Procedure for Exact Solutions of Nonlinear Pantograph Delay Differential Equations
Brahim Benhammouda
Abu Dhabi Men’s College, Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, United Arab Emirates.
Hector Vazquez-Leal *
Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltr ´an S/N, C.P. 91000, Xalapa, Veracruz, M´exico.
Luis Hernandez-Martinez
National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. Mar´Ä±a Tonantzintla 72840, Puebla, M´exico.
*Author to whom correspondence should be addressed.
Abstract
This work presents the application of the power series method (PSM) to find solutions of nonlinear delay differential equations of pantograph type (PDDEs). Three equations are solved to show that PSM can provide analytical solutions of PDDEs in convergent series form. The nonlinear pantograph cases study are: a first order equation, a second order equation, and a second order singular equation. Additionally, we present the post-treatment of the power series solutions with the Laplace-Pad´e (LP) resummation method as a powerful technique to find exact solutions. The proposed methodology possesses a simple procedure based on a few straightforward steps and it does not depend on a perturbation parameter.
Keywords: Pantograph equations, Power series method, Laplace transform, Pad´e approximant, Analytical solutions.