Chapter 3

Let us define $$ T_{n}(\cos \theta):=\cos n \theta, \quad n=0,1,2, \ldots, \quad-\pi \leq \theta \leq \pi $$ a Show that each \(T_{n}\) is a polynomial of degree \(n\) and that the \(T_{n}\) satisfy the three-term recurrence relation $$ T_{n+1}(t)=2 t T_{n}(t)-T_{n-1}(t), \quad n=1,2, \ldots $$ b Prove that \(T_{n}\) is an \(n\)th orthogonal polynomial with respect to the weight function \(\omega(t)=(1-t)^{-\frac{1}{2}},-1

Construct the Gaussian quadrature formulae for the weight function \(\omega(t) \equiv\) \(1,0 \leq t \leq 1\), of orders two, four and six.

Restricting your attention to scalar autonomous equations \(y^{\prime}=f(y)\), prove that the ERK method with the tableau \begin{tabular}{c|cccc} 0 & & & & \\ \(\frac{1}{2}\) & \(\frac{1}{2}\) & & & \\ \(\frac{1}{2}\) & 0 & \(\frac{1}{2}\) & & \\ 1 & 0 & 0 & 1 & \\ \hline & \(\frac{1}{6}\) & \(\frac{1}{3}\) & \(\frac{1}{3}\) & \(\frac{1}{6}\) \end{tabular} is of order four.

Derive the three-stage Runge-Kutta method that corresponds to the collocation points \(c_{1}=\frac{1}{4}, c_{2}=\frac{1}{2}, c_{3}=\frac{3}{4}\) and determine its order.