If the coefficients of the PDE $$\partial_t u = a(x,t)\partial_{xx}u(x,t) + b(x,t)\partial_x u(x,t) + c(x,t)u(x,t) + d(x,t)$$ are in some Hölder space, apparently we can solve this via separation of variables. But how can this be? For example $a(x,t)$ could be something horrible so we can't get terms depending on $x$ on one side and terms depending on $t$ on another side. So what is meant by separation of variables in this context? How does it work?
Separation of variables to solve PDE with variable coefficients
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2Do you have a reference to this statement of yours? Where did you learn this? – 2012-08-12