Assuming this to be so lets write the solution as rx

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Unformatted text preview: y(x) 1 eRx (1 ; ) eRx(1 ; ) c + ;10 ) e (1 + 2 where 1. (If = e determine c as ; and the solution as 2R x>0 then the above relation is exact at x = 1.) We would then 11 c R2 ;1 1 y(x) 2eR2 ;1 Rx x > 0: With R large, there is the possibility of having catastrophic growth in the solution even when is small. In order to demonstrate this, we solved this problem using the MATLAB Runge-Kutta procedure ode45. While an explicit Runge-Kutta code is not the most e cient solution procedure for this problem, e ciency was not our main concern. The maximum errors in the solution component y1 with R = 1 10 are reported in Table 7.1.1. The error has grown by six decades for a ten-fold increase in R. Indeed, the procedure failed to nd a solution of the BVP with R = 100. R ke1k =ky1k 1 2:445 10 8 10 2:940 10 2 Table 7.1.1: Maximum errors in y1 for Example 7.1.1 1 1 ; ; Let's pursue this di culty with a bit more formality. The exact solution of the linear 4 BVP (7.1.1) can be written as (Problem 1...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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