Initial Value Problem Review Questions

1. a) Derive the Euler approximation show it has a local truncation error of $O(h)$ of the Ordinary Differential Equation

(409)$$y^{{}^{\prime }}(x)=f(x,y),$$

with initial condition

(410)$$y(a)=\alpha .$$

[8 marks]

1. b) Suppose $f$ is a continuous and satisfies a Lipschitz condition with constant L on $D=\{(t,y)|a\le t\le b,-\mathrm{\infty }<y<\mathrm{\infty }\}$ and that a constant M exists with the property that

(411)$$|y^{{}^{″}}(t)|\le M.$$

Let $y(t)$ denote the unique solution of the IVP

(412)$$y^{{}^{\prime }}=f(t,y)a\le t\le by(a)=\alpha$$

and $w_0,w_1,...,w_N$ be the approx generated by the Euler method for some positive integer N. Then show for $i=0,1,...,N$

(413)$$|y(t_i)-w_i|\le \frac{Mh}{2L}|e^{L(t_i-a)}-1|$$

You may assume the two lemmas:\ If s and t are positive real numbers $\{a_i{\}}_{i=0}^N$ is a sequence satisfying $a_0\ge \frac{-t}{s}$ and $a_{i+1}\le (1+s)a_i+t$ then

(414)$$a_{i+1}\le e^{(i+1)s}(a_0+\frac{t}{s})-\frac{t}{s}$$

For all $x\ge 0.1$ and any positive m we have

(415)$$0\le (1+x{)}^m\le e^{mx}$$

[17 marks]

1. c) Use Euler’s method to estimate the solution of

(416)$$y^{{}^{\prime }}=(1-x)y^2-y;y(0)=1$$

at x=1, using $h=0.25$.

[8 marks]

2. a) Derive the difference equation for the Midpoint Runge-Kutta method\

(417)$$w_{n+1}=w_n+k_2$$

(418)$$k_1=hf(t_n,w_n)$$

(419)$$k_2=hf(t_n+\frac{1}{2}h,w_n+\frac{1}{2}{k}_1)$$

for solving the ordinary differential equation

(420)$$\frac{dy}{dt}=f(t,y)$$

(421)$$y(t_0)=y_0$$

by using a formula of the form

(422)$$w_{n+1}=w_n+a{k}_1+b{k}_2$$

where $k_1$ is defined as above,

(423)$$k_2=hf(t_n+\alpha h,w_n+\beta k_1)$$

and $a$, $b$, $\alpha$ and $\beta$ are constants are deteremined. Prove that $a+b=1$ and $b\alpha =b\beta =\frac{1}{2}$ and choose appropriate values to give the Midpoint Runge-Kutta method.

[18 marks]

2. b) Show that the midpoint Runge-Kutta method is stable.

[5 marks]

2. c)Use the Runge-Kutta method to approximate the solutions to the following initial value problem

(424)$$y^{{}^{\prime }}=1+(t-y{)}^2,2\le t\le 3,y(2)=1,$$

with $h=0.2$ with the exact solution $y(t)=t+\frac{1}{1-t}$.

[10 marks]

3. a) Derive the two step Adams-Bashforth method:

(425)$$w_{n+1}=w_n+(\frac{3}{2}hf(t_n,w_n)-\frac{1}{2}hf(t_{n-1},w_{n-1})),$$

and the local truncation error

(426)$${\tau }_{n+1}(h)=-\frac{5h^2}{12}{y}^3({\mu }_n).$$

[18 marks]

3. b) Apply the two step Adams-Bashforth method to approximate the soluion of the initial value problem:

(427)$$y^{\prime }=ty-y,(0\le t\le 2)y(0)=1.$$

Using $N=4$ steps, given that $y_1=0.6872$.

[15 marks]

4. a) Derive the Adams-Moulton two step method and its truncation error which is of the form

(428)$$w_0={\alpha }_0{w}_1={\alpha }_1$$

(429)$$w_{n+1}=w_n+\frac{h}{12}[5f(t_{n+1},w_{n+1})+8f(t_n,w_{n1})-f(t_{n-2},w_{n-2})]$$

and the local truncation error

(430)$${\tau }_{n+1}(h)=-\frac{h^3}{24}{y}^4({\mu }_n)$$

[23 marks]

4. b) Define the terms strongly stable, weakly stable and unstable with respect to the characteristic equation.

[5 marks]

4. c) Show that the Adams-Bashforth two step method is stongly stable.

[5 marks]

5. a) Given the initial value problem:

(431)$$y^{\prime }=f(t,y),y(t_0)=y_0$$

and a numerical method which generates a numerical solution $(w_n{)}_{n=0}^N$, explain what it means for the method to be convergent.

[5 marks]

5. b) Using the 2-step Adams-Bashforth method:

(432)$$w_{n+1}=w_n+\frac{3}{2}hf(t_n,w_n)-\frac{1}{2}hf(t_{n-1},w_{n-1})$$

as a predictor, and the 2-step Adams-Moulton method:

(433)$$w_{n+1}=w_n+\frac{h}{12}[5f(t_{n+1},w_{n+1})+8f(t_n,w_{n1})-f(t_{n-2},w_{n-2})]$$

as a corrector, apply the 2-step Adams predicitor-corrector method to approximate the solution of the initial value problem

(434)$$y^{\prime }=t{y}^3-y,(0\le t\le 2),y(0)=1$$

using N=4 steps, given $y_1=0.5$.

[18 marks]

5. c) Using the predictor corrector define a bound for the error by controlling the step size.

[10 marks]

6 a) Given the Midpoint point (Runge-Kutta) method

(435)$$w_0=y_0$$

(436)$$w_{i+1}=w_i+hf(x_i+\frac{h}{2},w_i+\frac{h}{2}f(x_i,w_i))$$

Assume that the Runge-Kutta method satisfies the Lipschitz condition. Then for the initial value problems

(437)$$y^{{}^{\prime }}=f(x,y)$$

(438)$$y(x_0)=Y_0$$

Show that the numerical solution $\{w_n\}$ satisfies

(439)$$\underset{a\le x\le b}{max}|y(x_n)-w_n|\le e^{(b-a)L}|y_0-w_0|+\left[\frac{e^{(b-a)L}-1}{L}\right]\tau (h)$$

where

(440)$$\tau (h)=\underset{a\le x\le b}{max}|{\tau }_n(y)|$$

If the consistency condition

(441)$$\delta (h)\to 0\text{ as }h\to 0$$

where

(442)$$\delta (h)=\underset{a\le x\le b}{max}|f(x,y)-F(x,y;h;f)|$$

is satisfied then the numerical solution $w_n$ converges to $Y(x_n)$.

[18 marks]

6. b) Consider the differential equation

(443)$$y^{{}^{\prime }}-y+x-2=0,0\le x\le 1,y(0)=0.$$

Apply the midpoint method to approximate the solution at $y(0.4)$ using $h=0.2.$\

[11 marks]

6. c) How would you improve on this result.

[4 marks]