Prove the following result: Let f(t, y) be Lipschitz in y, with Lipschitz constant L. and...

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Prove the following result: Let f(t, y) be Lipschitz in y, with Lipschitz constant L. and let the solution to the boundary-value problem y = f(t, y), y(a)= yo satisfy y € C²([a, b]) with ly"(t)| ≤M for a ≤t≤ b. Let y be the implicit Euler approximation to y(t.). Then, for any h such that Lh <, leil ≤ Mh (e²(t-to)-1) for 0≤ i ≤N. 2L modelling after the proof for explicit Euler in class: Approach the proof in 3 steps, (a) Derive the bound le+al ≤₁-leil + 1-hL Mh² 2(1-hL) (b) Use induction to bound le] in terms of the constants above. Hint: it may be easier to show the abstract bound that if eo = 0 and le+1] ≤ aleil + 3, then i-1 lal ≤BΣa² = 8(1-a). j-0 c) Prove that if 0 < x <, then ≤ 1+2r, and use this to conclude the bound, getting the exponential term as in class. Prove the following result: Let f(t, y) be Lipschitz in y, with Lipschitz constant L. and let the solution to the boundary-value problem y = f(t, y), y(a)= yo satisfy y € C²([a, b]) with ly"(t)| ≤M for a ≤t≤ b. Let y be the implicit Euler approximation to y(t.). Then, for any h such that Lh <, leil ≤ Mh (e²(t-to)-1) for 0≤ i ≤N. 2L modelling after the proof for explicit Euler in class: Approach the proof in 3 steps, (a) Derive the bound le+al ≤₁-leil + 1-hL Mh² 2(1-hL) (b) Use induction to bound le] in terms of the constants above. Hint: it may be easier to show the abstract bound that if eo = 0 and le+1] ≤ aleil + 3, then i-1 lal ≤BΣa² = 8(1-a). j-0 c) Prove that if 0 < x <, then ≤ 1+2r, and use this to conclude the bound, getting the exponential term as in class.

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Here is the proof a Derive the bound ci1 ci 1hL Mh221hL Where ci yti yi Then ci1 yt... View the full answer