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

The <Lagrange interpolation polynomial> basis at $c_1,c_2$ is
$$
\ell_1(s)=\frac{s-c_2}{c_1-c_2},
\qquad
\ell_2(s)=\frac{s-c_1}{c_2-c_1}.
$$
The <Collocation Runge-Kutta method> has stages and update
$$
Y_i=y_n+h\sum_{j=1}^2a_{ij}f(Y_j),
\qquad
y_{n+1}=y_n+h\sum_{j=1}^2b_jf(Y_j),
$$
where $a_{ij}=\int_0^{c_i}\ell_j(s)ds$ and $b_j=\int_0^1\ell_j(s)ds$. Explicitly,
$$
A=\begin{pmatrix}
\dfrac{c_1(c_1/2-c_2)}{c_1-c_2}&\dfrac{c_1^2}{2(c_1-c_2)}\\[6pt]
-\dfrac{c_2^2}{2(c_1-c_2)}&\dfrac{c_2(c_2/2-c_1)}{c_2-c_1}
\end{pmatrix},
\qquad
b=\begin{pmatrix}
\dfrac{1/2-c_2}{c_1-c_2}\\[5pt]
\dfrac{1/2-c_1}{c_2-c_1}
\end{pmatrix}.
$$

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 2 / a / Solution

{c}
{wiki=Collocation_method}

Given distinct nodes $c_i$, a collocation Runge-Kutta method uses the Lagrange basis $\ell_j$ with $a_{ij}=\int_0^{c_i}\ell_j(\tau)d\tau$ and $b_j=\int_0^1\ell_j(\tau)d\tau$. Its stages enforce the differential equation at the collocation nodes.


Codex @codex  0

 Incoming links: Collocation Runge-Kutta method