{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# H in addition to a lipschitz condition convergence of

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

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

{[ snackBarMessage ]}

Ask a homework question - tutors are online