Functional parameter estimation in partial differential equations

In this work we present a methodology of estimation of functional parameters that appear in models that are described by partial differential equations. We will focus on the following model: $$f\frac{\partial^2 u }{\partial t^2}+g\frac{\partial u }{\partial t}+hu=\frac{\partial }{\partial x}\left[{\mathcal{K}}\frac{\partial u}{\partial x}\right],$$ where the parameters $f,g,h$ and ${\mathcal{K}}$ are real valued functions of the real variable $x.$ We assume we know $N$ functions $v_1(x,t),...,v_N(x,t)$ that satisfy, for each $i$, $1\le i\le N,$ $v_i=u_i+\epsilon_i,$ where $u_i$ is a solution of the PDE and $\epsilon_i$ is small amplitude i.i.d. noise.