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Let's just say that I am a beginning practitioner of mathematics who never gave physics a serious shot and looking to get an intuition of how partial differential equation works in relation to the physical words.

In particular, I am considering the classical heat equation $$ \partial_{t} u - \mathit{\Delta}u = 0 \ \ \text{in $\mathit{\Omega}$}, \ \ u = f \ \ \text{on $ \mathit{\partial \Omega}$} $$ with initial condition $u_0$ for functions $ u = u(t,x)$ and $f=f(x,t)$, where $t > 0 $ is the time variable and $ x \in \mathit{\Omega} \subseteq \mathbb{R^{d}} $ is a bounded domain.

I want to understand this equation in relation to how it works physically.

In relation to the distribution of heat, my intuition for such a model is that, given an object with constant conductivity, if we can somehow control the temperature of the object on the boundary for all time (boundary condition), then based on the information at present (initial profile), we should be able to deduce the temperature inside the object for all time as well.

But how does such set up come in handy in real life? Intuitively, even if I take just a metal ball $B \subseteq \mathbb{R}^{3}$, and then heat up a small area on the surface of my metal ball. Intuitively I should be able to just sit there and wait it out in order to get a unique solution. However, the theory of heat equation says that if I do not already know how the flow will behave on the boundary, then I cannot expect to get a unique solution. But surly, all I had to do was just wait for the flow to go over the metal...

Some of my thoughts about this are the folloings:

  1. The environment will change and the heat could escape the boundary in certain way. Thus I need to know how the heat will behave when in contact with its surroundings in order to know how the interior behaves. However, this kind of thinking has a flaw, because if I take $\mathit{\Omega}$ to be say a closed loop $S^1$, then it will have no boundary and the theory says that I wouldn't have to take account of the boundary value.

  2. The heat flow is stochastic, meaning that if I fix everything else the same, every time I flow my heat, the distribution of heat through out the object as time increases will be different according to some probability density. This would make sense, because if the equation admits a solution in the probabilistic sense, then I need a priori a sample point, i.e. a solution to the heat equation. Restrict that sample point to the boundary, and then use the resulting function defined on the boundary to solve for the heat equation will then get me a unique solution.

But again I don't even know very well the first thing about physics. So I would appreciate if anyone could give a some clarification.

Many thanks in advance!

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    $\begingroup$ You consider the Dirichlet problem, but you can also consider the Neumann problem. If a heat flux is given depending on the temperature at the boundary, then there is a unique solution. In 1 it is not clear what kind of question is there? $\endgroup$ – Alex Trounev Oct 25 '19 at 13:51
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    $\begingroup$ How do you think the heat conduction equation is used in actual practice? Suppose you need to design an actual process (involving lots of $ investment. Do you throw your hands up in the air and say it just can't be done? Or do you just build the full-scale expensive system and hope it works? Or what? $\endgroup$ – Chet Miller Oct 25 '19 at 13:58
  • $\begingroup$ I suppose with a lot of expenses one could always try to build up a system controlling which ever conditions that need to be imposed. $\endgroup$ – Meagain Oct 25 '19 at 14:00
  • $\begingroup$ Do you mean without even doing any design calculations to decide whether you allowed a large enough system or enough time for the required amount of heat to be transferred? $\endgroup$ – Chet Miller Oct 25 '19 at 14:17
  • $\begingroup$ Obviously, you will need to calculate it out first. $\endgroup$ – Meagain Oct 25 '19 at 14:18
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As a fan of heat equation ,some times called heat transfer by conduction ,I know with some kind of certitude a few axioms that may help: i-The heat equation is some sort of the diffusion equation where the nature of the problem, medium properties such as diffusion combined with the boundary conditions often allow the simple solution by separation of variables x,t. ii-The solution to the physically defined heat partial differential equation with Dirichlet boundary conditions and zero source term (as the question asks) exists uniquely .Of course this spatio-temporal evolution is deterministic not probabilistic. iii- Recently,there are many analytic and numeric trials (some of them use stochastic methods ) to solve the heat equation in the general case (i.e. with heat source term distribution inside the heated object,such as microwave oven,).

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