Consider the following nonlinear boundary value problem f" = -(f')²-f+ lnx, vith x € [1,2], ƒ(1) = 0, and f(2)= ln 2. (a) Find the actual solution f(x).
Consider the following nonlinear boundary value problem f" = -(f')²-f+ lnx, vith x € [1,2], ƒ(1) = 0, and f(2)= ln 2. (a) Find the actual solution f(x).
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 following nonlinear boundary value problem
f" = -(f)² - f + lnx,
(2)
with a € [1,2], f(1) = 0, and f(2)= ln 2.
(a) Find the actual solution f(x).
(b) Assume an initial value f'(1) = 4. Use the nonlinear shooting method and
Newton's method to approximate the solution to (2). For each fixed number
of iterations N = {1,...,6}, plot the error norm || ||₁ at t = 2 versus h
{10-2, 10-3,..., 10-9} on a log-log scale.
=
-3
(c) For each step size h = {10-2, 10-³,..., 10-⁹}, determine the iteration step k > 2
such that || ||1 at t = 2 for k 1 iterations is less than ||· ||₁ at t = 2 for k
iterations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed051053-4bf3-451a-ac87-e1db40d07cdc%2F5289ab8e-eac9-4368-ac71-d495f03a735a%2Fsxdho4e_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following nonlinear boundary value problem
f" = -(f)² - f + lnx,
(2)
with a € [1,2], f(1) = 0, and f(2)= ln 2.
(a) Find the actual solution f(x).
(b) Assume an initial value f'(1) = 4. Use the nonlinear shooting method and
Newton's method to approximate the solution to (2). For each fixed number
of iterations N = {1,...,6}, plot the error norm || ||₁ at t = 2 versus h
{10-2, 10-3,..., 10-9} on a log-log scale.
=
-3
(c) For each step size h = {10-2, 10-³,..., 10-⁹}, determine the iteration step k > 2
such that || ||1 at t = 2 for k 1 iterations is less than ||· ||₁ at t = 2 for k
iterations.
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