  
The Difference ◽  
Robin Boundary ◽  
Outward Unit

We consider the eigenvalue problem with Robin boundary condition ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω, where Ω ⊂ Rn , n ≥ 2 is a bounded domain with a smooth boundary, ν is the outward unit normal, α is a real parameter. We obtain two terms of the asymptotic expansion of simple eigenvalues of this problem for α → +∞. We also prove an estimate to the difference between Robin and Dirichlet eigenfunctions.

scholarly journals On the principal eigenvalue of a Robin problem with a large parameter

2008 ◽  
Vol 281 (2) ◽  
pp. 272-281 ◽  
Author(s):  
Michael Levitin ◽  
Leonid Parnovski
Keyword(s):  
Principal Eigenvalue ◽  
Robin Problem ◽  
Large Parameter

scholarly journals On the eigenvalues of a Robin problem with a large parameter

2014 ◽  
Vol 139 (2) ◽  
pp. 341-352 ◽  
Author(s):  
Alexey Filinovskiy
Keyword(s):  
Robin Problem ◽  
Large Parameter

scholarly journals Linear evolution equations on the half-line with dynamic boundary conditions

2021 ◽  
pp. 1-33
Author(s):  
D. A. SMITH ◽  
W. Y. TOH
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Evolution Equations ◽  
Half Line ◽  
Robin Problem ◽  
Dynamic Boundary Conditions ◽  
Transform Method ◽  
Linear Evolution ◽  
Robin Condition ◽  
Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

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