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# Well have to investigate the stability of the bvp we

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Unformatted text preview: . 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 ill-conditioned. Assuming this to be so, let's write the solution as Rx eRx (1 + ) =R2...
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