Fourier Transforms

Consider the PDE∂ 2 u∂x 2 ∂2 u∂y 2 0on the domain defined by − x and y ≥ 0. This equation is called thetwo-dimensional Laplace Equation and the domain is often referred to as the upper halfplane. The solution is required to satisfy ux,y gx on y 0,for some given functiongx, andux,y → 0asy → . The latter condition known as a Far-field or Boundednesscondition.The strategy is to take the FT of the PDE and the boundary condition. This gives an ODEthat we attempt to solve subject to the transformed boundary condition. It is hoped that thefinal solution can be obtained by inversion but from previous experience we know that anintegral equation may arise.As the boundary conditions are given in terms of y, wetaketheFTinx. Define uk,y −ux,y e ikx dx gk −gx e ikx dx uk,0 .The FT of the two terms in the PDE are−−∂ 2 u∂x 2 e ikx dx − k 2 uk,y using (3.2.4)∂ 2 u∂y 2 e ikx dx ∂2∂y 2−ux,y e ikx dx ∂2 u∂y 2 .We regard the resulting equation as one in which y is the variable and and k is a constant,so partial differentiation in y can be replaced by full differentiation. Thus we need to solvethe equationd 2 u − k 2 u 0,dy 2subject to uk,0 gk and uk,y → 0asy → .It may seem that the solution of this ODE is straightforward to obtain, since equations ofthis type are associated with expontentially increasing or decreasing solutions. However,the parameter k lies within the range − k and so there are three cases: k 0,k 0and k 0. The first and third can be combined by using k 2 |k| 2 and then the questionarises as to how k 0 should be accommodated.For k ≠ 0, the solution of this ODE is easily shown to be14