Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
DOI:
https://doi.org/10.29059/cienciauat.v13i2.1119Keywords:
Laplace transform, perturbation method, nonlinear differential equationsAbstract
The field of differential equations has recently gained attention due to recent developments in science and technology. For this reason, the analysis for the use of new methodologies to solve them has become important. Based on the combination of Laplace Transform method (LT) and Perturbation Method (PM) this article proposes the Laplace transform-Perturbation Method (LT-PM) which finds its motivation on the application of LT to linear ordinary differential equations. The goal of this work is to propose a modification of PM - the LT-PM), in order to solve nonlinear perturbative problems with boundary conditions defined on finite intervals. The proposed methodology consisted on the application of LT to the differential equation to solve and then, assuming that its solutions can be expressed as a series of perturbative parameter powers. Thus, the solution of the problem is obtained by systematically applying the transformed inverse LT. The main results of this paper were shown through two case studies, where LT-PM is identified as potentially useful for finding multiple solutions to nonlinear problems. Additionally, the LT-PM enhances the applicability of PM, in some cases of mixed and Neumann boundary conditions, where PM is unsuitable to provide the results. With the purpose of verifying the accuracy of the obtained results, the Square Residual Error (SRE) was calculated. The resulting value was extremely low, which showed the precision and potential of LT-PM. We conclude that, although the proposed method resulted efficient for the case studies presented in this article, it is expected that LT-PM can be a potentially useful tool for other case studies. Particularly those related to the practical applications of science and engineering.
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