What do you call a more general boundary condition?

[Unrest]

Is there a term to describe something like a boundary condition but which can be applied within the domain, not just on the boundaries?

For example in a heat transfer problem you might specify a constant rate of heat generation in some region. Is that still called a boundary condition?

Ideally I'd like a term which also includes initial conditions, or things like initial conditions but which are applied over some arbitrary time interval, not just t=0.

 

[AlephZero]

For example in a heat transfer problem you might specify a constant rate of heat generation in some region. Is that still called a boundary condition?

If something is a term in the ODE or PDE that you are solving, it is not a boundary condition. The bounary conditions are the additional pieces of information that turn the general solution of the DE (containing arbitrary constants or arbitrary functions) into the unique solution to a particular problem.

A constant rate of heat generation, specified as an explicit function of position and/or time, is a term in the PDE, so it's not a boundary condition.

Such things are sometimes called "external" forces or heat fluxes or whatever, or "forcing functons", or even "right hand side functions" because conventionally they are often written on the right hand side of the equation.

Ideally I'd like a term which also includes initial conditions, or things like initial conditions but which are applied over some arbitrary time interval, not just t=0.

The initial conditions are part of the boundary conditions. The question of when you have to specify boundary conditions to get a unique solution depends on whether the DE is elliptic, parabolic, or hyperbolic, among other things.

 

[Unrest]

A constant rate of heat generation, specified as an explicit function of position and/or time, is a term in the PDE, so it's not a boundary condition.

Excuse me if I'm a bit mixed up here. I don't work directly with DEs...

Maybe what counts as "a term in the PDE" depends what you say the equation is? Suppose it's:

k d²T/dx² + q^. = 0

If we fix q^.=0 then we can apply two T=constant BCs and determine the T field.

But what if we fix T over part of the domain and leave q^. unknown, while we fix q^.=0 over the rest of the domain. Which of those are the BCs? Is that what the predetermined T field is, even though it's not just on the boundaries?

The initial conditions are part of the boundary conditions. The question of

I have a (free) book which says that BCs apply at t>0. And the wording implies that ICs are not a special case of BCs. Maybe this is just a personal choice? Or depends if you're a mathematician or an engineer?

 

[AlephZero]

Suppose it's:

k d²T/dx² + q^. = 0

If we fix q^.=0 then we can apply two T=constant BCs and determine the T field.

I don't think we have any disagreements about that.

But what if we fix T over part of the domain and leave q^. unknown, while we fix q^.=0 over the rest of the domain.

The way I would think about that is to say that you have two different equations on the two domains. In one equation the unknown variable is q, on the other one it is T.

So you need a complete set of boundary conditions for each equations. Of course on the interface between the domains, the boundary conditions presumably say that q and T (and possibly some of their derivatines as well) are consistent at the common boundary. That really means you have a coupled system of two equations, though it's a rather unusual type of coupled system.

I have a (free) book which says that BCs apply at t>0. And the wording implies that ICs are not a special case of BCs. Maybe this is just a personal choice?

It's hard to make any comment on that without knowing the context. FWIW I would have said the description "initial conditions" is just a subset of the more general term "boundary conditions".

If you are talking about the heat flow or diffusion equation, there is no reason why you must have explicit initial conditions of the form "the temperature everywhere in the region at time 0 is given by the function T(x,y,z)." You could have conditions like "the temperature distribution at time 0 is the same everywhere as the temperature distribution at time 1, and the mean temperature averaged over the whole region and between times 0 and 1 is zero". In other situations you may have boundary conditions like "the temperature is always finite", to exclude solutions where the temperature is proportional to 1/t when the time t is small, or temperatures that grow exposnentially as the time goes to infinity.

Or depends if you're a mathematician or an engineer?

Hm... I have a math degree, but I spent all my working life in engineering companies. So I don't know how to answer that!

 

[Unrest]

If you are talking about the heat flow or diffusion equation, there is no reason why you must have explicit initial conditions of the form "the temperature everywhere in the region at time 0 is given by the function T(x,y,z)." You could have conditions like "the temperature distribution at time 0 is the same everywhere as the temperature distribution at time 1, and the mean temperature averaged over the whole region and between times 0 and 1 is zero".

OK, and you can't have an IC like "The temperature everywhere at all times from 0 to 1 is some T(x,y,z). ? If you impose that, then (except for special cases) you have to change the equation (or make it coupled equations?) so the rate of heat generation within the domain is no longer fixed, but becomes an unknown to be solved for.

Let met just try to imagine out loud, how does this sound?
In a transient heat flow problem with a 3D solid object:
- A BC could be a temperature distribution imposed on a 2D surface for a continuous period of time.
- An IC could be a temperature distribution on a 3D region at a point in time.
- A strange-non-BC-equation-changer could be a temperature distribution T(x,y,z,t) that's imposed over a continuous region of both space and time. Effectively making the problem already solved in that region, and at the same time changing the equation to allow the unknown heat generation that's required to maintain that temperature distribution.

If it were a 1D problem we can think of it as a rectangular region with coordinates (x,t). BCs and ICs become indistinguishable and are applied only along 1D lines or curves in this rectangle or on its edges. We can't impose a temperature over any 2D area without changing the equation.