| Autor: |
Honegger, Reinhard, Lauxmann, Michael, Priwitzer, Barbara |
| Rok vydání: |
2024 |
| Předmět: |
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| Zdroj: |
Partial Differential Equations in Applied Mathematics 2024 Volume 9 (short version) |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1016/j.padiff.2024.100617 |
| Popis: |
Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a generalization of the primary wave (partial) differential equation. A short overview over three wave-like problems in physics and engineering is presented. The mathematical procedure for achieving positive, selfadjoint differential operators in an $\mathrm{L}^2$-Hilbert space is described, operators which then may be taken for wave-like differential equations. Also some general results from the functional analytic literature are summarized. The main part concerns the investigation of the free Euler--Bernoulli bending vibrations of a slender, straight, elastic beam in one spatial dimension in the $\mathrm{L}^2$-Hilbert space setup. Taking suitable Sobolev spaces we perform the mathematically exact introduction and analysis of the corresponding (spatial) positive, selfadjoint differential operators of $4$-th order, which belong to the different boundary conditions arising as supports in statics. A comparison with free wave swinging of a string is added, using a Laplacian as differential operator. |
| Databáze: |
arXiv |
| Externí odkaz: |
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