� Problem Set 2 : More on the Heat Problem 18.303 Linear Partial Differential Equations Matthew J. Hancock 1. Find the Fourier sine and cosine series of 1 f (x) = (1 − x) , 0 < x < 1. 2 (a) State a theorem which proves convergence of each series in (a). Graph the func-tions to which they converge. (b) Show that the Fourier sine series cannot be differentiated termwise (term-by-term). Show that the Fourier cosine series converges uniformly. 2. Prove uniqueness for Problem 4 on Assignment 1, ∂2∂u u ∂u = ; (0, t) = 0 = u (1, t) ; u (x, 0) = f (x)∂t ∂x2 ∂x where t > 0, 0 ≤ x ≤ 1 and f is a piecewise smooth function on [0, 1]. 3. Recall Problem 3 on Assignment 1, ∂2∂u u ∂u ∂u = ; (0, t) = 0 = (1, t) ; u (x, 0) = f (x)∂t ∂x2 ∂x ∂x where t > 0, 0 ≤ x ≤ 1 and f is a piecewise smooth function on [0, 1]. Prove that the average temperature 1 ¯u (t) = u (x, t) dx 0 is a constant for any solution of this problem. Why is this reasonable physically? Use your solution to Problem 3 (you don’t have to re-derive it) to show that limt→∞ u (x, t) = ¯ u is the constant average temperature. u, where ¯1 Fall 20064. A rod of homogeneous radioactive material lies along the x-axis, 0 ≤ x ≤ l. The neutron density n (x, t) at position x and time t is affected by two processes - fission and diffusion. Conservation of neutrons leads to the PDE, ∂2∂n n = D + kn ∂t ∂x2 where D is a diffusion coefficient and k is a fission constant, with D > 0, k > 0. Suppose that n = 0 at the ends of the rod. Show that the rod will explode (n → ∞) if and only if π2D k > . l2 5. Consider the inhomogeneous generalized heat equation ∂u ∂2u ∂u ∂t = ∂x2 + b ∂x + cu + g (x, t) (1) where b, c are constants. (a) Show that if u is a solution to (1), then v (x, t) = e αx+βt u (x, t) satisfies the standard heat equation ∂2∂v v = + h (x, t)∂t ∂x2 for suitable choices of the constants α, β and function h (x, t). In this way, more complicated heat problems can be simplified. (b) Now assume b = c = 0 and g = g0 is a constant. Suppose the BCs and IC are all homogeneous, u (0, t) = 0 = u (1, t) ; u (x, 0) = 0. Find the equilibrium solution uE (x) to (1) and, without using your results in part (a), transform (1) to a standard homogeneous problem for a temperature function w (x, t). (c) Continuing from part (b), show that for large t, u (x, t) ≈ uE (x) + Ce −π2t sin πx where C is some constant. Find C and comment on the physical significance of its sign. Illustrate the solution qualitatively by sketching typical spatial temperature profiles with t = constant and the temperature time profile at x = 1/2. 2�6. Consider the inhomogeneous heat problem ∂2∂u u = ; u (0, t) = a (t) , u (1, t) = b (t) ; u (x, 0) = f (x) (2) ∂t ∂x2 with inhomogeneous boundary conditions, where a (t) and b (t) are given continuous functions of time. (a) Show that (2) has at most one solution. (b) Transform (2) into a standard problem (i.e. one with homogeneous BCs) in terms of the unknown function v (x, t). (c) Now assume a (t), b (t) are constants and f (x) = 0. Find the equilibrium solution uE (x) to (2). (d) Continuing from part (c), show that for large t, u (x, t) ≈ uE (x) + Ce −π2t sin πx where C is some constant. Find C. Hint: use the approximate solution for the homogeneous heat problem we considered in class. 7. Fourier’s Ring. Consider a slender homogeneous ring which is insulated laterally. Let x denote the distance along the ring and let l be the circumference of the ring. From physics (see Haberman §2.4.2) , the temperature u (x, t) satisfies, in dimensionless form, ut = uxx; 0 < x < 2, t > 0 (3) u (x + 2, t) = u (x, t) ; t > 0 u (x, 0) = f (x) 0 < x < 2. The boundary condition (middle equation) merely states that the temperature is con-tinuous as you go around the ring. (a) Use separation of variables and Fourier Series to obtain the solution to (3): ∞ u (x, t) = A0 + e −n2π2t (An cos (nπx) + Bn sin (nπx)) n=1 Give formulae for the coefficients An, Bn in terms of f (x). 3� � (b) Prove that (3) has at most one solution. Hint: consider 2 Δ (t) = (u1 (x, t) − u2 (x, t))2 dx 0 where u1, u2 are solutions to (3). 8. Determine which of the following operators are linear: (a) L (u) = ut + x2uxx (b) L (u) = uuxx 2(c) L (u) = ex tuxx 1(d) L (u) = uxx − 0 ut (y, t) dy
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