Darboux problem

For the hyperbolic partial differential equation u_(x y) | = | F(x, y, u, p, q) p | = | u_x q | = | u_y on a domain Ω, Goursat's problem asks to find a solution u(x, y) of (-1) from the boundary conditions u(0, t) | = | ϕ(t) u(t, 1) | = | ψ(t) ϕ(1) | = | ϕ(0) for 0<=t<=1 that is regular in Ω and continuous in the closure Ω^_, where ϕ and ψ are specified continuously differentiable functions.

boundary value problem | function | hyperbolic partial differential equation | Riemann method