LAST UPDATE: Tue Dec 7 01:08:56 MST 2010 Office Hours: Tuesday 14:00-15:00; Thursday, 14:00-15:30, 16:30-17:00 ------------------------------------------------- Math. 312, Fall'10 - Homework Assignments ------------------------------------------------- Assignments based on Walter A. Strauss, Partial Differential Equations: An Introduction, 2nd Ed., Wiley, 2008 ------------------------------------------------------------------------------- ***Chapter 1, Sec.1-4 1( 8/24) Sec. 1.1 Introduction-background 2( 8/26) Sec. 1.2 First order linear PDE ------------------------- Set 1 (due 9/ 2): p.5, 1.1(4, 5e, 9), p.9, 1.2(5, 6, 9, 10) ------------------------- ------------------------------------------------------------------------------- 3( 8/31) Sec. 1.2 Problems 4( 9/ 2) Sec. 1.6 Types of 2nd order PDE; 1.3 Simple 1st order transport ------------------------- Set 2 (due 9/ 9): (1) Solve the PDE u_x + 2 x u_y + y u = 0; subject to the condition that u(0,y) = f(y), with f some given function. (2-5) 1.6(2, 3, 4, 6) (6) For the following second order linear PDE find the discriminant and characterise as elliptic, hyperbolic or parabolic. a. u_xx - 3 u_xy - 4 u_yy = u_x b. u_xx + u_xy + u_y = 0 c. u_x - u_y + u_yy = u *d. (Extra Credit) Factor the 2nd order part of the hyperbolic PDE among a-c above into a product of two linear first order operators. ------------------------------------------------------------------------------- ***Chapter 2, Sec. 1-5 5( 9/ 7) 2.1 The wave equation 6( 9/ 9) 2.1 d'Alembert formula; 2.2 Energy, causality Set 3 (due 9/16): Sec. 2.1(2, 5, 7, 10, 11), Sec. 2.2(2, 4) hint: for 2.1(10,11) follow the discussion of the wave equation with and without forcing; factor the operator as L = l1*l2, then work with L*u = f => l1*l2*u = f => { l2*u = v , l1*v = f} For the forced problem, consider changing variables: write as l1 = p_t + c1*p_x , l2 = p_t + c2*p_x (p_x = operator of partial differentiation by x, etc) and change coordinates to: r = x - c1*t , s = x - c2*t This means there are two wave speeds, c1 and c2. ------------------------------------------------------------------------------- 7( 9/14) 2.3 Diffusion: derivation from "first principles" 8( 9/16) 2.3 Maximum principle; 2.4 Solutions Set 4 (due 9/23): Sec. 2.3(1, 2, 6), Sec. 2.4(2, 4, 7, 10(a), 16) Instructions/hints: 2.3.6: consider w = v-u; use the minimum principle to show it cannot become < 0. 2.4.2: follow text derivation up to step 3, adapt to the initial conditions. 2.4.4: express the answer as some function of x,t times an integral of the form Int_(fn of x,t)^(Infinity) exp(-p^2) dp here _(.) is the lower limit, ^(.) upper limit of integration. 2.4.10: Answer: u(x,t) = x^2 + 2 k t Hint: use integration by parts to show that Int_(-inf)^(+inf) p^2 exp(-p^2) dp = sqrt(pi)/2 ------------------------------------------------------------------------------- ***Chapter 4, Sec. 1-3 9( 9/21) 4.1-3 Separation of variables; Dirichlet, Neumann, Robin BC's 10( 9/23) "" "" Set 5 (due 9/30): Sec. 4.1(4,5), Sec. 4.2(2,3), Sec. 4.3(6,13,17) ------------------------------------------------------------------------------- ***Chapter 5, Sec. 1-6 11( 9/28) 5.1 Fourier series 12( 9/30) 5.2 Even/odd functions Set 6 (due 10/07): Sec. 5.1(6,7,8), Sec. 5.2(2, 14) ------------------------------------------------------------------------------- 13(10/ 5) 5.3 Orthogonality 14(10/ 7) 5.4 Completeness; 5.6 Inhomogeneous BC ------------------------------------------------------------------------------- 15(10/12) Overview of Fourier Series Set 7 (due 10/28): Sec. 5.3(3,5), Sec. 5.4(5,7*,8), Sec.5.6(5,8,12) For problem 5.4.7, use the function f(x) = { -1+x^2, -1