Chapter 2, Problem 9E

Solve the differential equation in problem 8 for the following boundary conditions using the Galerkin method: $u\left(0\right)=1,u\left(1\right)=2$.

Assume the approximate solution as: $\tilde{u}\left(x\right)={\varphi }_0\left(x\right)+c_1\varphi \left(x\right)$, where ${\varphi }_0\left(x\right)$ is a function that satisfies the essential boundary conditions, and ${\varphi }_2\left(x\right)$ is the weight function that satisfies the homogeneous part of the essential boundary conditions, that is. ${\varphi }_1\left(0\right)={\varphi }_1\left(1\right)=0$. Hence, assume the functions as follows: ${\varphi }_0\left(x\right)=1+x,{\varphi }_1\left(x\right)=x\left(1-x\right)$.

Compare the approximate solution with the exact solution by plotting their graphs. The exact solution can be derived as: $u\left(x\right)=2.9231\sin x+\cos x-x$.