Chapter 2, Problem 17E

Consider a finite element with three nodes, as shown in the figure. When the solution is approximated using $u\left(x\right)=N_1\left(x\right)u_1+N_2\left(x\right)u_2+N_3\left(x\right)u_3$, calculate the interpolation functions $N_1\left(x\right),N_2\left(x\right),$ and $N_3\left(x\right)$. Hint: Start with an assumed solution in the following form: $u\left(x\right)=c_0+c_{1}x+c_2{x}^2$.