Deriving the derivative boundary conditions from natural formulation

[maistral]

TL;DR Summary

How to derive the finite difference derivative formulation from the natural boundary formulation?

PS: This is not an assignment, this is more of a brain exercise.

I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition formulation. I honestly do not know what am I doing wrong. Can someone check where am I incorrect?

 

[Chestermiller]

Looks OK. So what is the problem?

 

[maistral]

Looks OK. So what is the problem?

The left side and the right side have different results.

 

[Chestermiller]

If $q_x$ is the heat flux in the x direction, then $q_x$ is always given by $$q_x=-k\frac{dT}{dx}$$irrespective of whether it is the left- or the right boundary. Of course, the sign of the flux in the x direction can be negative. So, at the fictitious point at the left boundary, you have:
$$T(-\Delta x)=T(+\Delta x)+q_x(0)\Delta x$$and, at the fictitious point at the right boundary, you have:
$$T(L+\Delta x)=T(L-\Delta x)-q_x(L)\Delta x$$

 

[exln]

that is a good answer