<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.</p>
</abstract>

In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.

Solutions of the Nonlinear Evolution Problems and their Applications

Acta Mechanica et Automatica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Solutions of the Nonlinear Evolution Problems and their Applications"}

In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple...

Solutions of the Nonlinear Evolution Problems and their Applications

Data publikacji: 11 lip 2023

Zakres stron: 357 - 363

Otrzymano: 07 gru 2022

Przyjęty: 26 lut 2023

DOI: https://doi.org/10.2478/ama-2023-0040

Słowa kluczowe
Laplace variational iteration method, nonlinear problems, duffing equation, simple pendulum, mass and spring oscillator, Simulink

© 2023 Muhammad Amir et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Solutions of the Nonlinear Evolution Problems and their Applications

Data publikacji: 11 lip 2023

Zakres stron: 357 - 363

Otrzymano: 07 gru 2022

Przyjęty: 26 lut 2023

DOI: https://doi.org/10.2478/ama-2023-0040

Słowa kluczoweLaplace variational iteration method, nonlinear problems, duffing equation, simple pendulum, mass and spring oscillator, Simulink

© 2023 Muhammad Amir et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Słowa kluczowe
Laplace variational iteration method, nonlinear problems, duffing equation, simple pendulum, mass and spring oscillator, Simulink