H in addition to a lipschitz condition convergence of

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: .1b) if and only if it is consistent. Proof. Let z(t) satisfy the IVP z = (t z 0) z(0) = y0 0 (3.4.4) and let zn, n 0, satisfy zn = zn 1 + h (tn 1 zn 1 h) ; ; n0 ; z0 = y0: (3.4.5) Using the mean value theorem and (3.4.4) z(tn ) ; z(tn 1 ) = hz (tn 1 + h n ) = h (tn 1 + h n z(tn 1 + h n ) 0) 0 ; where n ; ; ; (3.4.6) 2 (0 1). Let en = z(tn ) ; zn (3.4.7) and subtract (3.4.5) from (3.4.6) to obtain en = en 1 + h (tn 1 + h n z(tn 1 + h n ) 0) ; (tn 1 zn 1 h)]: ; ; ; ; ; Adding and subtracting similar terms en = en ; 1 + h (tn 1 + h n z(tn 1 + h n ) 0) ; (tn ; + + (tn (tn 1 ; 1 ; ; ; 1 z(tn 1 ) 0) z(tn 1 ) h) ; (tn 1 zn 1 h) z(tn 1 ) 0) ; (tn 1 z(tn 1 ) h)]: ; ; ; ; ; ; ; (3.4.8a) Using the Lipschitz condition j (tn 1 z(tn 1 ) h) ; (tn 1 zn 1 h)j Ljenj: ; ; ; ; (3.4.8b) Since (t y h) 2 C 0 , it is uniformly continuous on the compact set t 2 0 T ], y = z(t), h 2 0 ^ ] thus, h (h) = tmax] j (tn 0T 1 ; 2 z(tn 1 ) 0) ; (tn 1 z(tn 1 ) h)j = O(h): ; ; 40 ; (3.4.8c) Similarly, (h) = tmax] j (tn 1 + h n z(tn 1 + h n ) 0) ; (tn 0T ; ; 1 ;...
View Full Document

This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online