Minimax approximate solutions of linear boundary value problems
Authors: Schmidt Darrell , Wiggins Kenneth L. 
Published: June 2010

Define the operator

D
:
C
[
0
,
τ
]
→
C
[
0
,
τ
]

D:C[0,\tau ] \to C[0,\tau ]

by

D
[
u
]
=
u
−

a
0

u
′
−

a
1

u

D[u] = u - {a_0}u\prime - {a_1}u

where

a
0

,

a
1

∈
C
[
0
,
τ
]

{a_0},{a_1} \in C[0,\tau ]

and consider the two point boundary value problem

(

BVP

)

D
[
y
]
(
x
)
=

a
2

(
x
)

({\text {BVP}})\;D[y](x) = {a_2}(x)

,

x
∈
[
0
,
τ
]

x \in [0,\tau ]

,

N
0

[
y
]
=

α
0

y
(
0
)
+

α
1

y
′
(
0
)
=

α
2

{N_0}[y] = {\alpha _0}y(0) + {\alpha _1}y\prime (0) = {\alpha _2}

,

N
τ

[
y
]
=

β
0

y
(
τ
)
+

β
1

y
′
(
τ
)
=

β
2

{N_\tau }[y] = {\beta _0}y(\tau ) + {\beta _1}y\prime (\tau ) = {\beta _2}

where

a
2

∈
C
[
0
,
τ
]

{a_2} \in C[0,\tau ]

,

α
0
2

+

α
1
2

≠
0

\alpha _0^2 + \alpha _1^2 \ne 0

and

β
0
2

+

β
1
2

≠
0

\beta _0^2 + \beta _1^2 \ne 0

. Let

Π
k

{\Pi _k}

denote the set of polynomials of degree at most k and define the approximating set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript k Baseline equals left-brace p element-of normal upper Pi Subscript k Baseline colon upper N 0 left-bracket p right-bracket equals alpha 2 comma upper N Subscript tau Baseline left-bracket p right-bracket equals beta 2 right-brace">

P

k

=
{
p
∈

Π
k

:

N
0

[
p
]
=

α
2

,

N
τ

[
p
]
=

β
2

}

{\mathcal {P}_k} = \{ p \in {\Pi _k}:{N_0}[p] = {\alpha _2},{N_\tau }[p] = {\beta _2}\}

. Then for each

k
⩾
3

k \geqslant 3

there exists

p
k

∈

P

k

{p_k} \in {\mathcal {P}_k}

satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper D left-bracket p Subscript k Baseline right-bracket minus a 2 double-vertical-bar equals inf Underscript p element-of script upper P Subscript k Baseline Endscripts double-vertical-bar upper D left-bracket p right-bracket minus a 2 double-vertical-bar equals delta Subscript k">

‖

D
[

p
k

]
−

a
2

‖

=

inf

p
∈

P

k

‖

D
[
p
]
−

a
2

‖

=

δ
k

\left \| {D[{p_k}] - {a_2}} \right \| = {\inf _{p \in {\mathcal {P}_k}}}\left \| {D[p] - {a_2}} \right \| = {\delta _k}

, where

‖
⋅
‖

\left \| \cdot \right \|

denotes the uniform norm on

C
[
0
,
τ
]

C[0,\tau ]

. If the homogeneous BVP

D
[
y
]
=
0

D[y] = 0

,

N
0

[
y
]
=

N
τ

[
y
]
=
0

{N_0}[y] = {N_\tau }[y] = 0

has no nontrivial solutions, then the nonhomogeneous BVP has a unique solution y and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript k right-arrow normal infinity Endscripts double-vertical-bar p Subscript k Superscript left-parenthesis i right-parenthesis Baseline minus y Superscript left-parenthesis i right-parenthesis Baseline double-vertical-bar equals 0">

lim

k
→
∞

‖

p
k

(
i
)

−

y

(
i
)

‖

=
0

{\lim _{k \to \infty }}\left \| {p_k^{(i)} - {y^{(i)}}} \right \| = 0

for

i
=
0
,
1
,
2

i = 0,1,2

. If X denotes a closed subset of

[
0
,
τ
]

[0,\tau ]

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript k comma upper X Baseline equals inf Underscript p element-of script upper P Subscript k Baseline Endscripts max Underscript x element-of upper X Endscripts StartAbsoluteValue upper D left-bracket p right-bracket left-parenthesis x right-parenthesis minus a 2 left-parenthesis x right-parenthesis EndAbsoluteValue comma">

δ

k
,
X

=

inf

p
∈

P

k

max

x
∈
X

|

D
[
p
]
(
x
)
−

a
2

(
x
)

|

,

{\delta _{k,X}} = \inf \limits _{p \in {\mathcal {P}_k}} \max \limits _{x \in X} |D[p](x) - {a_2}(x)|,

\]
then for each

ε
>
0

\varepsilon > 0

there exists

δ
>
0

\delta > 0

such that

d
(
x
)
⩽
δ

d(x) \leqslant \delta

implies that

0
⩽

δ
k

−

δ

k
,
X

⩽
ε

0 \leqslant {\delta _k} - {\delta _{k,X}} \leqslant \varepsilon

, where

d
(
X
)

d(X)

denotes the density of X in

[
0
,
τ
]

[0,\tau ]

. Several numerical examples are given.