Unnecessary Exact Solutions of Nonlinear Ordinary Differential Equations

TL;DR

This paper critiques a previous study claiming to find new solutions to a complex PDE, demonstrating that their solutions are actually well-known ODE solutions and highlighting common mistakes in such derivations.

Contribution

It introduces the concept of unnecessary exact solutions in nonlinear ODEs and clarifies common errors in solution derivation.

Findings

Previous solutions are reducible to known solutions

Authors made typical mistakes in solving nonlinear equations

The paper defines unnecessary exact solutions

Abstract

We analyze the paper by Wazwaz and Mehanna [Wazwaz A.M., Mehanna M.S., A variety of exact travelling wave solutions for the (2+1) -- dimensional Boiti -- Leon -- Pempinelli equation, Appl. Math. Comp. 217 (2010) 1484 -- 1490]. Using the tanh -- coth method and the Exp -- function method the authors claim that they have found exact solutions of the (2+1) -- dimensional Boiti -- Leon -- Pempinelli equation. We demonstrate that the authors have obtained the exact solutions of the well known nonlinear ordinary differential equation. We illustrate that all solutions presented by the authors can be reduced to the well-known solutions. Wazwaz and Mehanna made a number of typical mistakes in finding exact solutions of nonlinear differential equations. Taking the results of this paper we introduce the definition of unnecessary exact solutions for the nonlinear ordinary differential equations.