Graphics Programs Reference
In-Depth Information
cheap. Another complicationisthat iterative methods needreasonablygood starting
values in order to converge.Since there is no set formula fordetermining these, an
algorithm for solving nonlinear boundary value problems requires intelligent input;
itcannot betreatedas a“black box.”
8.2
Shooting Method
Second-Order Differential Equation
The simplest two-point boundary value problemis a second-orderdifferentialequa-
tionwith onecondition specifiedat
x
=
a
and another one at
x
=
b
. Here is an exam-
ple of a second-orderboundary value problem:
y
=
y
)
f
(
x
,
y
,
,
y
(
a
)
=
α,
y
(
b
)
=
β
(8.1)
Let us nowattempttoturn Eqs. (8.1) into the initial value problem
y
=
y
)
y
(
a
)
f
(
x
,
y
,
,
y
(
a
)
=
α,
=
u
(8.2)
The key to success is finding the correct valueof
u
. Thiscouldbe done by trial and
error: guess
u
and solve the initial value problem by marching from
x
=
a
to
b
. If
the solution agrees with the prescribedboundary condition
y
(
b
)
, we are done;
otherwise wehavetoadjust
u
and try again. Clearly,this procedure is very tedious.
Moresystematic methods become available to us if we realize that the determi-
nation of
u
is aroot-finding problem. Because the solution of the initial value problem
dependson
u
, the computedboundary value
y
(
b
) is a function of
u
; that is
=
β
y
(
b
)
=
θ
(
u
)
Hence
u
is aroot of
=
θ
−
β
=
r
(
u
)
(
u
)
0
(8.3)
where
r
(
u
) is the
boundary residual
(difference between the computed and specified
boundary values).Equation (8.3)canbe solvedby any oneoftheroot-findingmethods
discussedinChapter 4.We reject themethod of bisectionbecause it involves toomany
evaluationsof
θ
(
u
). In the Newton-Raphsonmethodwe runinto the problemof having
to compute
d
du
, which can be done, but not easily. Thatleaves Brent's algorithm
asour method of choice.
Here is the procedure we use in solving nonlinear boundary value problems:
θ/
1. Specify the starting values
u
1
and
u
2
which
must bracket the root u
of Eq. (8.3).
2. ApplyBrent's method to solve Eq. (8.3)for
u
. Note thateach iterationrequires
evaluation of
θ
(
u
) by solving the differentialequationas an initial value problem.
Search WWH ::
Custom Search