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,
$$
A=\begin{pmatrix}
\dfrac{c_1(c_1/2-c_2)}{c_1-c_2}&\dfrac{c_1^2}{2(c_1-c_2)}\\[6pt]
-\dfrac{c_2^2}{2(c_1-c_2)}&\dfrac{c_2(c_2/2-c_1)}{c_2-c_1}
\end{pmatrix},
\qquad
b=\begin{pmatrix}
\dfrac{1/2-c_2}{c_1-c_2}\\[5pt]
\dfrac{1/2-c_1}{c_2-c_1}
\end{pmatrix}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/2/a/solution