**1**. Take the problem to be:

(*) [$]u_t = \rho \, u_{xy}[$] with

bc: u is given on two adjacent edges and

ic: [$]u(t=0,x,y)=u_0(x,y)[$] is given (arbitrary, as long as compatible with the bc).

**2**. Take the Laplace transform with respect to time of (*), with [$]U(x,y;s) = \int_0^{\infty} e^{-s t} u(t,x,y) \, dt[$]. This yields

(**) [$]U_{xy} + c \, U = f(x,y)[$], where [$]c = s/\rho[$] and [$]f = -u_0/\rho[$], and where the function [$]U(x,y)[$] is known on two adjacent edges.

**3**. So, that's a Goursat problem again, and the solution to (**) is given at the link or various books in terms of the edge and initial functions, using the Riemann function [$]R = J_0(\sqrt{4 c (x-\xi)(y-\eta)})[$].

**4**. Do a numerical Laplace inversion w.r.t. [$]s[$] of the solution in Step

**3**to get numbers.

Or don't, as the argument shows what bc and ic's `work', which was actually the original question.