Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.6 DIFFERENTIAL EQUATIONS

x h y(exact)

0. 01 0.1 0.5 1.0 1.5 2 3

0 (1) (1) (1) (1) (1) (1) (1) (1)

0.5 0.605 0.590 0.500 0 − 0. 500 − 1 − 2 0.607

1.0 0.366 0.349 0.250 0 0. 250 1 4 0.368

1.5 0.221 0.206 0.125 0 − 0. 125 − 1 − 8 0.223

2.0 0.134 0.122 0.063 0 0. 063 1 16 0.135

2.5 0.081 0.072 0.032 0 − 0. 032 − 1 − 32 0.082

3.0 0.049 0.042 0.016 0 0. 016 1 64 0.050

Table 27.10 The solutionyof differential equation (27.61) using the Euler

forward difference method for various values ofh. The exact solution is also

shown.

27.6.1 Difference equations

Consider the differential equation

dy

dx

=−y, y(0) = 1, (27.61)

and the possibility of solving it numerically by approximatingdy/dxby a finite

difference along the lines indicated in section 27.5. We start with the forward

difference
(
dy
dx

)

xi

≈

yi+1−yi

h

, (27.62)

where we use the notation of section 27.5 but withfreplaced byy.Inthis

particular case, it leads to the recurrence relation

yi+1=yi+h

(

dy

dx

)

i

=yi−hyi=(1−h)yi. (27.63)

Thus, sincey 0 =y(0) = 1 is given,y 1 =y(0 +h)=y(h) can be calculated, and

so on (this is theEulermethod). Table 27.10 shows the values ofy(x) obtained

if this is done using various values ofhand for selected values ofx. The exact

solution,y(x)=exp(−x), is also shown.

It is clear that to maintain anything like a reasonable accuracy only very small

steps,h, can be used. Indeed, ifhis taken to be too large, not only is the accuracy

bad but, as can be seen, forh>1 the calculated solution oscillates (when it

should be monotonic), and forh>2 it diverges. Equation (27.63) is of the form

yi+1=λyi, and a necessary condition for non-divergence is|λ|<1, i.e. 0<h<2,

though in no way does this ensure accuracy.

Part of this difficulty arises from the poor approximation (27.62); its right-

hand side is a closer approximation tody/dxevaluated atx=xi+h/2thanto

dy/dxatx=xi. This is the result of using a forward difference rather than the