I have the following PDE:
$k_{xx}(x, y) - k_{yy}(x, y) = \lambda(y)k(x, y)$
$k(x, 0) = 0$
$k(x, x) = -\frac{1}{2} \int_0^x \lambda(y) dy$
defined on the triangle $0 < y \leq x \leq 1$. ($k_{xx}$ is second derivative with respect to x) I am trying to derive a finite difference scheme to implement in some numerical simulations. Naturally, I obtain the following:
$k_{xx} = \frac{k_{i+1, j} -2k_{i, j} + k_{i-1, j}}{(dx)^2}$
$k_{yy} = \frac{k_{i, j+1} -2k_{i, j} + k_{i, j-1}}{(dy)^2}$
Assuming an equal dx = dy, we obtain:
$k_{i, j} = \frac{k_{i+1, j} + k_{i-1, j} - k_{i, j+1} - k_{i, j-1}}{(dx)^2 \lambda_j}$
However, we cannot calculate the point $k_{i+1, j}$ since we have no boundary condition on that side of our triangle. Does anyone know any schemes or ways to solve this condition? I know these PDEs can be solved via finite differences, but cannot find an exact approach.