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)#y(x)=f(x,y),

with initial condition

(413)#y(a)=α.

[8 marks]

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

(414)#|y(t)|M.

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

(415)#y=f(t,y)   atb   y(a)=α

and w0,w1,...,wN be the approx generated by the Euler method for some positive integer N. Then show for i=0,1,...,N

(416)#|y(ti)wi|Mh2L|eL(tia)1|

You may assume the two lemmas:\ If s and t are positive real numbers {ai}i=0N is a sequence satisfying a0ts and ai+1(1+s)ai+t then

(417)#ai+1e(i+1)s(a0+ts)ts

For all x0.1 and any positive m we have

(418)#0(1+x)memx

[17 marks]

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

(419)#y=(1x)y2y;   y(0)=1

at x=1, using h=0.25.

[8 marks]

Question 2#

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

(420)#wn+1=wn+k2
(421)#k1=hf(tn,wn)
(422)#k2=hf(tn+12h,wn+12k1)

for solving the ordinary differential equation

(423)#dydt=f(t,y)
(424)#y(t0)=y0

by using a formula of the form

(425)#wn+1=wn+ak1+bk2

where k1 is defined as above,

(426)#k2=hf(tn+αh,wn+βk1)

and a, b, α and β are constants are deteremined. Prove that a+b=1 and bα=bβ=12 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)#y=1+(ty)2,  2t3,  y(2)=1,

with h=0.2 with the exact solution y(t)=t+11t.

[10 marks]

Question 3#

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

(428)#wn+1=wn+(32hf(tn,wn)12hf(tn1,wn1)),

and the local truncation error

(429)#τn+1(h)=5h212y3(μn).

[18 marks]

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

(430)#y=tyy,  (0t2)   y(0)=1.

Using N=4 steps, given that y1=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)#w0=α0   w1=α1
(432)#wn+1=wn+h12[5f(tn+1,wn+1)+8f(tn,wn1)f(tn2,wn2)]

and the local truncation error

(433)#τn+1(h)=h324y4(μn)

[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)#y=f(t,y),    y(t0)=y0

and a numerical method which generates a numerical solution (wn)n=0N, explain what it means for the method to be convergent.

[5 marks]

Question 5#

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

(435)#wn+1=wn+32hf(tn,wn)12hf(tn1,wn1)

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

(436)#wn+1=wn+h12[5f(tn+1,wn+1)+8f(tn,wn1)f(tn2,wn2)]

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

(437)#y=ty3y,    (0t2),  y(0)=1

using N=4 steps, given y1=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)#w0=y0
(439)#wi+1=wi+hf(xi+h2,wi+h2f(xi,wi))

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

(440)#y=f(x,y)
(441)#y(x0)=Y0

Show that the numerical solution {wn} satisfies

(442)#maxaxb|y(xn)wn|e(ba)L|y0w0|+[e(ba)L1L]τ(h)

where

(443)#τ(h)=maxaxb|τn(y)|

If the consistency condition

(444)#δ(h)0 as h0

where

(445)#δ(h)=maxaxb|f(x,y)F(x,y;h;f)|

is satisfied then the numerical solution wn converges to Y(xn).

[18 marks]

  1. b) Consider the differential equation

(446)#yy+x2=0,  0x1,  y(0)=0.

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]