11a where 2 3 2 3 y1t f1t y1 ym 6 y t 7 6 f t

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Unformatted text preview: ) where 2 3 2 3 y1(t) f1(t y1 : : : ym) 6 y (t) 7 6 f (t y : : : y ) 7 y(t) = 6 2.. 7 f (t y) = 6 2 1 .. m 7 6 7 6 7 4.5 4 5 . ym(t) fm (t y1 : : : ym) and y := dy=dt. Data for IVPs would is speci ed as 3 2 y10 6 y20 7 y(0) = y0 = 6 .. 7 : 6 5 4.7 (1.1.1b) 0 ym0 6 (1.1.1c) Boundary data is more complex. The most general boundary conditions specify a nonlinear relationship between y(0) and y(l) of the form 2 3 g1(y(0) y(l)) 6 g (y(0) y(l)) 7 7 = 0: g(y(0) y(l)) = 6 2 .. (1.1.1d) 6 7 4 5 . gm(y(0) y(l)) In many cases the extension of a scalar to a vector IVP will be obvious and we can study methods for a scalar IVP y (t) = f (t y) t>0 0 y(0) = y0 (1.1.2) Example 1.1.1. Higher-order ODEs can be written as rst-order systems. Consider the m th order equation z(m) = g(t z z : : : z(m 1) ) 0 ; where z(i) := diz=dti, and introduce the new variables y1 = z y2 = z y3 = z 0 ::: 00 ym = z(m 1) : ; Then, we have a system in the form (1.1.1) with y1 = y2 y2 = y3 0 ::: 0 ym = g(t y1 y2 : : : ym): 0 For an IVP, that data would prescribe as z(i) (0) = ci i = 0 1 ::: m; 1 with the understanding that z(0) = z. Written in terms of y, we have y1(...
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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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