The nonlinear heat equation with state-dependent parameters and its connection to the Burgers' and the potential Burgers' equation

Abstract

In this work the stability properties of a nonlinear partial differential equation (PDE) with state-dependent parameters is investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (Potential) Burgers' Equation. We show that for certain forms of coefficient functions, the PDE converges to a stationary solution given by (fixed) boundary conditions that make physical sense. We illustrate the results with numerical simulations.

Optimal boundary control of a contact thawing process for foodstuff

“…For freezing, (5) has extensively been studied, for example in Backi et al (2015), where a stability investigation as well as an observer design have been conducted, and in Backi et al (2014), where in addition similarities to the (potential) Burgers' equation were presented.…”

Estimation of inner-domain temperatures for a freezing process

“…The 1-dimensional version of this parabolic PDE has been described earlier in [4], where the PDE was subject to an optimal control problem in order to find an optimal boundary input. Further it has been described in [2], where stability in terms of L 2 -and H 1 -norms has been proven for classes of input and parameter distribution functions.…”

“…In [3] the composition of fish tissue as an alloy of different substances is introduced. In [4] and [2] different continuous representations of the parameter functions have been defined, which take the phenomenon of thermal arrest caused by latent heat of fusion into account. This is achieved by defining a temperature range ±∆T around the freezing point T F where c(T ) is increased significantly in order to slow down heat conduction.…”

“…This is achieved by defining a temperature range ±∆T around the freezing point T F where c(T ) is increased significantly in order to slow down heat conduction. The parameter functions in [2] are the basis for this paper and illustrated in Figure 2 for completeness. We remark that ρ(T ) is assumed constant over the considered temperature range.…”

“…3 the design relies on a nonlinear model running in parallel to the plant. Both, plant and model base on the same spatially discretized equations of the PDE (2) and are fed with the same inputs, namely the boundary conditions at x = 0, x = (T Ammonia ) and y = 0 (T Air ). The boundary conditions at y = h are directly embedded in the spatially discretized equations of the plant and the model.…”