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We write the fundamental solution as Rx sinh
Y(x) = cosh Rx cosh Rx :
sinh
Rx
From (7.1.2a), the solution of this BVP has the form Rx sinh
=R2
y(x) = cosh Rx cosh Rx c + ;10 :
sinh
Rx
The initial and terminal conditions (7.1.3) are c1 + ;1=R2 = 0 :
c2
;1=R2
0 1
0
cosh R sinh R
Thus,
and the solution of the BVP is 1
c = R2 1 1;cosh R
sinh R 1
Rx sinh
y(x) = R2 cosh Rx cosh Rx
sinh
Rx
Di culties arise when R
by 1 1;cosh R
sinh R =R2 :
+ ;10 1. In this case, the solution y1(x) is asymptotically given y1 1e
R2 Rx + e R(1 x) ; 1]: ; ; 3 ; Thus, it is approximately ;1=R2 away from the boundaries at x = 0 and 1 with narrow
boundary layers near both ends.
When Rx is large, we would not be able to distinguish between sinh Rx and cosh Rx
and, instead of the exact result, we would compute
Rx Rx
=R2
y(x) 1 eRx eRx c + ;10
2e e x > 0: The matrix Y(x) would be singular in this case. With small discernable di erences
between sinh Rx and cosh Rx, Y would be illconditioned. Assuming this to be so, let's
write the solution as
Rx
eRx (1 +
)
=R2...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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