A sequence approach to solve the Burgers' equation

TL;DR

This paper introduces a semi-analytic sequence transformation method that converts the nonlinear Burgers' equation into a series of linear diffusion equations, enabling recursive solutions for complex initial conditions.

Contribution

A novel sequence-based approach that transforms the Burgers' equation into linear problems, providing explicit recursive integral solutions for arbitrary initial and boundary conditions.

Findings

The method produces solutions that converge absolutely and uniformly.

It matches numerical solutions with arbitrary precision.

A new analytic solution using Bell polynomials is derived for exponential initial conditions.

Abstract

The Burgers' equation is a one-dimensional momentum equation for a Newtonian fluid. The Cole-Hopf transformation solves the equation for a given initial and boundary condition. However, in most cases the resulting integral equation can only be solved numerically. In this work a new semi-analytic solving method is introduced for analytic and bounded series solutions of the Burgers' equation. It is demonstrated that a sequence transformation can split the non-linear Burgers' equation into a sequence of linear diffusion equations. Each consecutive sequence element can be solved recursively using the Green's function method. The general solution to the Burgers' equation can therefore be written as a recursive integral equation for any initial and boundary condition. For a complex exponential function as initial condition we derive a new analytic solution of the Burgers' equation in terms of the Bell polynomials. The new solution converges absolutely and uniformly and matches a numerical solution with arbitrary precision. The presented semi-analytic solving method can be generalized to a larger class of non-linear partial differential equations which we leave for future work.