Consider the initial value problem (y' (t) = 2(t- y) +1, 0≤ t ≤ 2 ly(0) = 1, for which the exact solution is y(t) = e-²t + t. (a) Show that this IVP is well-posed on D = {(t,y)|0 ≤ t ≤ 2, -∞ < y<∞0}. (b) Using the error bound at t = 2, hM YN - YN ≤ -[e¹(b-a) – 1] 2L where yN = y(2), YN is the numerical approximation to y(2), L is the Lipschitz constant, M = max ly"(t)], and [a, b] = [0,2], find the value of h necessary for the error to be at most 10-5. Ost≤2
Consider the initial value problem (y' (t) = 2(t- y) +1, 0≤ t ≤ 2 ly(0) = 1, for which the exact solution is y(t) = e-²t + t. (a) Show that this IVP is well-posed on D = {(t,y)|0 ≤ t ≤ 2, -∞ < y<∞0}. (b) Using the error bound at t = 2, hM YN - YN ≤ -[e¹(b-a) – 1] 2L where yN = y(2), YN is the numerical approximation to y(2), L is the Lipschitz constant, M = max ly"(t)], and [a, b] = [0,2], find the value of h necessary for the error to be at most 10-5. Ost≤2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. Consider the initial value problem
(y' (t) = 2(t- y) + 1, 0 ≤ t ≤2
{o=1,
for which the exact solution is y(t) = e−²t + t.
(a) Show that this IVP is well-posed on D = {(t, y)|0 ≤ t ≤ 2,-∞ < y < ∞0}.
(b) Using the error bound at t = 2,
where YN
YN YN ≤ -[eL(b-a) – 1]
hM
2L
= y(2), Y is the numerical approximation to y(2), L is the Lipschitz constant,
M = max ly" (t)], and [a, b] = [0,2], find the value of h necessary for the error to be at most 10–5.
0≤t≤2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2F2836320a-a846-4243-bbc8-0f2e53a32bf7%2Fx00b3oi_processed.png&w=3840&q=75)
Transcribed Image Text:3. Consider the initial value problem
(y' (t) = 2(t- y) + 1, 0 ≤ t ≤2
{o=1,
for which the exact solution is y(t) = e−²t + t.
(a) Show that this IVP is well-posed on D = {(t, y)|0 ≤ t ≤ 2,-∞ < y < ∞0}.
(b) Using the error bound at t = 2,
where YN
YN YN ≤ -[eL(b-a) – 1]
hM
2L
= y(2), Y is the numerical approximation to y(2), L is the Lipschitz constant,
M = max ly" (t)], and [a, b] = [0,2], find the value of h necessary for the error to be at most 10–5.
0≤t≤2
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