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(a) Approximate y(0.1) using 1.821400

(a) Approximate y(0.1) using 1.821400

Advanced Engineering Mathematics
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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Consider the initial-value problem y' = 6y, y(0) = 1. The analytic solution is y(x) = ex. (a) Approximate y(0.1) using one step and the fourth-order RK4 method. (Round your answer to six decimal places.) 1.821400 (b) Find a bound for the local truncation error in y1. (Round your answer to nine decimal places.) 0.001180732 (c) Compare the error in yı with your error bound. The actual error, rounded to nine decimal places, is 0.000718801 which is less than the local truncation error found in part (b). (d) Approximate y(0.1) using two steps and the RK4 method. (Round your answer to six decimal places.) (e) Verify that the global truncation error for the RK4 method is O(h*) by comparing the errors in parts (a) and (d). The actual error for the answer in (d), rounded to nine decimal places, is 0.00005754: X expect the error for h = 0.05 to be approximately one-sixteenth v With global truncation error O(h4), when the step size is halved we 0.1. Comparing the two errors we see that this is the error when h the case.
Transcribed Image Text:Consider the initial-value problem y' = 6y, y(0) = 1. The analytic solution is y(x) = ex. (a) Approximate y(0.1) using one step and the fourth-order RK4 method. (Round your answer to six decimal places.) 1.821400 (b) Find a bound for the local truncation error in y1. (Round your answer to nine decimal places.) 0.001180732 (c) Compare the error in yı with your error bound. The actual error, rounded to nine decimal places, is 0.000718801 which is less than the local truncation error found in part (b). (d) Approximate y(0.1) using two steps and the RK4 method. (Round your answer to six decimal places.) (e) Verify that the global truncation error for the RK4 method is O(h*) by comparing the errors in parts (a) and (d). The actual error for the answer in (d), rounded to nine decimal places, is 0.00005754: X expect the error for h = 0.05 to be approximately one-sixteenth v With global truncation error O(h4), when the step size is halved we 0.1. Comparing the two errors we see that this is the error when h the case.
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