Sample Questions

1.

State and prove the classical Peano existence theorem for

$$y' = f(t,y),\ y(t_0) = y_0,$$

where f is continuous.

2.

State and prove the classical Cauchy existence and uniqueness theorem for

$$y' = f(t,y),\ y(t_0) = y_0,$$

where f and $\partial f/ \partial y$ are continuous.

3.

Verify that the system

$$\begin{eqnarray*}x'&=& cos(x y) - x,\\ y'&=&-y + x^2 + 1 \end{eqnarray*}$$

has an equilbrium point at

$$(x,y)\doteq (0.632639, 1.4003)$$

and write down the linearization of the system around this equilibrium solution. Analyze the behavior of the linearization and relate it to the behavior of non-linear system near the equilibrium point.

4.

Consider the Runge-Kutta numerical scheme

$$\begin{eqnarray*}k_1&=&h f(x_n,y_n),\\ k_2&=&h f(x_n + h/2, y_n + k_1/2),\\ k_... ...+ k_3),\\ y_{n+1}&=&y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4). \end{eqnarray*}$$

(a)

Compute the local truncation error.

(b)

Count the number of function evaluations required for a single step of this scheme.

(c)

Make appropriate assumptions on the function f and estimate the global error involved in using this scheme to solve the IVP

$$y' =f(t,y),\ y(0)=y_0$$

on the interval [0, L].

(d)

Perform a stability analysis of the scheme.

5.

Consider the linear multistep method

$$y_{n+1} =y_n + \frac{h}{12}( 23 y'_n - 16 y'_{n-1} + 5 y'_{n-2} ).$$

(a)

Compute the local truncation error.

(b)

Count the number of function evaluations required for a single step of this scheme.

(c)

Make appropriate assumptions on the function f and estimate the global error involved in using this scheme to solve the IVP

$$y' = f(t,y),\ y(t_0) = y_0$$

on the interval [0, L].

6.

Give an example of a stiff system of ODE. Explain why a general numerical scheme is unlikely to provide an accurate solution to a stiff system.