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?

  • $\begingroup$ 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

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: $U(x,t)=\sum_{n=0}^\infty c_n U_n(x,t)$.

  • $\begingroup$ 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

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