Chapter 2, Problem 7E

Solve the one-dimensional heat conduction problem 6 using the Rayleigh-Ritz method. For the heat conduction problem, the total potential can be defined as

$\pi ={\int }_0^L\left[\frac{1}{2}K{\left(\frac{dT}{dx}\right)}^2-QT\right]dx.$

Use the approximate solution $T\left(x\right)=T_1{\varphi }_1\left(x\right)+T_2{\varphi }_2\left(x\right)+T_3{\varphi }_3\left(x\right)$, where the trial functions are given in eq. (2.37) with $N_p=3$ and $x_1=0,x_2=\frac{L}{2}$, and $x_3=L$. Compare the approximate temperature with the exact one by plotting them on a graph.