View the step-by-step solution to:

Question

2.Consider a metal rod (0<x<l), insulated along its sides but not at its ends, which is initially at

temperature = 1. Suddenly both ends are plunged into a bath of temperature = 0. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature u(x,t) at later times. In this problem, assume the infinite series expansion


1=(4/pi)*(sin(pix)/l + 1/3 sin(3pix)/l + 1/5 sin(5pix)/l +....)






In Exercise 2 above, adopt l=1, and show that, for long times, the solution has the shape u(x,t) ~ sin(pix), in the sense that there exists a function a(t) such that


lim(t ->infinity) ((u(x,t)-a(t)sin(pix))/((a(t)sin(pix)) -> 0.


Can you modify the initial data so that this is no longer the case? What is the minimal change that will work (in the sense of changing the norm of the initial data as little as possible)?







May you help me with the question below starting from "In Exercise 2 above..." all the way to "as little as possible"? The "Exercise 2 above" in the sentence refers to Exercise 2 on top starting from "Consider a metal rod...." to "1/5 sin(5pix)/l +...)"

Recently Asked Questions

Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.

  • -

    Study Documents

    Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

    Browse Documents
  • -

    Question & Answers

    Get one-on-one homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!

    Ask a Question
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes