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Derivation and physical interpretation of general boundary conditions. (English) Zbl 1107.35010

Summary: 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.

MSC:

35A15 Variational methods applied to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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