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It's a shame that, though I've taken the "Equations of Mathematical Physics" class for one semester and solved numbers of PDEs with Mathematica, I'm still unclear about how many initial conditions(ICs) or boundary conditions(BCs) are needed for getting the determine solution of a PDE or a set of PDEs: I never met a book which mentioned that.

I roughly know that there's a theorem for ODEs which tells us that for a n-th order ordinary differential equation, we need n ICs or BCs which have lower order than the ODE to get the determine solution, and usually it's the principle I followed when I tried to solve PDEs numerically with NDSolve in Mathematica (Of course in this case the number of IC or BC is considered seperately for every argument), but it's inaccurate, right? A popular case is the d'Alembert's formula, the 1D wave equation

$${ \partial^2 u \over \partial t^2 } = c^2 { \partial^2 u \over \partial x^2 } $$ with only 2 initial conditions $$u(x,0)=f(x) $$ $$u_t(x,0)=g(x) $$

gives the determine solution

$$u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds$$

while with my "principle" we need 4 ICs or BCs(2 for x, 2 for t).

I also encountered several equations that don't follow my "principle" when I wandered in Mathematica.SE, for example, it seems that this set of equations needs 6 BCs, but it can be solved with only 5 BCs in fact:

And this one, which actually needs only 4 BCs in total, no matter which variables the BCs are given to, while with my "principle", we need 1 for ρg, 1 for u, 2 for te:

So, as my title said, is there a theorem or something for the decision of the number of IC and BC? Any help would be appreciated.

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For equations that can be shown to be hyperbolic, you can use Cauchy-Kowalevski. But in general the number depends on the "type" of the equation, and there are very few general rules. For evolution equations a good rule of thumb (though not always accurate) is to pretend your equation is actually an ODE on infinite dimensional space and apply your intuition from ODE theory. – Willie Wong Jul 24 '13 at 7:58
At each instant of time, the solution belongs to a function space, which is infinite dimensional. But I do not have the time to explain in more detail. For your question of IC and BC, see, perhaps, pages 5 through 7 of this document. – Willie Wong Jul 24 '13 at 11:41
(BTW, that is from Olver's upcoming new book Introduction to Partial Differential Equations. ) – Willie Wong Jul 24 '13 at 15:35
See this extreme example you will understand for the solution(s) of a PDE exist, the proportion between "B.C.s" and "I.C.s" can even be not restricted, the nature of the solution(s) are only depending on the number of conditions. – doraemonpaul Jul 24 '13 at 23:47
@doraemonpaul In fact I was also planning to ask a question which might be titled as "Is there a mathematical distinction between IC and BC?", but now it seems to be not that necessary :) – xzczd Jul 25 '13 at 7:36

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