Question 5.67: The differential equation y″ + y = 0, with initial condition......
The differential equation y^{\prime \prime}+y=0, with initial conditions y(0)=0, y(h)=K, is solved by the Numeröv method.
(a) For which values of h is the sequence \left\{y_{n}\right\}_{0}^{\infty} bounded ?
(b) Determine an explicit expression for y_{n}. Then, compute y_{6} when h=\pi / 6 and K=1 / 2.
The Numeröv method
y_{n+1}-2 y_{n}+y_{n-1}=\frac{h^{2}}{12}\left(y_{n+1}^{\prime \prime}+10 y_{n}^{\prime \prime}+y_{n-1}^{\prime \prime}\right)
is applied to the equation y^{\prime \prime}=-y yielding
y_{n+1}-2 B y_{n}+y_{n-1}=0
where B=\left(1-\frac{5}{12} h^{2}\right) /\left(1+\frac{1}{12} h^{2}\right).
The characteristic equation is
\xi^{2}-2 B \xi+1=0
whose roots are \xi=B \pm \sqrt{B^{2}-1} .
(a) The solution y_{n} will remain bounded if
B^{2} \leq 1, \quad \text { or } \quad\left(1-\frac{5}{12} h^{2}\right)^{2} \leq\left(1+\frac{h^{2}}{12}\right)^{2} \quad \text { or } \quad-\frac{h^{2}}{6}\left(6-h^{2}\right) \leq 0 \text {. }
Hence, we obtain 0<h^{2} \leq 6.
(b) Since, |B| \leq 1, let B=\cos \theta. The roots of the characteristic equation are given by \xi=\cos \theta \pm i \sin \theta, and the solution can be written as
y_{n}=C_{1} \cos n \theta+C_{2} \sin n \theta.
Satisfying the initial conditions, we obtain
\begin{aligned} & y_{0}=C_{1}=0, \\ & y_{1}=K=C_{2} \sin \theta, \quad \text { or } \quad C_{2}=K / \sin \theta. \end{aligned}
We have y_{n}=K \frac{\sin n \theta}{\sin \theta}.
For n=6, h=\pi / 6 and K=1 / 2, we have
B=\cos \theta=\left[1-\frac{5}{12} \cdot \frac{\pi^{2}}{36}\right] /\left[1+\frac{1}{12} \cdot \frac{\pi^{2}}{36}\right]=0.865984,
and \theta=0.523682.
Hence, y_{6}=\frac{1}{2} \frac{\sin 6 \theta}{\sin \theta} \approx-0.0005.