Methods of solution by Green's function are developed for the design of two-dimensional unbranched channels with prescribed velocities as a function of arc length along the channel walls. The methods apply to incompressible and linearized compressible, nonviscous irrotational flow. One numerical example is presented for an accelerating elbow shape obtained from the solution by Green's function is the same as that obtained from a solution by relaxation methods for the same prescribed conditions. The time required for the calculations is considerably less for solutions by Green's functions.