H in addition to a lipschitz condition convergence of

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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 ;...
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