5
$\begingroup$

A common step in solving many PDEs is to write a multivariable solution function as a product of two more or single variable functions. For example, if given,

$$ \alpha\,\frac{\partial U(x,t)}{\partial x} = \frac{\partial U(x,t)}{\partial t}, $$

one may begin be writing,

$$ U(x,t) = X(x) T(t). $$

My question is, what are the necessary conditions for writing a multi variable function as a product of several single variable functions? Furthermore, are there any cases where this cannot be done?

$\endgroup$
  • $\begingroup$ Pretty sure you could say that if your pde looks like this (just taking two independent variables - the generalization would work as well): $$f_1(x,U,U_x,U_{xx},\cdots)+f_2(x,U,U_x,U_{xx},\cdots)+\cdots+f_n(x,U,U_x,U_{xx},\cdots)+g_1(y,U,U_y,U_{yy},\cdots)+g_2(y,U,U_y,U_{yy},\cdots) +\cdots+g_m(y,U,U_y,U_{yy},\cdots)=0,$$ those would be sufficient conditions. I'm less sure about necessary conditions. You might also need the pde to be linear. $\endgroup$ – Adrian Keister Sep 25 at 19:10
  • $\begingroup$ There are certainly loads of non-separable pde's out there. $\endgroup$ – Adrian Keister Sep 25 at 19:12
2
$\begingroup$

Of course most two-variable functions $U(x,t)$ cannot be written as $X(x) T(t)$. Those that can are, well, those that happen to be of that form...

The point in the context of PDEs like the heat equation is that you want to find a whole sequence of such separable solutions $U_n(x,t)=X_n(x) T_n(t)$ that you can use in order to write the general solution as a linear combination of these simple solutions: $U(x,t)=\sum_{n=0}^\infty c_n U_n(x,t)$.

$\endgroup$
  • $\begingroup$ Yes, but it is important to note that the linearity of the differential operators are important for this to work. $\endgroup$ – mathreadler Sep 26 at 13:07
  • $\begingroup$ @mathreadler: Certainly! $\endgroup$ – Hans Lundmark Sep 26 at 13:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.