The unified transform for Burgers' equation: Application to unsaturated flow in finite interval

TL;DR

This paper applies the Unified Transform Method to solve Burgers' equation derived from Richards' equation for vertical infiltration, providing explicit solutions with improved convergence and stability demonstrated through hydrological examples.

Contribution

It introduces the use of the Unified Transform Method for solving Burgers' equation in hydrology, offering explicit integral solutions with enhanced numerical properties.

Findings

Solutions match classical Fourier series results exactly.

Method offers better convergence and stability.

Validated with hydrological application examples.

Abstract

In this paper, we focus on one-dimensional vertical infiltration, assuming constant diffusivity and a quadratic relationship between hydraulic conductivity and water content. Under these assumptions, Richards' equation reduces to Burgers' equation, which we then linearize via the Hopf-Cole transformation. This turns the initial boundary value problem into a diffusion equation on a finite interval with mixed boundary conditions. To solve it, we use the Unified Transform Method (also known as the Fokas method). This approach gives an explicit integral representation of the solution, and when evaluated numerically, the results match classical Fourier series solutions exactly, but with better convergence and stability. Two examples from hydrological applications are examined.