Graphics Programs Reference
In-Depth Information
Two-Point Boundary Value Problems
8
Solve
y
=
y
)
f
(
x
,
y
,
,
y
(
a
)
=
α,
y
(
b
)
=
β
8.1
Introduction
In two-point boundary value problems the auxiliary conditions associatedwith the
differentialequation,called the
boundary conditions
, arespecifiedattwo different
values of
x
. This seemingly small departurefrom initial value problemshas a major
repercussion—itmakes boundaryvalue problemsconsiderablymore difficult to solve.
In an initial value problemwe were able to start at the point where the initial values
were given and march the solution forward asfar as needed. This technique does not
workforboundary value problems, because there are not enoughstarting conditions
available ateither end pointtoproduce a unique solution.
One way to overcome the lack of starting conditions istoguess themissing values.
The resulting solutionis very unlikely to satisfyboundary conditions at the other end,
but by inspecting the discrepancywecan estimate whatchanges to make to the initial
conditions before integrating again. This iterative procedure isknown as the
shooting
method
. The name is derived from analogy with target shooting—take a shot and
observe where ithits the target, then correct the aim and shoot again.
Anothermeans of solving two-point boundary value problems is the
finite differ-
ence method
, where the differentialequations are approximatedbyfinite differences
atevenly spacedmesh points.As a consequence, adifferentialequationistransformed
into set of simultaneous algebraicequations.
The two methodshave a commonproblem: they give rise to nonlinear sets of
equations if thedifferentialequationisnot linear.AswenotedinChapter 4, allmethods
of solving nonlinear equations are iterative procedures thatcan consume a lot of
computational resources. Thus solution of nonlinear boundary value problems is not
297
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