Question: Problem 2 (eigenfunction expansion, Haberman 3.4.11) Consider (x,t) = (1 -e-*3/4) sin(xx/2) and wr,0) -esin(ar) on (0.2) x (0, co). (a) Find the initial/boundary value

Problem 2 (eigenfunction expansion, Haberman 3.4.11) Consider (x,t) = (1 -e-*3/4) sin(xx/2) and wr,0) -esin(ar) on (0.2) x (0, co). (a) Find the initial/boundary value problem (for a forced heat equation) satisfied by (b) Find the initial/boundary value problem satisfied by w. (c) Find the initial/boundary value problem satisfied by v + w. (d) Find a point I E (0, 2) at which the temperature v( z, () +wr(x, t) first decreases and then increases. Find a point & E (0, 2) at which the temperature v(3, !) + w(x, () only increases. Can you increase the forcing to ensure the temperature at all points only increases? Hint(s): This means to consider av( r, t) + w(x, t) for a > 1. If you're stuck you might want to look at part (e) first and then come back to this part

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