A Symbolic Algorithm for Solving Two-Point BVPs on

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Nov 22, 2003 - (Bergman, 19E 9). The reduction system should have the following five ¤ ey ! ualities for ! ualities for Contracting. Algebraic implifi cationвб.
A Symbolic Algorithm for Solving Two-Point BVPs on the Operator Level

 

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The rules DA and AMA yield the S|polynomial: ` f p A - D aÙ

*

f q A + D A aÙ

` f p A - D aÙ

*

f q A + D A aÙ

H¼L

fq

*

*

f q + ` f p A - D aÙ

*



*

f q + ` f p A - b IÙ

f M’ r A - aÙ



*

f q - aÙ

fq D A

x

=

HDML

*

*

HDAL

fq



*

fq A

x

=

x

=

HdaL

fq D A

*

x

=

HDAL

x

=

ƒ

0 ¼

The rules RA and AMA yield the S|polynomial: A ` f p A + B ` f p A - R aÙ

*

f q A + R A aÙ

A ` f p A + B ` f p A - R aÙ

*

f q A + R A aÙ

*

fq *

H¼L

x

=

fq

HRAL

x

=

A aÙ

*

f q + B aÙ

*

f q + A ` f p A + B ` f p A - R aÙ

A aÙ

*

f q + B aÙ

*

f q - IÙ

A aÙ

*

f q + B aÙ

*

f q - H˜ f L R A + A ` f p A + B ` f p A

fM

®

*

*

x

=

R A + A ` fp A + B ` fp A

-H˜ f L A - H˜ f L B + A aÙ

*

f q + B aÙ

*

-H˜ f L A - H˜ f L B + B aÙ

*

f q + aÙ

fq A + B ` fp A

-H˜ f L A - H˜ f L B + B aÙ

*

f q + B b Ù* f r + aÙ

0

HRML

fq A

*

HraL

x

=

HRAL

x

=

fq + A ` fp A + B ` fp A

*

HAMAL

HBMAL

x

=

f q A + b Ù* f r A

ƒ

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x

=

HbL

x

=

The rules BR and RR yield the S|polynomial: -B R + aÙ* 1q R2

H¼L

x

=

-B R + b Ù* 1 r R2

HbL

x

=

HRRL

-B R + H˜ 1L R2 - aÙ 1q R2

x

*

HRRL

H˜ 1L R - B R - aÙ 1q R2

x

*

H˜ 1L R - B R - aÙ 1q R *

=

HBRL

x

=

H˜ 1L R - b Ù* 1 r R - aÙ 1q R *

0

=

HbL

x

=

ƒ

´ Computed 233 S|polynomials in 129 seconds. ´ Reduced them in 3144 seconds. ´ All of them reduced to zero!

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