Classes of second order nonlinear partial differential equations reducible to first order

TL;DR

This paper introduces new techniques for reducing second order nonlinear PDEs to first order equations like Bernoulli, Ricatti, and Abel, enabling the derivation of exact solutions through combined methods.

Contribution

It presents a novel approach that combines parameter variations and characteristic methods to solve classes of second order nonlinear PDEs, extending to more complex equations.

Findings

New reduction techniques for second order nonlinear PDEs

Explicit solutions for various classes of PDEs

Method extensible to higher order PDEs

Abstract

In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and Abel equations. The main idea is based on implementing new techniques by combining variations of parameters with characteristic methods to obtain many new and general exact solutions. In each class of PDE's, we give illustrated examples. Moreover, the method presented in this paper can be easily extended to classes of second order nonlinear PDEs.