Problems

01

Runge-Kutta practice

Find an approximate solution to $y^{\prime }=5t-3\sqrt{y}$ with $y(0)=2$ using the standard fourth-order Runge-Kutta method described above, up to $t=0.2$, with a step size $h=0.05$.

02

Three-step Adams method

Compute the formula for the three-step Adams method.

03

Euler method accuracy: small enough step size $h$

For the IVP of Question 06-01, find (with justification!) a value of $h$ that is small enough to ensure that the Euler method gives a solution at $t=0.5$ which is accurate to within $0.001$ of the exact value.

04

Backward Euler method

There is a family of procedures called “backward differentiation methods.” The backward Euler method specifies $y_{n+1}$ implicitly by the following equation:

$$y_{n+1}=y_n+hf(t_{n+1},y_{n+1}).$$

It is called backward because the slope used here is the slope of the future step $t_{n+1}$. There are backward versions for all the Adams methods. (The forward Adams methods are also called Adams-Bashforth methods, while the backward versions are called Adams-Moulton methods.)

For this problem, use the backward Euler method to approximate the solution to the IVP $y^{\prime }=0.5-t+2y,y(0)=1$ up to $t=0.3$, with $h=0.05$. (See Boyce et al. text for examples if desired.)