Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions

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 Metho...

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Main Authors:	Filobello-Niño, Uriel Antonio, Vázquez-Leal, Héctor, Sandoval-Hernández, Mario Alberto, Huerta-Chua, Jesús, Jiménez-Fernández, Víctor Manuel
Format:	Online
Language:	spa
Published:	Universidad Autónoma de Tamaulipas 2019
Online Access:	https://revistaciencia.uat.edu.mx/index.php/CienciaUAT/article/view/1119
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id	oai:ojs.pkp.sfu.ca:article-1119
record_format	ojs
institution	CIENCIA UAT
collection	OJS
language	spa
format	Online
author	Filobello-Niño, Uriel Antonio Vázquez-Leal, Héctor Sandoval-Hernández, Mario Alberto Huerta-Chua, Jesús Jiménez-Fernández, Víctor Manuel
spellingShingle	Filobello-Niño, Uriel Antonio Vázquez-Leal, Héctor Sandoval-Hernández, Mario Alberto Huerta-Chua, Jesús Jiménez-Fernández, Víctor Manuel Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
author_facet	Filobello-Niño, Uriel Antonio Vázquez-Leal, Héctor Sandoval-Hernández, Mario Alberto Huerta-Chua, Jesús Jiménez-Fernández, Víctor Manuel
author_sort	Filobello-Niño, Uriel Antonio
title	Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
title_short	Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
title_full	Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
title_fullStr	Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
title_full_unstemmed	Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions
title_sort	laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and neumann boundary conditions
description	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.
publisher	Universidad Autónoma de Tamaulipas
publishDate	2019
url	https://revistaciencia.uat.edu.mx/index.php/CienciaUAT/article/view/1119
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spelling	oai:ojs.pkp.sfu.ca:article-11192020-01-29T08:29:57Z Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions Método de perturbación con transformada de Laplace para resolver problemas no lineales de múltiples soluciones, con condiciones a la frontera mixtas y Neumann Filobello-Niño, Uriel Antonio Vázquez-Leal, Héctor Sandoval-Hernández, Mario Alberto Huerta-Chua, Jesús Jiménez-Fernández, Víctor Manuel Laplace transform perturbation method nonlinear differential equations transformada de Laplace método de perturbación ecuaciones diferenciales no lineales 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. El campo de las ecuaciones diferenciales ha cobrado auge en la actualidad por el desarrollo científico y tecnológico. Por esta situación, el estudio de nuevas metodologías para solucionarlas se ha vuelto importante. A partir de la combinación del método de Laplace Transform (LT) y el método de perturbación (PM) este trabajo presenta el método LT-PM, y su motivación se encuentra en la aplicación conocida de la LT a ecuaciones diferenciales ordinarias lineales. El objetivo de este trabajo fue presentar una modificación del método de perturbación (PM), el método de perturbación con transformada de Laplace (LT-PM), con el fin de resolver problemas perturbativos no lineales, con condiciones a la frontera definidas en intervalos finitos. La metodología consistió en aplicar LT a la ecuación diferencial por resolver y después de asumir que la solución de la misma se puede expresar como una serie de potencias de un parámetro perturbativo, se obtiene la solución del problema aplicando sistemáticamente la transformada inversa de Laplace. Los principales resultados de este trabajo se muestran a partir de dos casos de estudio presentados, donde se observa que LT-PM es potencialmente útil para encontrar soluciones múltiples de problemas no lineales. Además, LT-PM mejora la aplicabilidad del método de perturbación en algunos casos de condiciones a la frontera mixtas y de Neumann, donde PM simplemente no funciona. Con el fin de verificar la exactitud de los resultados obtenidos, se calculó su error residual cuadrático (SRE), el cual resultó muy bajo, de donde se dedujo su precisión y la potencialidad de LT-PM. Se concluye que si bien el método propuesto resulta eficiente en los casos particulares presentados, se espera que sea una herramienta potencialmente eficiente y útil para otros casos de estudio, particularmente, en aquellos relacionados con aplicaciones prácticas en ciencias e ingeniería. Universidad Autónoma de Tamaulipas 2019-01-31 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion application/pdf text/html application/xml https://revistaciencia.uat.edu.mx/index.php/CienciaUAT/article/view/1119 10.29059/cienciauat.v13i2.1119 CienciaUAT; Vol 13 No. 2. January-June 2019; 06-17 CienciaUAT; Vol. 13 No. 2: Enero-Junio 2019; 06-17 2007-7858 2007-7521 spa https://revistaciencia.uat.edu.mx/index.php/CienciaUAT/article/view/1119/550 https://revistaciencia.uat.edu.mx/index.php/CienciaUAT/article/view/1119/563 https://revistaciencia.uat.edu.mx/index.php/CienciaUAT/article/view/1119/682 Derechos de autor 2019 CienciaUAT

Laplace transform-perturbation method to solve nonlinear perturbative multiple solutions problems with mixed and Neumann boundary conditions

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