MM354 Assignment 1 This is the first of two assigments which form the assessment for the second part of the module. Each will be marked out of 40,...
1. Inversion of the Laplace transform
The Laplace transform of a function f (t) of a function of exponential order is defined as
F(s) = ∫∞
0
e−st f (t) dt.
This transform had many applications in applied mathematics. It can be shown that the inverse
transform can be obtained as the Bromwich integral,
f (t) = 1
2πi ∫γ + i∞
γ − i∞
ezt F(z) dz.
The integrand here is over the line where is chosen sufficiently large that all
singularities of lie to the left of the line. In practice we evaluate the integral as the limit as
of a contour integral over the contour shown below. Note that the curved part is a
semicircle of radius centred on the point , not . We assume throughout that is
real.
Re (z) = γ γ
F(s)
R → ∞
R z = γ z = 0 t
R
−R
Re( z )
Im( z )
γ
The Bromwich contour
We are going to use this contour to evaluate the inverse transform of two functions. Both of
these can be done easily by elementary methods, but they serve to illustrate the method.
(i) Let
F(s) = 1
(s + 1)2 + 1
.
Since the poles of this are in the left half plane, we can chose γ = 0.
Use the Residue Theorem to find f (t).
Note that to obtain full marks you must give a proper proof that the integral around the curved
section of the Bromwich contour tends to zero as goes to infinity. In particular, you need to
find the maximum value of on the contour.
MM354 Assignment 1This is the first of two assigments which form the assessment for the second part of the module.Each will be marked out of 40, but contribute 20% of the total module mark.Hand in the work to the CMIS office by 8.15am on Friday 21st May 2010.1. Inversion of the Laplace transformThe Laplace transform of a function of a function of exponential order is defined asf(t)F(s) =∫∞0e−stf(t)dt.This transform had many applications in applied mathematics. It can be shown that the inversetransform can be obtained as the Bromwich integral,f(t) =12πi∫γ+i∞γ−i∞eztF(z)dz.The integrand here is over the linewhereis chosen sufficiently large that allsingularities oflie to the left of the line. In practice we evaluate the integral as the limit asof a contour integral over the contour shown below. Note that the curved part is asemicircle of radiuscentred on the point, not. We assume throughout thatisreal.Re(z) =γγF(s)R→ ∞Rz=γz=0tRRRe( )zIm( )zγThe Bromwich contourWe are going to use this contour to evaluate the inverse transform of two functions. Both ofthese can be done easily by elementary methods, but they serve to illustrate the method. (i) LetF(s) =1(s+1)2+1.Since the poles of this are in the left half plane, we can chose . γ=0Use the Residue Theorem to find . f(t)Note that to obtain full marks youmustgive a proper proof that the integral around the curvedsection of the Bromwich contour tends to zero asgoes to infinity. In particular, you need tofind the maximum value of on the contour. R eztContinued over the pagePage 1MM354 Assignment 1Fri, Apr 30, 2010S.W.Ellacott
RecalleRe(w). ew =(12 marks)(ii) Now consider the caseF(s) =1s+1.Again the pole is in the left half plane, so we can chose . γ=0What problem arises when we try to show that the integral ontends at zero asgoes toinfinity? There is a theorem which overcomes this problem and which is found in most complexvariable textbooks. Look up and quote the theorem, and use it to complete the argumentcorrectly.ΓR(13 marks)2. Numerical evaluation of complex integralsNumerical evaluation around smooth curves in the complex plane is usually carried out usingthe trapezium rule. (You will find out later why the trapezium rule is particularly good for thispurpose.) For our purposes we just need integrals around the unit circle. So we requireto evaluate z =1∫ z =1f(z) dz.Set and show that the point composite trapezium rule givesz=eiθN+1∫ z =1f(z) dz=∫2π0f(eiθ)ieiθdθ≈2πiN∑N−1k=0f(eiθk)eiθkθk=2πkN.where For the case, write an Excel spreadsheet to implement this rule. Note that Excel doesnot support complex arithmetic, so you will need to calculate the real and imaginary parts of thesummand explicitly.N=16Test your spreadsheet on at two integrals evaluated exactly using the Cauchy formula or theresidue theorem. For at least one of your test problems, both the real and imaginary parts of theintegral should be nonzero. You should find that the trapezium rule is very accurate. Youshould submit printouts for both of your test problems, but you need not submit an electronicversion of the spreadsheet. (15 marks)You will need this spreadsheet for the second assignment.S.W.EllacottFri, Apr 30, 2010MM354 Assignment 1Page 2
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