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

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.

Figures (6)

• Figure 1: The water content $\theta$ given by \ref{['ex1sol']} at fixed time steps (left) and for $x \in [0,\,L]$ and $t\in[0,\,135]$min (right).
• Figure 2: The conductivity $\tfrac{K}{K_1}$ given by \ref{['sol_robin']} for different flow velocities at fixed time $t=\tfrac{L K_1}{R}$ (left) and for $x \in [0,\,L]$ and $t\in[0.5,\,2.5] \tfrac{L K_1}{R}$ (right).
• Figure 3: The control $v(t)$ (left) and the solution $\theta(x,t)$ (right) for $N=4$ of the first example.
• Figure 4: The control $v(t)$ (left) and the solution $\theta(x,t)$ (right) for $N=2$ of the third example.
• Figure 5: The exact (left) and the regularized (right) controls of the second example for different values of $N$.

Theorems & Definitions (1)

• Remark 2.1