Chapter 2, Problem 2E

Use the Galerkin method to solve the following boundary-value problem using a: (a) one-term approximation and (b) two-term approximation. Compare your results with the exact solution by plotting them on the same graph.

$\begin{array}{l}\frac{d^{2}u}{d{x}^2}+x^2=0,0\le x\le 1\\ \begin{array}{c}u\left(0\right)=1\\ u\left(1\right)=0\end{array}\}\text{ Boundary conditions}\end{array}$

Hint: Use the following one- and two-term approximations

One-term approximation:

$\begin{array}{l}\tilde{u}\left(x\right)=\left(1-x\right)+c_1{\varphi }_1\left(x\right)\\ =\left(1-x\right)+c_{1}x\left(1-x\right).\end{array}$

Two-term approximation:

$\begin{array}{l}\tilde{u}\left(x\right)=\left(1-x\right)+c_1{\varphi }_1\left(x\right)+c_2{\varphi }_2\left(x\right)\\ =\left(1-x\right)+c_{1}x\left(1-x\right)+c_2{x}^2\left(1-x\right).\end{array}$

The exact solution is $u\left(x\right)=1-\frac{x\left(x^2+11\right)}{12}$

The approximate solution is split into two parts. The first term satisfies the given essential boundary conditions exactly, i.e., $u\left(0\right)=1$ and $u\left(1\right)=1$ .The rest of the solution containing the unknown coefficients vanishes at the boundaries.