An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

Ali K Joohy,Mohammed S Mechee,Ghassan A Al-Juaifri

doi:10.1155/2020/5276024

Abstract

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

Highlights

In the last five decades, fascinating methods have been made in identifying nonlinear integrable dynamical systems in the real line
Different methods have been developed or modified to investigate innovative integrable cases and experience the potential dynamics that could be related to the finite-dimensional nonlinear dynamical systems, which is defined on the real numbers [1]. e most extensively mathematical tools that are being used to solve ODEs are Painleve Analysis [1, 2], Direct Method, [3] Lie Symmetry Analysis [1, 4], Noether’s theorem [1, 4], Direct Linearization [5], λ-symmetries, adjoint symmetries, and Jacobi last multiplier technique [6, 7]
The authors are moving a step forward to generalize one particular method, namely, Prelle–Singer method, and present a related procedure on the complex plane; the authors have generalized a class of nonlinear ODEs [9], in which it has a real interval of definition. en, they construed a class of (NLOCDE (Nonlinear Ordinary Complex Differential Equations (NLOCDE, where the general solution to the mentioned))) is an algebraic combination of complex elementary functions that are analytic on a particular region in the complex plane

Summary

Introduction

In the last five decades, fascinating methods have been made in identifying nonlinear integrable dynamical systems in the real line. The authors are moving a step forward to generalize one particular method, namely, Prelle–Singer method, and present a related procedure on the complex plane; the authors have generalized a class of nonlinear ODEs [9], in which it has a real interval of definition. En, they construed a class of (NLOCDE (Nonlinear Ordinary Complex Differential Equations (NLOCDE, where the general solution to the mentioned))) is an algebraic combination of complex elementary functions that are analytic on a particular region in the complex plane. E design of the paper is as follows: in Section 2, authors developed the theory of the extended Prelle–Singer method to make it applicable to the third-order ordinary nonlinear CDEs. In Sections 1–4, the theoretical definitions have been shown; in Section 5, authors considered an example and constructed complex integrals of motion. Definition 6. e ordinary complex differential equation is called nonlinear complex differential equation when it is nonlinear [24]

Analysis of Prelle–Singer Method for ThirdOrder OCDEs

Example 1

Method of Generating Complex Integrals of Motion Analogue

Applications

Conclusion