Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.9 EXERCISES

(b) Try to repeat the processes described in (a) for the integrals

Jn=

∫ 5

2

1

√

7 x−x^2 − 10

dx.

WhyisitverydifficulttoestimateJ∞?

27.13 Given a random numberηuniformly distributed on (0,1), determine the function
ξ=ξ(η) that would generate a random numberξdistributed as
(a) 2ξ on 0≤ξ<1,
(b)^32

√

ξ on 0≤ξ<1,

(c)

π

4 a

cos

πξ

2 a

on −a≤ξ<a,

(d)^12 exp(−|ξ|)on−∞<ξ<∞.

27.14 A,BandCare three circles of unit radius with centres in thexy-plane at
(1,2),(2. 5 , 1 .5) and (2,3), respectively. Devise a hit or miss Monte Carlo calculation
to determine the size of the area that lies outsideCbut insideAandB,aswell
as inside the square centred on (2, 2 .5), that has sides of length 2 parallel to the
coordinate axes. You should choose your sampling region so as to make the
estimation as efficient as possible. Take the random number distribution to be
uniform on (0,1) and determine the inequalities that have to be tested using the
random numbers chosen.
27.15 Use a Taylor series to solve the equation

dy

dx

+xy=0,y(0) = 1,

evaluatingy(x)forx=0.0 to 0.5 in steps of 0.1.
27.16 Consider the application of the predictor–corrector method described near the
end of subsection 27.6.3 to the equation
dy
dx

=x+y, y(0) = 0.

Show, by comparison with a Taylor series expansion, that the expression obtained
foryi+1in terms ofxiandyiby applying the three steps indicated (without any
repeat of the last two) is correct to O(h^2 ). Using steps ofh=0.1 compute the
value ofy(0.3) and compare it with the value obtained by solving the equation
analytically.
27.17 A more refined form of the Adams predictor–corrector method for solving the
first-order differential equation
dy
dx

=f(x, y)

is known as the Adams–Moulton–Bashforth scheme. At any stage (say thenth)

in anNth-order scheme, the values ofxandyat the previousNsolution points

are first used topredictthe value ofyn+1. This approximate value ofyat the

next solution point,xn+1, denoted by ̄yn+1, is then used together with those at the

previousN−1 solution points to make a more refined (corrected) estimation of

y(xn+1). The calculational procedure for a third-order scheme is summarised by

the two following two equations:

y ̄n+1=yn+h(a 1 fn+a 2 fn− 1 +a 3 fn− 2 )(predictor),

yn+1=yn+h(b 1 f(xn+1,y ̄n+1)+b 2 fn+b 3 fn− 1 ) (corrector).

(a) Find Taylor series expansions forfn− 1 andfn− 2 in terms of the function

fn=f(xn,yn) and its derivatives atxn.