Quasi-exact solutions of nonlinear differential equations

TL;DR

This paper introduces the concept of quasi-exact solutions for nonlinear differential equations, providing approximate solutions that are close to exact ones for specific parameter values, demonstrated on several important equations.

Contribution

It defines quasi-exact solutions, expanding the scope of exact solutions, and applies this concept to key nonlinear differential equations.

Findings

Quasi-exact solutions are close approximations to exact solutions.

The method is applied successfully to Kuramoto--Sivashinsky, Korteweg--de Vries--Burgers, and Kawahara equations.

The approach broadens the toolkit for solving nonlinear differential equations.

Abstract

The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate solutions of nonlinear differential equations but they are close to exact solutions. Quasi-exact solutions of the the Kuramoto--Sivashinsky, the Korteweg--de Vries--Burgers and the Kawahara equations are founded.