Chapter 2, Problem 6E

A one-dimensional heat conduction problem can be expressed by the following differential equation: $k\frac{d^{2}T}{d{x}^2}+Q=0,0\le x\le L$. where k is the thermal conductivity, $T\left(x\right)$ is the temperature, and Q is heat generated per unit length. Q, the heat generated per unit length, is assumed constant. Two essential boundary conditions are given at both ends: $T\left(0\right)=T\left(L\right)=0$. Calculate the approximate temperature $T\left(x\right)$ using the Galerkin method. Compare the approximate solution with the exact one. Hint: Start with an assumed solution in the following form: $\tilde{T}\left(x\right)=c_0+c_{1}x+c_2{x}^2$, and then make it satisfy the two essential boundary conditions.