Differential-algebraic solutions of the heat equation

TL;DR

This paper introduces a differential-algebraic ansatz for solving the heat and Burgers equations, explicitly constructing solutions from specific nonlinear ODEs, advancing analytical solution methods for these PDEs.

Contribution

It proposes the n-ansatz, a novel approach linking solutions of nonlinear ODEs to PDE solutions, providing explicit constructions for the heat and Burgers equations.

Findings

Constructed explicit solutions for heat and Burgers equations.

Introduced the n-ansatz method for PDE solutions.

Linked nonlinear ODE solutions to PDE solutions.

Abstract

In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.