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. 2022 May 17:2022:2357258.
doi: 10.1155/2022/2357258. eCollection 2022.

An Elementary Solution to a Duffing Equation

Alvaro H Salas  1
Affiliations

An Elementary Solution to a Duffing Equation

Alvaro H Salas. ScientificWorldJournal. .

Abstract

In this work, we study the Duffing equation. Analytical solution for undamped and unforced case is provided for any given arbitrary initial conditions. An approximate analytical solution is given for the damped or trigonometrically forced Duffing equation for arbitrary initial conditions. The analytical solutions are expressed in terms of elementary trigonometric functions as well as in terms of the Jacobian elliptic functions. Examples are added to illustrate the obtained results. We also introduce new functions for approximating the Jacobian and Weierstrass elliptic functions in terms of the trigonometric functions sine and cosine. Results are high accurate.

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Conflict of interest statement

The authors declare that they have no conflicts of interest.

Figures

Figure 1
Figure 1
Comparison between the exact solution and the numerical solution for Example 1.
Figure 2
Figure 2
Comparison between the exact solution and the numerical solution for Example 2.
Figure 3
Figure 3
Comparison between the exact solution and the approximate analytical solution for Example 3.
Figure 4
Figure 4
Comparison between the exact solution and the approximate analytical solution for Example 4.
Figure 5
Figure 5
Comparison between the approximate analytical solution and the numerical solution for Example 5.
Figure 6
Figure 6
Comparison between the approximate analytical solution and the numerical solution for Example 6.
Figure 7
Figure 7
Comparison between the approximate analytical solution and the numerical solution for Example 7.
Figure 8
Figure 8
Comparison between the approximate analytical solution and the numerical solution for Example 8.
Figure 9
Figure 9
Comparison between the approximate analytical solution and the numerical solution for Example 9.
Figure 10
Figure 10
Comparison between the approximate analytical solution and the numerical solution for Example 10.
Figure 11
Figure 11
Comparison between the approximate analytical solution and the homotopy solution for Example 10.
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References

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