Laplace transform homotopy perturbation method for the approximation of variational problems

U Filobello-Nino; H Vazquez-Leal; M M Rashidi; H M Sedighi; A Perez-Sesma; M Sandoval-Hernandez; A Sarmiento-Reyes; A D Contreras-Hernandez; D Pereyra-Diaz; C Hoyos-Reyes; V M Jimenez-Fernandez; J Huerta-Chua; F Castro-Gonzalez; J R Laguna-Camacho

doi:10.1186/s40064-016-1755-y

Laplace transform homotopy perturbation method for the approximation of variational problems

Springerplus. 2016 Mar 5:5:276. doi: 10.1186/s40064-016-1755-y. eCollection 2016.

Affiliations

¹ Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico.
² Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, 4800 Cao An Rd., Jiading, Shanghai, 201804 China ; ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai, China.
³ Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran.
⁴ Doctorado en Ciencia, Cultura y Tecnología, Universidad de Xalapa, Km 2 Carretera Xalapa-Veracruz, 91190 Xalapa, Veracruz Mexico.
⁵ National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla, 72840 Puebla, Mexico.
⁶ Facultad de Ingeniería Electrónica y Comunicaciones, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolución, 93390 Poza Rica, Veracruz Mexico.
⁷ Facultad de Ingeniería Mecánica Eléctrica, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolución, 93390 Poza Rica, Veracruz Mexico.

Abstract

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

Keywords: Approximate solutions; Euler equation; Homotopy perturbation method; Laplace transform; Laplace transform homotopy perturbation method; Nonlinear differential equation; Variational calculus.