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Analytical closed-form solution of the Bagley-Torvik equation

Juan Luis González-Santander, Alexander Apelblat

TL;DR

This work provides a closed-form solution to the Bagley-Torvik equation with arbitrary initial data and external forcing by decomposing $y(t)$ into a forcing-driven term $y_f(t)$ and an initial-condition term $y_c(t)$. The forcing term is a convolution with a kernel formed from exponentials multiplied by $\operatorname{erfc}$, while $y_c(t)$ is a finite sum of similar terms whose coefficients depend on the four roots of the quartic polynomial $P(r)=a r^4+b r^3+c$. A key contribution is a new inverse Laplace transform for $\dfrac{s^{\lambda}}{a s^m + b s^{p/q} + c}$, enabling explicit expressions in terms of Mittag-Leffler functions evaluated at $r_k t^{1/q}$, where $r_k$ are the roots of $P$. The paper also derives closed-form solutions for potential and sinusoidal forcings, provides detailed asymptotics as $t\to 0^+$ and $t\to \infty$, and demonstrates substantial computational speedups and stability relative to prior methods, making the results valuable as both exact benchmarks and practical solvers for fractional differential equations in engineering.

Abstract

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.

Analytical closed-form solution of the Bagley-Torvik equation

TL;DR

This work provides a closed-form solution to the Bagley-Torvik equation with arbitrary initial data and external forcing by decomposing into a forcing-driven term and an initial-condition term . The forcing term is a convolution with a kernel formed from exponentials multiplied by , while is a finite sum of similar terms whose coefficients depend on the four roots of the quartic polynomial . A key contribution is a new inverse Laplace transform for , enabling explicit expressions in terms of Mittag-Leffler functions evaluated at , where are the roots of . The paper also derives closed-form solutions for potential and sinusoidal forcings, provides detailed asymptotics as and , and demonstrates substantial computational speedups and stability relative to prior methods, making the results valuable as both exact benchmarks and practical solvers for fractional differential equations in engineering.

Abstract

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 and . The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature.
Paper Structure (15 sections, 5 theorems, 107 equations, 11 figures, 1 table)

This paper contains 15 sections, 5 theorems, 107 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some properties of the Mittag-Leffler function
  4. Analytical solutions
  5. Numerical solutions
  6. A useful inverse Laplace transform
  7. The general solution of the Bagley-Torvik equation
  8. The potential solution
  9. The sinusoidal solution
  10. Asymptotic behaviour
  11. Asymptotic behaviour of $y_{c}\left( t\right)$
  12. Asymptotic behaviour of $y_{f}\left( t\right)$
  13. Numerical examples
  14. Conclusions
  15. Auxiliary integrals

Key Result

Theorem 5

For $p/q\in \mathbb{Q} ^{+}$, $m\in \mathbb{N}$, $\lambda \in \mathbb{R}$, the following inverse Laplace transform holds true: where $r_{k}$$\left( k=1,\ldots ,N\right)$ are the $N=\max \left( mq,p\right)$ different roots of the polynomial:

Figures (11)

  • Figure 1: Immersed plate in a Newtonian fluid connected by a massless spring.
  • Figure 2: Solution of the Bagley-Torvik equation for a Bessel-type force with $y\left( 0\right) =y^{\prime }\left( 0\right) =0$.
  • Figure 3: Solution of the Bagley-Torvik equation for a Bessel-type force with $y\left( 0\right) =y^{\prime }\left( 0\right) =1$.
  • Figure 4: Solution of the Bagley-Torvik equation for $f\left( t\right) =1+\sqrt{t}$ with $y\left( 0\right) =y^{\prime }\left( 0\right) =0$.
  • Figure 5: Solution of the Bagley-Torvik equation for $f\left( t\right) =1+\sqrt{t}$ with $y\left( 0\right) =y^{\prime }\left( 0\right) =1$.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Theorem 10