Minimization of eigenvalues for the Camassa–Holm equation

2020 ◽  
pp. 2050021
Author(s):  
Hao Feng ◽  
Gang Meng
Keyword(s):  
Boundary Condition ◽  
Lower Bound ◽  
Minimization Problem ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Smallest Eigenvalue ◽  
Holm Equation ◽  
The One ◽  
The Relationship ◽  
Optimal Lower Bound

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem [Formula: see text] We will study in this paper the optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation with the Neumann boundary condition when the [Formula: see text] norm of potentials is given. First, we will study the optimal lower bound for the smallest eigenvalue in the measure differential equations to make our results more applicable. Second, Based on the relationship between the minimization problem of the smallest eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation.

scholarly journals Classical Lie Point Symmetry Analysis of a Steady Nonlinear One-Dimensional Fin Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
R. J. Moitsheki ◽  
M. D. Mhlongo
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Group Classification ◽  
One Dimensional ◽  
Fin Efficiency ◽  
The One ◽  
The Heat Transfer Coefficient ◽  
Preliminary Group

We consider the one-dimensional steady fin problem with the Dirichlet boundary condition at one end and the Neumann boundary condition at the other. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended. Some invariant solutions are constructed. The effects of thermogeometric fin parameter and the exponent on temperature are studied. Also, the fin efficiency is analyzed.

Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Agil K. Khanmamedov ◽  
Nigar F. Gafarova
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Inverse Spectral Problem ◽  
Anharmonic Oscillator ◽  
Neumann Boundary ◽  
Effective Algorithm ◽  
Inverse Spectral Problems ◽  
Main Equation ◽  
Inverse Spectral ◽  
Half Axis

AbstractAn anharmonic oscillator {T(q)=-\frac{d^{2}}{dx^{2}}+x^{2}+q(x)} on the half-axis {0\leq x<\infty} with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

scholarly journals Pointwise observation of the state given by complex time lag parabolic system

2017 ◽  
Vol 27 (1) ◽  
pp. 77-89
Author(s):  
Adam Kowalewski
Keyword(s):  
Boundary Condition ◽  
Parabolic System ◽  
Neumann Boundary Condition ◽  
Optimization Problems ◽  
Time Lag ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
The State ◽  
Constant Time ◽  
Time Lags

AbstractVarious optimization problems for linear parabolic systems with multiple constant time lags are considered. In this paper, we consider an optimal distributed control problem for a linear complex parabolic system in which different multiple constant time lags appear both in the state equation and in the Neumann boundary condition. Sufficient conditions for the existence of a unique solution of the parabolic time lag equation with the Neumann boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme [13], necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional with pointwise observation of the state and constrained control are derived. The example of application is also provided.

An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition

2018 ◽  
Vol 356 ◽  
pp. 115-126 ◽  
Author(s):  
Yesom Park ◽  
Jeongho Kim ◽  
Jinwook Jung ◽  
Euntaek Lee ◽  
Chohong Min
Keyword(s):  
Boundary Condition ◽  
Poisson Equation ◽  
Neumann Boundary Condition ◽  
Neumann Boundary
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