For $(t,x)\in[0,T]\times[0,X]$ let there be a partial differential equation (PDE) of the generic form $$\partial_ty=f(t,x,y,\partial_xy)$$ along with some initial condition (i.c.) $$y_i(x):=y(0,x)$$ but no boundary condition (b.c.).
The above initial-boundary value problem (IBVP) is profoundly underdetermined.
However one may come to think that for some infinitesimal $h$ there are two distinct approximations for the $x$-partial derivative: $$\partial_xy\approx\frac{y(t,x+h)-f(t,x)}{h}\qquad\qquad\partial_xy\approx\frac{y(t,x)-f(t,x-h)}{h}$$
Based on these approximations one can
- evaluate $\partial_xy(0,x)$ for any $x\in[0,X]$ and then
- evaluate $\partial_ty(0,x)$ for any $x\in[0,X]$ and then
- evaluate $y(0+dt,x)=y(0,x)+dt\cdot \partial_ty(0,x)$
In this way one is able to construct a unique solution $y(t,x)$ although the above IBVP is underdetermined. It is as if the missing boundary condition $y(t,X)$ is constructed in parallel with the evolution of initial data.
I would appreciate any recommendations of bibliography or ideas concerning this kind of integration.