Method Of Characteristics-PDE

Question

I am haveing trouble understanding the method of characteristics. I understand the derivation of the fact that the system $ \frac{dx}{dt}=a $,... gives us the characteristic curves and I understand what they represent.

My question is (geometrically): Why do we need to add an initial condition to these equations? i.e. - why isn't the system above enough ? (I think it is because we need the charactistic curves not to intersect each other, but I can't understand why the initla condition helps us in such a case... Can't we have the intersection in a point that is not on the initial curve?)

Thanks in advance

Answer 1

You did not state what equation you are solving, but since you mention ``initial curve'', I guess you mean a single pde like $au_x+bu_y=c$, with data $u(x,y)$ specified on some initial curve. We know from the chain rule that along any other curve where $\frac{dx}{dt}=a$ and $\frac{dy}{dt}=b$, you will have $\frac{du}{dt} = c$. So, the point of the method is to use the initial data to determine the characteristic curves, and thus find out the values of $u$ along the characteristics. That is why you add the initial condition, so that $u$ will hopefully be determined from the data.

But, if two characteristics intersect at some point not on the initial curve, as you mentioned, then you have a value for $u$ at that point coming from two directions, which means trouble because the two values probably do not match. This is unavoidable with many pde, so it requires more advanced analysis to find out what happens then.