For the nodes $c_1=\alpha$ and $c_2=1$, the <Lagrange interpolation polynomial>[Lagrange basis] is
$$
\ell_1(t)=\frac{1-t}{1-\alpha},
\qquad
\ell_2(t)=\frac{t-\alpha}{1-\alpha}.
$$
Direct integration gives
$$
a_{ij}=\int_0^{c_i}\ell_j(t)\,dt
$$
and therefore
$$
A=\begin{pmatrix}
\dfrac{\alpha(2-\alpha)}{2(1-\alpha)}&-\dfrac{\alpha^2}{2(1-\alpha)}\\[6pt]
\dfrac1{2(1-\alpha)}&\dfrac{1-2\alpha}{2(1-\alpha)}
\end{pmatrix}.
$$
The <Collocation Runge-Kutta method> also requires
$$
b_j=\int_0^1\ell_j(t)\,dt,
\qquad
b^T=\left(\frac1{2(1-\alpha)},\frac{1-2\alpha}{2(1-\alpha)}\right).
$$
This exposes a sign error in the printed tableau: its lower-right entry is shown as $(-1+2\alpha)/[2(1-\alpha)]$. With $1-2\alpha$ in that position, the tableau is exactly the claimed collocation method. Taken literally, the printed weights satisfy $b^T\mathbf1=\alpha/(1-\alpha)$, so the method is not even consistent unless $\alpha=1/2$ and cannot be a collocation method for general $\alpha$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section A / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-341/section-a/2/a/solution