Wiener–Hopf method

The Wiener-hopf technique was developed to solve Fourier transform problems with mixed boundary conditions. For example, consider the linear PDE

${\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}=y{\frac {\partial f}{\partial y}}$

subject to the $y=0$ mixed boundary conditions

$f=0$ for $x\geq 0$, $f_{y}=0$ when $x<0$.