Initial Value Problem Review Questions#

Question 1#

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

(412)#\[\begin{equation} y^{'}(x)=f(x,y), \end{equation}\]

with initial condition

(413)#\[\begin{equation}y(a)=\alpha. \end{equation}\]

[8 marks]

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

(414)#\[\begin{equation} |y^{''}(t)|\leq M. \end{equation}\]

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

(415)#\[\begin{equation}y^{'}=f(t,y) \ \ \ a\leq t \leq b \ \ \ y(a)=\alpha \end{equation}\]

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

(416)#\[\begin{equation} |y(t_i)-w_i| \leq \frac{Mh}{2L}|e^{L(t_i-a)}-1| \end{equation}\]

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

(417)#\[\begin{equation}a_{i+1} \leq e^{(i+1)s}\left(a_0+\frac{t}{s}\right)-\frac{t}{s} \end{equation}\]

For all \( x \geq 0.1\) and any positive m we have

(418)#\[\begin{equation}0\leq (1+x)^m \leq e^{mx}\end{equation}\]

[17 marks]

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

(419)#\[\begin{equation} y^{'}=(1-x)y^2-y; \ \ \ y(0)=1 \end{equation}\]

at x=1, using \(h=0.25\).

[8 marks]

Question 2#

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

(420)#\[\begin{equation} w_{n+1}=w_n+k_2\end{equation}\]
(421)#\[\begin{equation}k_1=hf(t_n,w_n)\end{equation}\]
(422)#\[\begin{equation}k_2=hf(t_n+\frac{1}{2}h,w_n+\frac{1}{2}k_1)\end{equation}\]

for solving the ordinary differential equation

(423)#\[\begin{equation}\frac{dy}{dt}=f(t,y)\end{equation}\]
(424)#\[\begin{equation}y(t_0)=y_0 \end{equation}\]

by using a formula of the form

(425)#\[\begin{equation}w_{n+1}=w_n+ak_1+bk_2 \end{equation}\]

where \(k_1\) is defined as above,

(426)#\[\begin{equation}k_2=hf(t_n+\alpha h,w_n+\beta k_1)\end{equation}\]

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]

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

[5 marks]

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

(427)#\[\begin{equation}y^{'}=1+(t-y)^2, \ \ 2\leq t \leq 3, \ \ y(2)=1,\end{equation}\]

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

[10 marks]

Question 3#

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

(428)#\[\begin{equation} w_{n+1}=w_n+(\frac{3}{2}hf(t_{n},w_{n})-\frac{1}{2}hf(t_{n-1},w_{n-1})),\end{equation}\]

and the local truncation error

(429)#\[\begin{equation} \tau_{n+1}(h)=-\frac{5h^2}{12}y^{3}(\mu_n).\end{equation}\]

[18 marks]

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

(430)#\[\begin{equation} y'=ty-y, \ \ (0\leq t \leq 2) \ \ \ y(0)=1.\end{equation}\]

Using \(N=4\) steps, given that \(y_1=0.6872\).

[15 marks]

Question 4#

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

(431)#\[\begin{equation}w_0=\alpha_0 \ \ \ w_1=\alpha_1 \end{equation}\]
(432)#\[\begin{equation}w_{n+1}=w_n + \frac{h}{12}[5f(t_{n+1},w_{n+1})+8f(t_{n},w_{n1})-f(t_{n-2},w_{n-2})] \end{equation}\]

and the local truncation error

(433)#\[\begin{equation} \tau_{n+1}(h)=-\frac{h^3}{24}y^{4}(\mu_n)\end{equation}\]

[23 marks]

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

[5 marks]

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

[5 marks]

  1. a) Given the initial value problem:

(434)#\[\begin{equation} y'=f(t,y), \ \ \ \ y(t_0)=y_0 \end{equation}\]

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]

Question 5#

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

(435)#\[\begin{equation} w_{n+1}=w_n+\frac{3}{2}hf(t_n,w_n)-\frac{1}{2}hf(t_{n-1},w_{n-1})\end{equation}\]

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

(436)#\[\begin{equation}w_{n+1}=w_n + \frac{h}{12}[5f(t_{n+1},w_{n+1})+8f(t_{n},w_{n1})-f(t_{n-2},w_{n-2})] \end{equation}\]

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

(437)#\[\begin{equation}y'=ty^3-y, \ \ \ \ (0\leq t \leq 2), \ \ y(0)=1 \end{equation}\]

using N=4 steps, given \(y_1=0.5\).

[18 marks]

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

[10 marks]

Question 6#

  1. a) Given the Midpoint point (Runge-Kutta) method

(438)#\[\begin{equation}w_0=y_0\end{equation}\]
(439)#\[\begin{equation}w_{i+1}=w_{i}+hf(x_i+\frac{h}{2},w_i+\frac{h}{2}f(x_i,w_i) ) \end{equation}\]

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

(440)#\[\begin{equation}y^{'}=f(x,y)\end{equation}\]
(441)#\[\begin{equation} y(x_0)=Y_0 \end{equation}\]

Show that the numerical solution \(\{ w_n\}\) satisfies

(442)#\[\begin{equation} \max_{a\leq x\leq b}|y(x_n)-w_n| \leq e^{(b-a)L}|y_0-w_0|+\left[\frac{e^{(b-a)L}-1}{L} \right]\tau(h)\end{equation}\]

where

(443)#\[\begin{equation}\tau(h) = \max_{a\leq x\leq b}|\tau_n(y)|\end{equation}\]

If the consistency condition

(444)#\[\begin{equation} \delta(h) \rightarrow 0 \mbox{ as } h\rightarrow 0 \end{equation}\]

where

(445)#\[\begin{equation} \delta(h) = \max_{a \leq x \leq b}|f(x,y)-F(x,y;h;f)| \end{equation}\]

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

[18 marks]

  1. b) Consider the differential equation

(446)#\[\begin{equation} y^{'}-y+x-2=0, \ \ 0\leq x \leq 1, \ \ y(0)=0.\end{equation}\]

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

[11 marks]

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

[4 marks]