082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

20.5 Numerical methods 615
Combining these equations and usingEquation (20.13) toeliminateq i
andq o
gives
dh 2
= ρg ( ρg
h
dt R 1
A 1
− + ρg )
h
2
R 1
A 2
R 2
A 2
+ 1 q
2
A 2
(20.16)
2
Equations (20.14) and (20.16) are the state-space equations for the system and can
be written inmatrix formas
⎛
−ρg ρg
⎞ ⎛ ⎞
1
)
(ḣ1
R 1
A 1
R 1
A 1
( )
0
h1
A ( )
1 = ⎜
ḣ 2 ⎝ ρg
− ρg − ρg
⎟ +
q1
⎠ h ⎜ ⎟
2 ⎝ 1 ⎠
q 2
0
R 1
A 2
R 1
A 2
R 2
A 2 A 2
Note that in this case the input vector is two-dimensional as there are two inputs to
the system.The output equation isgiven by
q o
= ( ) ( )
h
0 ρg/R 1
2
h 2
and isobtained directly from Equation (20.15).
20.5 NUMERICALMETHODS
All the techniques we have so far met for solving differential equations are known as
analytical methods, and these methods give rise to a solution in terms of elementary
functionssuchassinx,e x ,x 3 ,etc.Inpractice,mostengineeringproblemsinvolvingdifferential
equations are too complicated to be solved easily using analytical techniques.
It is therefore frequently necessary to make use of computers. One such approach is to
use analogue simulation, which we discussed in Section 20.2. Another, more common,
approachistouse digital computers.
There are a variety of computer software packages available for solving differential
equations,includingMATLAB ® whichhasalreadybeenintroduced.Oftenthedetailof
howthecomputersolvestheequationishiddenfromtheuseranditusesoneofarange
of numericalmethods.Wewillexaminesomeofthemethodsthatmightbeappliedby
the computer inthissection.
Whenusingnumericalmethodsitisimportanttonotethattheyresultinapproximate
solutions to differential equations and solutions are only calculated at discrete intervals
oftheindependentvariable,typicallyxort.Weshallbeginbyexaminingthefirst-order
differential equation
dy
dx =f(x,y)
subject to the initial condition y(x 0
) = y 0
. Usually the solution is obtained at equally
spaced values of x, and we call this spacing the step size, denoted by h. By choosing
a suitable value ofhwe can control the accuracy of the approximate solution obtained.
Weshallwritey n
forthisapproximatesolutionatx =x n
,whereaswewritey(x n
)forthe
true solution atx = x n
. Generally these values will not be the same although we try to
ensure the difference issmallfor obvious reasons.
The simplest numerical method forthe solution of the differential equation
dy
dx =f(x,y)