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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.03383 |
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| _version_ | 1866916236744458240 |
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| author | Honegger, Reinhard Lauxmann, Michael Priwitzer, Barbara |
| author_facet | Honegger, Reinhard Lauxmann, Michael Priwitzer, Barbara |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Wave-Like Differential Equations in General Hilbert Space. The Functional Analytic Investigation of Euler-Bernoulli Bending Vibrations of a Beam as an Application in Engineering Science Honegger, Reinhard Lauxmann, Michael Priwitzer, Barbara Mathematical Physics Analysis of PDEs 35G10 35Q74 46E35 74H20 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. |
| title | On Wave-Like Differential Equations in General Hilbert Space. The Functional Analytic Investigation of Euler-Bernoulli Bending Vibrations of a Beam as an Application in Engineering Science |
| topic | Mathematical Physics Analysis of PDEs 35G10 35Q74 46E35 74H20 |
| url | https://arxiv.org/abs/2405.03383 |