A new method to find exact solution of nonlinear ordinary differential equations: Application to derive thermophoretic waves in graphene sheets

TL;DR

This paper introduces a novel iterative method for finding exact solutions to nonlinear ordinary differential equations, demonstrated through thermophoretic wave equations in graphene sheets, yielding precise closed-form solutions.

Contribution

The paper presents a new efficient iterative approach for deriving exact solutions of nonlinear ODEs, capable of producing closed-form solutions in various functions, including elliptic functions.

Findings

Successfully derived closed-form solutions for thermophoretic waves in graphene sheets.

The method converges rapidly and provides exact, unique solutions unlike previous approximate methods.

Demonstrated solutions include exponential, hyperbolic, trigonometric, algebraic, and elliptic functions.

Abstract

This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed form solution. The proposed approach is very efficient and essentially perfect for determining exact solutions of nonlinear equations. To demonstrate the effectiveness of this method, we examined the extended (2 + 1) dimensional equation for thermophoretic motion, which is based on wrinkle wave movements in graphene sheets supported by a substrate. The implementation of the suggested approach effectively yielded closed-form solutions in terms of exponential functions, hyperbolic functions, trigonometric functions, algebraic functions, and Jacobi elliptic functions, respectively. Three generated solutions illustrated to examine the characteristics of thermophoretic waves in graphene sheets. The proposed method's benefits and drawbacks are also examined. Consequently, unlike previous solutions obtained via the variation of parameters method for nonlinear issues, the solutions presented here are exact and unique.