Math 126 - Homework 4 (Due Friday Sept 25) 1. Maximum principle: Consider the heat equation (H) ut uxx = 0, u(x, 0) = (x), < x < , t > 0 < x < . As it turns out, there are innitely many solutions u of the above heat equation. All but one solution are \"non-physical\" and grow exponentially fast as x (see, e.g., Fritz John, Partial Dierential Equations, Chapter 7). In this question, you will show that if M , then any bounded solution u of (H) satises u M . This is a maximum principle for (H), and can be used to establish uniqueness of bounded solutions of (H). Throughout the question let u be a bounded solution of (H); this means there exists C > 0 such that |u(x, t)| C for all (x, t). (a) Show that w(x, t) = x2 + 2t solves the heat equation. (b) For every > 0 show that u(x, t) w(x, t) + M for all x R and t > 0, where M > 0 is any number satisfying (x) M for all x R. [Hint: For N > 0 let RN denote the rectangle RN = [N, N ] [0, N ]. Show that there exists N > 0 such that for all N > N , u w + M on the sides and base of RN . Then apply the comparison principle from HW2, problem 7.] (c) Let M > 0 such that (x) M for all x R. Show that u M . (d) Show that there is at most one bounded solution u of (H) when is bounded. [Hint: Take two bounded solutions u, v and consider w = u v.] It is possible to prove a stronger result; namely that there is at most one solution u of (H) satisfying the exponential growth estimate u(x, t) Cex 2 for a constant C > 0. The proof is similar to this exercise, except that w has a dierent form (given in HW2 #6). This means that the \"non-physical\" solutions all grow faster 2 than ex as x . 2. (a) Use the Fourier expansion to explain why the note produced by a violin string rises by one octave when the string is clamped exactly at its midpoint. [Each increase by an octave corresponds to a doubling of the frequency.] (b) Explain why the pitch of the note rises when the string is tightened. 3. A quantum-mechanical particle on the line with an innite potential outside the interval (0, l) (\"particle in a box\") is given by Schrdinger's equation ut = iuxx on (0, l), with Dirichlet conditions at the ends. Separate variables to nd its representation as a series. 4. Consider waves in a resistant medium, which satisfy utt c2 uxx + rut = 0, (W) u(x, 0) = (x), u(0, t) = u(l, t) = 0 the equation 0 < x < l, t > 0 0 0, where r is a constant, 0 < r < 2c/l. Write down the series expansion of the solution. 5. Consider the equation utt = c2 uxx for 0 < x < l with boundary conditions ux (0, t) = u(l, t) = 0. 1 (a) Show that the eigenfunctions are Xn (x) = cos (n + 2 )x/l . (b) Write down the series expansion for a solution u(x, t). 6. Consider diusion inside an enclosed circular tube. Let its length (circumference) be 2l. Let x denote the arc length parameter where l < x < l. Then the concentration of the diusion substance satisfes ut kuxx = 0, l < x < l, t > 0 u(l, t) = u(l, t) t > 0. These are called periodic boundary conditions. (a) Show that the eigenvalues are = (n/l)2 for n = 0, 1, 2, 3, . . . . (b) Show that the concentration is 1 u(x, t) = A0 + 2 An cos n=1 nx nx + Bn sin l l exp n2 2 kt l2 . 7. Let (x) = x2 for 0 x 1 = l. (a) Calculate the Fourier sine series for . (b) Calculate the Fourier cosine series for . 8. Find the Fourier cosine series of the function | sin(x)| in the interval (, ). Use it to nd the sums 1 (1)n and . 4n2 1 4n2 1 n=1 n=1 2