Section 1.2 2. Given the following initial boundary value problem, u(x, 0) = /(x) Du (0,1) = 1; (L,t) = 8, a. determine an equilibrium temperature distribution, if one exists, and b. find the values of / for which there are such equilibrium solutions. Section 1.3 3. a. Solve the following initial boundary value problem for the heat equation du = D- 00 u(I, 0) = f(F) u(0, t) = u(L,1) =0, 1>0. when f(x) = 6sin = b. Solve the following initial boundary value problem for the diffusion equation Ou = D; 72. 00 u(x,0) = f(x) "(0,t) = (L,t) =0, 1>0, when f (x) = x L/2. Partial Diferential Equations Section 1.3 4. Determine the equilibrium temperature distribution for the thin circular ring described in Figure 2.4.1 (p. 59). a. by finding the steady state solution directly. Hint: See Section 1.4 of the textbook. b. by computing the limit as f + co of the time-dependent solution. Section 1.4 5. a. Sketch the Fourier series (on the interval -L { c > L/2