CS3331: Numerical Methods
Quiz 4
May. 30, 1997

1.

(3%) Let y' = y² + e^t. Write down the recurrent relationship between y_n+1 (=y(t_n+1)) and y_n (= y(t_n)) for each of the following methods:

(a)

Forward Euler method.

(b)

Modified Euler method.

(c)

Backward Euler method.

(You don't need to express y_n+1 as an explicit function of y_n and t_n.)

2.

(2%) The modified Euler method is derived by applying the trapezoidal rule to solution of y'=f(y,t):

$$y_{n+1} = y_n + \frac{\textstyle h}{\textstyle 2} [f(y_{n+1}, t_{n+1})+f(y_n, t_n)].$$

If we want to use the successive substitution method to find y_n+1, what is the range of h for convergence?

3.

(3%) Please fill up the following table in terms of the integration step h:

Forward Euler method Modified Euler method Backward Euler method Order of local error Order of global error

(RK method = Runge-Kutta method)

4.

(4%) Put the following higher-order ODEs into the standard format ${\bf y}' = {\bf f}({\bf y}, t)$, where ${\bf y}$ is a vector of variables and ${\bf f}$ is a vector of functions. (Remember to specify the initial conditions ${\bf y}(0)$.)

(a)

y'''(t)+2y''(t)+3y'(t)+4y(t)=g(t), where y(0)=y₀, y'(0)=y'₀, and y''(0)=y''₀.

(b)

$y''(t)+y'(t)+y(t)+\int_0^t y(s)ds=g(t)$, where y(0)=y₀, and y'(0)=y'₀.

5.

(4%) Let y' = f(y, t) = y²+e^t.

(a)

Express y'' in terms of y and t.

(b)

Express y''' in terms of y and t.

6.

(2%) Let y' = y²+e^t. What are the k₁ and k₂ (in terms of y_n and t_n) for the second-order Runge-Kutta method?

7.

(3%) Please fill up the following table in terms of the integration step h:

2nd-order RK method 3rd-order RK method 4th-order RK method Order of local error Order of global error

8.

(2%) What is the shooting method good for?

9.

The exact solution for $y'=-\alpha y$ is $y=y_0 e^{-\alpha t}$. Suppose that y₀ and $\alpha$ is positive, so y(t) is positive for all t.

(a)

(1%) Write down the forward Euler formula for $y'=-\alpha y$.

(b)

(2%) What is the range for h such that the y_n remains positive for all n? Does y_n converge or diverge for a given h in the range?

(c)

(2%) What is the range for h such that the sign of y_n alternates at each step, but still converge to as n goes to infinity?

(d)

(2%) What is the range for h such that the sign of y_n alternates at each step, and the absolute value of y_n diverge to infinity as n goes to infinity?