Boundary Value Problems
Numerical Methods for BVPs
Scientific Computing: An Introductory Survey
Chapter 10 – Boundary Value Problems for
Ordinary Differential Equations
Prof. Michael T. Heath
Department of Computer Science
University of Illinois at UrbanaChampaign
Copyright c 2002. Reproduction permitted
for noncommercial, educational use only.
Michael T. Heath
Scientific Computing
1 / 45
Boundary Value Problems
Numerical Methods for BVPs
Outline
1
Boundary Value Problems
2
Numerical Methods for BVPs
Michael T. Heath
Scientific Computing
2 / 45
Boundary Value Problems
Numerical Methods for BVPs
Boundary Values
Existence and Uniqueness
Conditioning and Stability
Boundary Value Problems
Side conditions prescribing solution or derivative values at
specified points are required to make solution of ODE
unique
For initial value problem, all side conditions are specified at
single point, say
t
0
For
boundary value problem
(BVP), side conditions are
specified at more than one point
k
th order ODE, or equivalent firstorder system, requires
k
side conditions
For ODEs, side conditions are typically specified at
endpoints of interval
[
a, b
]
, so we have
twopoint boundary
value problem
with boundary conditions (BC) at
a
and
b
.
Michael T. Heath
Scientific Computing
3 / 45
Boundary Value Problems
Numerical Methods for BVPs
Boundary Values
Existence and Uniqueness
Conditioning and Stability
Boundary Value Problems, continued
General
firstorder twopoint BVP
has form
y
=
f
(
t,
y
)
,
a < t < b
with BC
g
(
y
(
a
)
,
y
(
b
)) =
0
where
f
:
R
n
+1
→
R
n
and
g
:
R
2
n
→
R
n
Boundary conditions are
separated
if any given component
of
g
involves solution values only at
a
or at
b
, but not both
Boundary conditions are
linear
if they are of form
B
a
y
(
a
) +
B
b
y
(
b
) =
c
where
B
a
,
B
b
∈
R
n
×
n
and
c
∈
R
n
BVP is
linear
if ODE and BC are both linear
Michael T. Heath
Scientific Computing
4 / 45
Boundary Value Problems
Numerical Methods for BVPs
Boundary Values
Existence and Uniqueness
Conditioning and Stability
Example: Separated Linear Boundary Conditions
Twopoint BVP for secondorder scalar ODE
u
=
f
(
t, u, u
)
,
a < t < b
with BC
u
(
a
) =
α,
u
(
b
) =
β
is equivalent to firstorder system of ODEs
y
1
y
2
=
y
2
f
(
t, y
1
, y
2
)
,
a < t < b
with separated linear BC
1
0
0
0
y
1
(
a
)
y
2
(
a
)
+
0
0
1
0
y
1
(
b
)
y
2
(
b
)
=
α
β
Michael T. Heath
Scientific Computing
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Boundary Value Problems
Numerical Methods for BVPs
Boundary Values
Existence and Uniqueness
Conditioning and Stability
Existence and Uniqueness
Unlike IVP, with BVP we cannot begin at initial point and
continue solution step by step to nearby points
Instead, solution is determined everywhere simultaneously,
so existence and/or uniqueness may not hold
For example,
u
=

u,
0
< t < b
with BC
u
(0) = 0
,
u
(
b
) =
β
with
b
integer multiple of
π
, has infinitely many solutions if
β
= 0
, but no solution if
β
= 0
Michael T. Heath
Scientific Computing
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Boundary Value Problems
Numerical Methods for BVPs
Boundary Values
Existence and Uniqueness
Conditioning and Stability
Existence and Uniqueness, continued
In general, solvability of BVP
y
=
f
(
t,
y
)
,
a < t < b
with BC
g
(
y
(
a
)
,
y
(
b
)) =
0
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 Fall '11
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 Mechanical Engineering, Boundary value problem, Michael T. Heath, BVP

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