Rainfall infiltration: Direct and Inverse problems on a linear evolution equation

TL;DR

This paper models rainfall infiltration using a linear evolution PDE, solving it with the Fokas method, and develops algorithms for boundary control and regularisation to manage the infiltration process effectively.

Contribution

It introduces a novel application of the Fokas method to rainfall infiltration modeling and presents an algorithm for null-controllability with regularisation schemes.

Findings

Numerical solutions for rainfall infiltration problems.

An algorithm for boundary control to achieve desired infiltration states.

A regularisation scheme for small boundary controls.

Abstract

Originating from the mathematical modelling of rainfall infiltration, we derive the solution of an initial-boundary value problem of a linear evolution partial differential equation, by using the Fokas method. We present numerical examples which correspond to specific physical rainfall problems. Based on this formalism we present an effective algorithm for the associated null-controllability problem, namely we numerically derive a family of boundary controls that steer the solution to the desired flat final state. Finally, a regularisation scheme allows the derivation of relatively small controls, in cases where this is necessary.