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<p>We calculate the solution of the Bagley-Torvik equation for arbitrary initial conditions and arbitrary external force as a 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t\rightarrow 0^{+}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t\rightarrow +\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature.</p>

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Analytical closed-form solution of the Bagley-Torvik equation

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  • 发表日期：

    2026年2月17日

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  • Juan Luis González-Santander,
  • Alexander Apelblat

摘要

We calculate the solution of the Bagley-Torvik equation for arbitrary initial conditions and arbitrary external force as a 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 \(t\rightarrow 0^{+}\) t → 0 + and \(t\rightarrow +\infty \) t → + ∞ . The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature.