A heat transfer problem might be suitable here. Temperature is an intuitive parameter and a simple scalar, and energy balances are easy to write down but often challenging to solve analytically because of temperature-dependent material properties, for example.
I wrote here about the parabolic and catenary temperature profiles that arise in a simple 1-D geometry with heat generation. One practical application is described here: a microfabricated suspended silicon beam that heats up from an applied current, expands, and deflects—thus, a microscale linear actuator:
Let's consider the temperature profile only and forget about the motion. In the second link, I write about how the time-dependent analysis diverges from the experimental results because the analytical solution doesn't incorporate the temperature dependence of certain material properties.
As you request, there are at least two ways to obtain a more accurate temperature distribution numerically:
(1) One could use a lookup table for the temperature-dependent material properties (for simplicity, maybe just one material property, say, the thermal conductivity of silicon) and perform a 1-D finite-difference heat transfer analysis by discretizing the beam into segments, each with a uniform temperature.
(2) One could fit the temperature-dependent material property of interest by an analytical function and solve the resulting differential heat transfer equation (namely, $k(T)\frac{\partial^2T(x,t)}{\partial x^2}+J^2r=c\rho\frac{\partial T(x,t)}{\partial t}$, where $k$ is the thermal conductivity, $T$ is temperature, $x$ is the position, $t$ is time, $J$ is the current density, $r$ is the resistivity, $c$ is the heat capacity, and $\rho$ is the density) using a preferred numerical scheme. Or for simplicity, one could drop the time dependence and simply solve $k(T)\frac{\partial^2T(x,t)}{\partial x^2}+J^2r=0$. (Here, if $k$ were constant, we'd simply obtain a parabola.)