Analytical closed-form solution of the Bagley-Torvik equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908838282657792 |
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| author | González-Santander, Juan Luis Apelblat, Alexander |
| author_facet | González-Santander, Juan Luis Apelblat, Alexander |
| contents | We calculate the solution of the Bagley-Torvik equation for arbitrary initial conditions and arbitrary external force as the sum of two terms. The first one is a linear combination of exponentials with error functions, and the second one is a convolution integral whose kernel is a linear combination of exponentials with error functions. The derivation of the solution is carried out by using the Laplace transform method and the calculation of a new inverse Laplace transform. The aforementioned convolution integral can be calculated for the cases of a sinusoidal- or a potential-type external force. In addition, we calculate the asymptotic behaviour of the solution for $t\rightarrow 0^{+}$ and $t\rightarrow +\infty$. The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytical closed-form solution of the Bagley-Torvik equation González-Santander, Juan Luis Apelblat, Alexander Classical Analysis and ODEs 26A33, 44A10, 34A08 We calculate the solution of the Bagley-Torvik equation for arbitrary initial conditions and arbitrary external force as the sum of two terms. The first one is a linear combination of exponentials with error functions, and the second one is a convolution integral whose kernel is a linear combination of exponentials with error functions. The derivation of the solution is carried out by using the Laplace transform method and the calculation of a new inverse Laplace transform. The aforementioned convolution integral can be calculated for the cases of a sinusoidal- or a potential-type external force. In addition, we calculate the asymptotic behaviour of the solution for $t\rightarrow 0^{+}$ and $t\rightarrow +\infty$. The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature. |
| title | Analytical closed-form solution of the Bagley-Torvik equation |
| topic | Classical Analysis and ODEs 26A33, 44A10, 34A08 |
| url | https://arxiv.org/abs/2501.05972 |