## Problem

The problem is a heat conduction problem:

$$frac{partial U}{partial t}= frac{partial^2u}{partial x^2}, 0<x<1,, , t>0 , ,, (1)$$

$$u(0,t) = 0, , , frac{partial u}{partial x}(1,t) = 0, , , t>0$$

$$u(x,0) = x^2 – 2x, , , 0 le x le 1$$

My task is to derive an explicit numerical method for this **heat conduction problem**

## My attempt

I used the Finite Difference Method. By using Forward Euler Approximation I arrived at:

$$ k = 1/N, h = 1/J$$

$$frac{partial u}{partial t}(x_j, t_n) approx frac{U_{j, n+1}- U_{j, n+1}}{k}$$

$$frac{partial ^2 u}{partial x^2}(x_j, t_n) approx frac{U_{j, n} – 2U_{j,n}+ U_{j-1, n}}{h^2}$$

By using $(1)$ I got:

$$U_{j, n+1} = (1-2r)U_{j,n} + r(U_{j+1, n} + U_{j-1, n}), , , r=frac{k}{h^2}$$

The condition become:

$$ U_{j, 0} = x_j^2 – 2x_j$$

$$U_{0, n} = 0$$

However how can I use this? $$frac{partial u}{partial x}(1,t) = 0$$

My attempt at using this by centered difference approximation :

$$frac{partial u}{partial x}(1,t) = frac{U_{2,n}-U_{0,n}}{2h} = 0$$

However I do not know how to proceed from here. My goal is to just derive the equations for the difference scheme. I feel like I am only missing the use of the Neumann Condition. All advice will be much appreciated.