Derivation and physical interpretation of general boundary conditions

By G. Goldstein. Advances in Differential Equations, 2006

269 citations recorded by the source corpus.

In this paper we give new derivations of the heat and wave equation which incorporate the boundary conditions into the formulation of the problems. The principle of least action and the inclusion of a kinetic energy contribution on the boundary are used to derive the wave equation together with kinetic boundary conditions. The methods described for both equations admit all of the standard boundary conditions as well as general Wentzell and dynamic boundary conditions; in addition the boundary conditions arise naturally as part of the formulation of the problems. The physical interpretation for general Wentzell boundary conditions is given for both the heat and wave equations.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02e40949736f6b007937fae6f6fd152e50aef07a

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In Computational Mechanics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

An extremal eigenvalue problem for the Wentzell-Laplace operator (2014, background): “A general derivation of Ventcel boundary conditions can be found in [14].”
A Cahn–Hilliard model in bounded domains with permeable walls (2006, background): “For a complete discussion of the physical interpretation of (11), in the context of heat and wave equations, see Reference [16].”
Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation (2008, background): “Instead of the zero-flux boundary condition (7), the Wentzell boundary condition (3) is derived from mass conservation laws that include boundary sources, following an argument devised in [14].”
Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type (2013, background): “Dynamic surface and interface processes have attracted a lot of attention in recent years in the mathematical and applied literature, see [3, 5, 7, 8, 14, 15, 16, 17, 18, 26, 30, 34].”
Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions (2008, background): “In the two dimension space, as showed in [26] and in the references therein, these boundary conditions arise when we consider the transverse motion of a flexible membrane Ω whose boundary may be affected by the vibrations only in a region.”
A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions (2021, background): “…b → + ∞ ), but a new conservation law for the “total mass” defined in the measure space ( Ω , d ν ) = ( Ω , d x ) ⊕ ( ∂ Ω , d S / b ) (see [98, 115]): The boundary condition (3.12) describes the situation that there is certain mass source (or leak) on the boundary and thus the system…”