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 +...)"