The initial value problem for a nonlinear semi-infinite string

1978 ◽  
Vol 82 (1-2) ◽  
pp. 19-26 ◽  
Author(s):  
R. W. Dickey
Keyword(s):  
Boundary Value Problem ◽  
Classical Solution ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Galerkin Procedure ◽  
Initial Boundary Value ◽  
Infinite String

SynopsisThe existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Initial-Boundary Value Problem on Unsteady Flows in a Canal

2015 ◽  
Vol 723 ◽  
pp. 136-139
Author(s):  
Da Yong Nie
Keyword(s):  
Boundary Value Problem ◽  
Classical Solution ◽  
Boundary Value ◽  
Unsteady Flows ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value

In this paper we consider initial-boundary value problem on unsteady flows in a canal, which is an important model in hydrodynamics, under certain assumptions, the global resolvability of classical solution is obtained by using the method of global extension.

scholarly journals CLASSICAL SOLUTION FOR INITIAL–BOUNDARY VALUE PROBLEM FOR WAVE EQUATION WITH INTEGRAL BOUNDARY CONDITION

2012 ◽  
Vol 17 (3) ◽  
pp. 309-329 ◽  
Author(s):  
Victor Korzyuk ◽  
Victor Erofeenko ◽  
Julia Sheika
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Boundary Value Problem ◽  
Classical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Initial Boundary Value

The unique existence of classical solution of initial–boundary value problem for wave equation with a special integral boundary condition is proved in the work. Classical solution of the problem in analytical form is also found in the article. This problem arises at the modeling of electromagnetic fields with arbitrary time dependence when interaction between the field and solids is simulated with impedance boundary conditions.

scholarly journals Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hailiang Li ◽  
Houzhi Tang ◽  
Haitao Wang
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Classical Solution ◽  
Half Space ◽  
Stokes Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
One Dimensional ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

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