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On the Initial‐boundary‐value Problem for the Extensible Beam with Attached Load
Author(s) -
Dalsen Marié GrobbelaarVan
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199608)19:12<943::aid-mma804>3.0.co;2-f
Subject(s) - mathematics , boundary value problem , beam (structure) , partial differential equation , mathematical analysis , extensibility , boundary (topology) , vibration , initial value problem , galerkin method , construct (python library) , finite element method , physics , computer science , quantum mechanics , optics , thermodynamics , operating system , programming language
When one end of an extensible beam whose ends are a fixed distance apart, is hinged while to other end a load is attached, the mathematical model describing the vibrations of the beam contains a non‐linearity both in the partial differential equation as well as in the dynamical boundary condition. We introduce weak damping and consider the resulting boundary‐value problem within the framework of the theories of B ‐evolutions and fractional powers of a closed pair of operators. This approach enables us to construct a unique weak solution which exhibits exponential decay by employing Faedo–Galerkin approximations.