# Laplace Equation -- Heated Plate -- Heat Flux Boundary Condition

I am having a problem with transferring the heat flux boundary conditions into a temperature to be able to put it into a matrix. I understand that deltat = deltax*q''/k but I do not know how to code it so that I can loop it into the matrix in MATLAB. I have not had heat transfer and it is a steady state problem, so it should be relatively simple.

How could you index code to be able to loop through since deltat = T(i,j) - T(I+1,J)?

Thank you for any help!

• Your second for-loop has a semicolon in it which will lead to problems Nov 29, 2017 at 11:59
• Could you please put the code in text form rather than as images? If you indent it with four spaces it will get syntax highlighting. Nov 29, 2017 at 14:59

The vertical lines are the faces of you cells and the circles are the centers of you cells. Your left boundary is located at the face at $x=0$ and your right boundary is located at the face at $x=L$. The domain of size $L$ is divided into $N$ cells. To simplify the boundary treatment, I have included two ghost / virtual nodes $i=0$ and $i=N+1$. I assume a uniform gridspacing $\Delta x=L/N.$
Now how do you impose a flux condition at $x=0$ in terms of local temperatures? Well from a Taylor expansion around $x=0$ we find: $$T_{i=0} = T_{x=0} - \left.\frac{dT}{dx}\right|_{x=0}\left(\frac{\Delta x}{2}\right) + \frac{1}{2}\left.\frac{d^2T}{dx^2}\right|_{x=0}\left(\frac{\Delta x}{2}\right)^2 + O\left(\Delta x^3\right)$$ and: $$T_{i=1} = T_{x=0} + \left.\frac{dT}{dx}\right|_{x=0}\left(\frac{\Delta x}{2}\right) + \frac{1}{2}\left.\frac{d^2T}{dx^2}\right|_{x=0}\left(\frac{\Delta x}{2}\right)^2 + O\left(\Delta x^3\right)$$ Subtracting: $$T_{i=1} - T_{i=0} = \left.\frac{dT}{dx}\right|_{x=0}\Delta x + O\left(\Delta x^3\right)$$ If $\Delta x \ll 1$ we can neglect the $O\left(\Delta x^3\right)$ terms and we find the flux at $x=0$ in terms of the local temperatures at $i=0$ and $i=1$: $$\left.q"\right|_{x=0}=-\lambda\left.\frac{dT}{dx}\right|_{x=0} = -\lambda\frac{T_{i=1} - T_{i=0}}{\Delta x}$$
So to impose the flux at $x=0$ you need to impose the local temperature of the ghost node at $i=0$ according to above equation.
A similar analysis can be done for the boundary at $x=L$, I'll leave that for OP.