Solve initial value problem

Question

I have an exercise that I do not understand. We have to solve an initial value problem:

$$ \begin{array}{ccl} y'(t) &=& f(t,y(t)) \\ y(a) &=& y_0 \end{array} $$

We have to derive an effective method to calculate $\alpha$, $\beta$ and $\gamma$ with the highest possible accuracy. The method has the form below: $$ y_{k+1} = \alpha y_k + \beta y_{k-1} + \gamma h f(t_k, y_k) $$ We can even assume that $y_1 = y(a+h)$ and that the interval is equidistant.

I can guess that we have to derive a Euler forward, but I cannot solve it? Besides, they ask the order of accuracy of the found method. Shall I write all the $y(a+h)$ as a Taylor polynomial? I would greatly appreciate some guidance.

Answer 1

For the test problem $f(t,y)=y$ you get the exact solution $y=Ce^x$ and with the constant chosen so that $y_k=1$ the equation reduces to $$ e^h=α+βe^{-h}+γh $$ which has to be correct up to some power of $h$.

Inserting the power series of the exponential and comparing coefficients of equal degree gives \begin{align} γh&=(1+h+\tfrac12h^2+\tfrac16h^3+...)-α-β(1-h+\tfrac12h^2-\tfrac16h^3\pm...) \\[1em]\hline 0&=1-α-β\\ γ&=1+β\\ 0&=\tfrac12(1-β)\\ 0&=\tfrac16(1+β) \end{align} which have to be satisfied from top down as far as possible. Obviously, the last equation contradicts the next-to-last one. Thus $β=1$, $γ=2$ and $α=0$.

You will need to confirm that this holds true for any differential equation, not just this (quite standard) test case.