The governing differential equation for steady state one dimensional conduction heat transfer with internal heat generation is given by $\frac{d}{dx}[K \frac{dT}{dx}]=q \hspace{1cm} \text{for} \hspace{1cm} 0 \leq x \leq 1$ were develop the finite element termulation of linear element. use R-R method, mapped over general element.

Given D.E. $ \frac{d}{dx}[K \frac{dT}{dx}] = q $ for 0 $\leq$ x $\leq$ i.e. $ \frac{d}{dx}[K \frac{dT}{dx}] - q = 0 $

In local co-ordinates, replace $dx$ by $d \bar{x}$

$ \frac{d}{d \bar{x}}[K \frac{dT}{d \bar{x}}] - q = 0 $

With local boundary conditions as,

$ K \frac{dT}{d \bar{x}}|_{\bar{x}=0} = …