Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

other exact expressions are possible, e.g. the integral off(xi+y) over the range

0 ≤y≤h, but we will find (27.35) the most useful for our purposes.

Although the preceding discussion has implicitly assumed that both of the

limitsaandbare finite, with the consequence thatNis finite, the general method

can be adapted to treat some cases in which one of the limits is infinite. It is

sufficient to consider one infinite limit, as an integral with limits−∞and∞can

be considered as the sum of two integrals, each with one infinite limit.

Consider the integral

I=

∫∞

a

f(x)dx,

whereais chosen large enough that the integrand is monotonically decreasing

forx>aand falls off more quickly thanx−^2. The change of variablet=1/x

converts this integral into

I=

∫ 1 /a

0

1

t^2

f

(

1

t

)

dt.

It is now an integral over a finite range and the methods indicated earlier can be

applied to it. The value of the integrand at the lower end of thet-range is zero.

In a similar vein, integrals with an upper limit of∞and an integrand that is

known to behave asymptotically asg(x)e−αx,whereg(x) is a smooth function, can

be converted into an integral over a finite range by settingx=−α−^1 lnαt. Again,

the lower limit,a, for this part of the integral should be positive and chosen

beyond the last turning point ofg(x). The part of the integral forx<ais treated

in the normal way. However, it should be added that if the asymptotic form

of the integrand is known to be a linear or quadratic (decreasing) exponential

then there are better ways of estimating it numerically; these are discussed in

subsection 27.4.3 on Gaussian integration.

We now turn to practical ways of approximatingI, given the values offi,ora

means to calculate them, fori=0, 1 ,...,N.

27.4.1 Trapezium rule

In this simple case the area shown in figure 27.4(a) is approximated as shown in

figure 27.4(b), i.e. by a trapezium. The areaAiof the trapezium is

Ai=^12 (fi+fi+1)h, (27.36)

and if such contributions from all strips are added together then the estimate of

the total, and hence ofI,is

I(estim.) =

N∑− 1

i=0

Ai=

h

2

(f 0 +2f 1 +2f 2 +···+2fN− 1 +fN). (27.37)