Consider the initial-value problem y'= 5y, y(0) = 1. The analytic solution is y(x) = ²x (a) Approximate y(0.1) using one step and the fourth-order RK4 method. (Round your answer to six decimal places.) 1.648438 (b) Find a bound for the local truncation error in y₁. (Round your answer to nine decimal places.) 0.00002678 X (c) Compare the error in y, with your error bound. The actual error, rounded to nine decimal places, is 0.00028327 truncation error found in part (b). , which is less than (d) Approximate y(0.1) using two steps and the RK4 method. (Round your answer to six decimal places.) 1.679282 x one-sixteenth the case. the local (e) Verify that the global truncation error for the RK4 method is O(n) by comparing the errors in parts (a) and (d). The actual error for the answer in (d), rounded to nine decimal places, is ]. With global truncation error O(4), when the step size is halved we expect the error for h = 0.05 to be approximately the error when h = 0.1. Comparing the two errors we see that this is
Consider the initial-value problem y'= 5y, y(0) = 1. The analytic solution is y(x) = ²x (a) Approximate y(0.1) using one step and the fourth-order RK4 method. (Round your answer to six decimal places.) 1.648438 (b) Find a bound for the local truncation error in y₁. (Round your answer to nine decimal places.) 0.00002678 X (c) Compare the error in y, with your error bound. The actual error, rounded to nine decimal places, is 0.00028327 truncation error found in part (b). , which is less than (d) Approximate y(0.1) using two steps and the RK4 method. (Round your answer to six decimal places.) 1.679282 x one-sixteenth the case. the local (e) Verify that the global truncation error for the RK4 method is O(n) by comparing the errors in parts (a) and (d). The actual error for the answer in (d), rounded to nine decimal places, is ]. With global truncation error O(4), when the step size is halved we expect the error for h = 0.05 to be approximately the error when h = 0.1. Comparing the two errors we see that this is
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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Transcribed Image Text:Consider the initial-value problem y' = 5y, y(0) = 1. The analytic solution is y(x) = e³x.
(a) Approximate y(0.1) using one step and the fourth-order RK4 method. (Round your answer to six decimal
places.)
1.648438
(b) Find a bound for the local truncation error in y₁. (Round your answer to nine decimal places.)
0.00002678 X
(c) Compare the error in y₁ with your error bound.
The actual error, rounded to nine decimal places, is 0.00028327
truncation error found in part (b).
1
which is less than
(d) Approximate y(0.1) using two steps and the RK4 method. (Round your answer to six decimal places.)
1.679282 X
one-sixteenth
the case.
the local
(e) Verify that the global truncation error for the RK4 method is O(n) by comparing the errors in parts (a) and (d).
The actual error for the answer in (d), rounded to nine decimal places, is
With global
truncation error O(4), when the step size is halved we expect the error for h = 0.05 to be approximately
the error when h = 0.1. Comparing the two errors we see that this is
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