Can I use Separation of Variables like this? (3 terms)

[TylerH]

$$\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$$
$$\mbox{Let }z=T(t)X(x)Y(y)$$
$$T'(t)X(x)Y(y)=T(t)X'(x)Y(y)+T(t)X(x)Y'(y) \Rightarrow$$
$$\frac{T'(t)}{T(t)}=\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)} \Rightarrow$$
$$\frac{T'(t)}{T(t)}=A \wedge \left(\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)}=B \rightarrow \frac{X'(x)}{X(x)}=C \wedge \frac{Y'(y)}{Y(y)}=D \right)$$

... and so on.

 

[LCKurtz]

Yes, separation of variables like that sometimes works. For example, the 2d wave equation for a vibrating drumhead is solved that way. Remember, when you separate the variables like that it will only work if the PDE happens to have solutions like that. So you can always try. Maybe it will work on a particular problem and maybe it won't.

 

[TylerH]

How about this?

$$f(x) \frac{\partial z}{\partial x} = g(y) \frac{\partial z}{\partial y}$$
$$z=X(x)Y(y)$$
$$f(x)X'(x)Y(y)=g(y)X(x)Y'(y)$$
$$\frac{f(x)X'(x)}{X(x)}=\frac{g(y)Y'(y)}{Y(y)}$$
$$\frac{f(x)X'(x)}{X(x)}=A \wedge \frac{g(y)Y'(y)}{Y(y)}=B$$
... and so on.

 

[TylerH]

... anyone?

 

[LCKurtz]

How about this?

$$f(x) \frac{\partial z}{\partial x} = g(y) \frac{\partial z}{\partial y}$$

In this case the left side can't depend on y and the right on x so they are both constant. If f and g are not zero then z_x=C/f(x) and z_y=C/g(y).

Integrating the first gives z(x,y) = F(x) + G(y) where F is the antiderivative of C/f(x).
Differentiating this with respect to y gives G'(y) = C/g(y) so your solution is

z(x,y)=F(x)+G(y) +D

 

[TylerH]

OH! I was going about it the wrong way, but with the right technique, I see.

Can you, please, give an example of a PDE that can't be solved by separation (or the general form of one)?

Thanks for your help.

 

[LCKurtz]

OH! I was going about it the wrong way, but with the right technique, I see.

Can you, please, give an example of a PDE that can't be solved by separation (or the general form of one)?

Thanks for your help.

There is no reason to expect most pde's are solvable that way. Most can't be solved analytically in the first place. I don't think there is a general form for such but likely most any random PDE you might write down wouldn't be solvable that way, or any other way except numerically. Here's a simple first order one I just made up:

u_x + u_y = sin(xy²)

I don't know it can't be solved by separation of variables, but I doubt it. In fact, I don't know whether it can be solved analytically at all.