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:
https://mathematica.stackexchange.com/questions/13795/problem-with-ndsolve-for-a-system-of-equations
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.
Best Answer
I believe that your intuition that you need two boundary/initial conditions per derivation degree and variable is actually quite helpful for elliptic and parabolic PDEs. However, in the case of hyperbolic PDEs in an infinite domain, you needn't set BCs. Intuitively, this is related to the fact that in hyperbolic systems local perturbations are "felt" or propagated only locally, but in elliptic or parabolic systems they are propagated instantaneously to the whole domain, so local hyperbolic solutions "don't care" about what is happening non-locally.
However, note that in a finite domain, the wave equation does require BCs in addition to the ICs you wrote (as stated in the Cauchy problem). For example, $u(0,t)=u(L,t)=0$ is typical, and it would comply with your rule. This is done by using the d'Alembert formula and judiciously extending f and g outside the solution domain. This example insinuates that intuitions break down at infinity.