Problem 2. (20 pts)

Find from your favorite applied math book the solution to the diffusion equation

du d² u

𐐀t 𐐀х2

on the domain 0 < x < 1, t≥0 subject to initial conditions

π.Χ.

u(x,0): = sin

and boundary conditions

u(0,t) = 0, u(1,t) = 1

(Hint: Your solution should contain a sine or cosine series summation).

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a) [10 pts] What is the long-term steady state solution, and what is the steady p.d.e.

and boundary conditions that it satisfies?

b) [10 pts] Use the AE5031_HMW1_2.m code with different combinations of 4t, 4x

and plot the numerical solution at t = 0.1 and at steady state for one combination

of 4t, 4x that produces a stable solution.