Consider the initial value problem y(1) = 2. y' = 1 + 1 ≤ t ≤ 3, (a) Use the second order Runge-Kutta modified Euler method h Yi+1 · Yi + z [ƒ (ti, Yi) + ƒ (ti+1, Y¡ + hf (ti, Y;))] to approximate the solution to the IVP with h = 1. (b) By approximately what factor would the error in your approximation decrease if instead you were to use the second order RK modified Euler method with h = 0.1? (Do not do the actual numerical calculations as in (a), just use the fact that the truncation error of this method is 0(h²).)
Consider the initial value problem y(1) = 2. y' = 1 + 1 ≤ t ≤ 3, (a) Use the second order Runge-Kutta modified Euler method h Yi+1 · Yi + z [ƒ (ti, Yi) + ƒ (ti+1, Y¡ + hf (ti, Y;))] to approximate the solution to the IVP with h = 1. (b) By approximately what factor would the error in your approximation decrease if instead you were to use the second order RK modified Euler method with h = 0.1? (Do not do the actual numerical calculations as in (a), just use the fact that the truncation error of this method is 0(h²).)
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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![6. Consider the initial value problem
y' = 1 + ²/₁
1 ≤ t ≤ 3,
(a) Use the second order Runge-Kutta modified Euler method
y(1) = 2.
Yi+1 = Y; +
h
2
− [ƒ (tį, Yį) + f (ti+1, Y; + hf (t₁, Y;))]
to approximate the solution to the IVP with h = 1.
(b) By approximately what factor would the error in your approximation decrease if instead you
were to use the second order RK modified Euler method with h = 0.1? (Do not do the actual
numerical calculations as in (a), just use the fact that the truncation error of this method is
0(h²).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2Fac68aa39-5c8d-42e4-8e7c-cfb7da33fd95%2F55q8io_processed.png&w=3840&q=75)
Transcribed Image Text:6. Consider the initial value problem
y' = 1 + ²/₁
1 ≤ t ≤ 3,
(a) Use the second order Runge-Kutta modified Euler method
y(1) = 2.
Yi+1 = Y; +
h
2
− [ƒ (tį, Yį) + f (ti+1, Y; + hf (t₁, Y;))]
to approximate the solution to the IVP with h = 1.
(b) By approximately what factor would the error in your approximation decrease if instead you
were to use the second order RK modified Euler method with h = 0.1? (Do not do the actual
numerical calculations as in (a), just use the fact that the truncation error of this method is
0(h²).)
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