Y(t) is the exact solution of an initial value problem J (T, Y) Jor t E [to, 1 ], Y(to) Yo. Assume y(t) is arbitrary times continuously differentiable: (1) If Euler's method is used to approximate y(t), the local truncation error, Ti+1 is defined as y(ti+1) – (y(t:) + hf(t;, y(t;))) ti+1 – ti. Using the Taylor series expansion of Tit1 = - where h is the step size, so that h y(ti+1) at ti, derive a series expansion for Ti+1. (You only need to write down the first 4 terms for the Taylor expansion)
Y(t) is the exact solution of an initial value problem J (T, Y) Jor t E [to, 1 ], Y(to) Yo. Assume y(t) is arbitrary times continuously differentiable: (1) If Euler's method is used to approximate y(t), the local truncation error, Ti+1 is defined as y(ti+1) – (y(t:) + hf(t;, y(t;))) ti+1 – ti. Using the Taylor series expansion of Tit1 = - where h is the step size, so that h y(ti+1) at ti, derive a series expansion for Ti+1. (You only need to write down the first 4 terms for the Taylor expansion)
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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![y(t) is the exact solution of an initial value problem = f(t, y) for t E \to, T], y(to) = yo-
Assume y(t) is arbitrary times continuously differentiable:
(1) If Euler's method is used to approximate y(t), the local truncation error, T;41 is defined
as
Tit1
y(ti+1) – (y(t:) + hf(t;, y(t;)))
where h is the step size, so that h
y(ti+1) at ti, derive a series expansion for Ti+1. (You only need to write down the first 4
terms for the Taylor expansion)
ti+1 – ti. Using the Taylor series expansion of
(2) If |y"| < M < ∞ for t E [to, T], what is the local truncation error bound?
h
then tit1
2
t; +5, give the Taylor series expansion
(3) If change the step size from h to
of y(t;+1) = y(t; + ) about t;. (You only need to write down the first 4 terms for the
Taylor expansion)
(4) If the step size is , the local truncation error of Euler's method is
= y(ti+1) -
h
y(t:) +f(ti, y(t;))
Tit1
Using the Taylor series expansion from (3), derive a series expansion for Ti+1. (You
only need to write down the first 4 terms for the Taylor expansion)
(5) If the step size is ;, y"| < M < ∞ for t e [to, T], what is the local truncation error
bound?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2Fb60b1239-ddac-4672-83b2-eabd9c5ff228%2Fgamvbrp_processed.png&w=3840&q=75)
Transcribed Image Text:y(t) is the exact solution of an initial value problem = f(t, y) for t E \to, T], y(to) = yo-
Assume y(t) is arbitrary times continuously differentiable:
(1) If Euler's method is used to approximate y(t), the local truncation error, T;41 is defined
as
Tit1
y(ti+1) – (y(t:) + hf(t;, y(t;)))
where h is the step size, so that h
y(ti+1) at ti, derive a series expansion for Ti+1. (You only need to write down the first 4
terms for the Taylor expansion)
ti+1 – ti. Using the Taylor series expansion of
(2) If |y"| < M < ∞ for t E [to, T], what is the local truncation error bound?
h
then tit1
2
t; +5, give the Taylor series expansion
(3) If change the step size from h to
of y(t;+1) = y(t; + ) about t;. (You only need to write down the first 4 terms for the
Taylor expansion)
(4) If the step size is , the local truncation error of Euler's method is
= y(ti+1) -
h
y(t:) +f(ti, y(t;))
Tit1
Using the Taylor series expansion from (3), derive a series expansion for Ti+1. (You
only need to write down the first 4 terms for the Taylor expansion)
(5) If the step size is ;, y"| < M < ∞ for t e [to, T], what is the local truncation error
bound?
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