Math 4263 Homework Set 1

Transcription

1 Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that cross the line y = 1 at t = 0. (b) Solve the following PDE/BVP dx dt = x, dy dt = y xφ x + yφ y = y φ(x, 1) = x + 1 by first finding solutions of the PDE/BVP along the curves γ determined in Part (a) and then extending these solutions coherently to arbitrary points in R (a) Find curves γ : t [x (t), y (t)] such that (b) Solve the following PDE/BVP: dx dt dy dt = y = 2y y φ φ + 2y x y = xy φ (x, 1) = x Use the Method of Characteristics to show that the solution of uu x + u y = 0, u(x, 0) = f(x) is given implicitly by u = f (x uy) and verify this result by direct differentiation. 6. Use the Method of Characteristics to solve φ x + φ y + φ = ex+wy φ (x, 0) = 0 1

2 2 Homework Set 2 1. Apply Separation of Variables to the Wave Equation φ tt c 2 φ xx = 0 to obtain four distinct one-parameter families of linearly independent solutions. 2. Solve u tt c 2 u xx = 0, u (x, 0) = e x, u t (x, 0) = sin (x) 3. Solve u xx 3u xt 4u tt = 0, u (x, 0) = x 2, u t (x, 0) = e x. (Hint: find a change of variables that factors the differential operator as we did for the wave equation.). 4. Suppose φ (x) and ψ (x) are both odd functions of x: that is, to say φ ( x) = φ (x) for all x and similarly for ψ (x). Show that the solution of u tt c 2 u xx = 0, x < u (x, 0) = φ (x), x < u t (x, 0) = ψ (x), x < is an odd function of x for all t; i.e. u ( x, t) = u (x, t) for all t. 5. Solve u tt c 2 u xx = 0, 0 x < u (x, 0) = φ (x), 0 x < u t (x, 0) = ψ (x), 0 x < u t (0, t) = 0 t 6. Solve u tt c 2 u xx = xt, x < u (x, 0) = 0, x < u t (x, 0) = 0, x <

3 Homework Set 3 1. Use the Maximum Principle for the Heat Equation to demonstrate that there is a unique solution to 2. Prove the following identities π u t k 2 u xx = f (x, t), 0 x L, t > 0 (1a) u (0, t) = g (t), t > 0 (1b) u (L, t) = h (t), t > 0 (1c) u (x, 0) = φ (x), 0 x L (1d) π π π π π sin (mx) sin (nx) dx = { π if m = n 0 if m n 3 (2a) sin (mx) cos (nx) dx = 0 (2b) cos (mx) cos (nx) dx = 3. Consider the following Heat Equation boundary value problem: { π if m = n 0 if m n (2c) u t k 2 u xx = 0, 0 x L, t > 0 (3a) u (0, t) = 0, t > 0 (3b) u (L, t) = 0, t > 0 (3c) u (x, 0) = φ (x), 0 x L (3d) (a) Apply the method of Separation of Variables to find a family of solutions of (3a) the form u (x, t) = X (x) T (t). (b) Impose the boundary conditions (3b) and (3c) to find a more specialized family of solutions u n (x, t) = X n (x) T n (t) satisfying (1a) (1c). (c) Set u (x, t) = a n u n (x, t) n where the u n (x, t) are the solutions found in (b), impose (3d), and then use properties of Fourier expansions to determine the coefficients a n. 4. Find the solution of the following PDE/BVP: u t u xx = 0, 0 x 1, t > 0 (4a) u (0, t) = 0, t > 0 (4b) u (1, t) = 0, t > 0 (4c) u (x, 0) = 1 x 2, 0 x 1 (4d)

4 4 Homework Set 4 1. Prove the Maximum Principle for solutions of the homogeneous Laplace equation: i.e., show that any solution of u xx + u yy = 0, 0 x a, 0 y b attains its maximal value on one of the four boundary lines l 1 = {(x, 0) 0 x a} l 2 = {(a, y) 0 y b} l 3 = {(x, b) 0 x a} l 4 = {(0, y) 0 y b} 2. Use the result of the preceding problem to prove that there exists at most one solution to u xx + u yy = f (x, y) u (x, 0) = φ 1 (x) u (a, y) = φ 2 (y) u (x, b) = φ 3 (x) u (0, y) = φ 4 (y) 3. Solve u xx + u yy = 0, x a u (a cos θ, a sin θ) = sin θ 4. Construct the solution to Laplace s equation on the annular region R = { (x, y) R 2 a 2 x 2 + y 2 b 2} subject to the boundary conditions u (a cos θ, a sin θ) = g (θ) u (b cos θ, b sin θ) = h (θ)

5 5 Homework Set 5 1. Determine if the following ODEs are of the Sturm-Liouville type [ d p (x) dy ] q (x) y + λr (x) y = 0, p (x) > 0, r (x) > 0 dx dx and if so identify the functions p (x), q (x) and r (x). (a) ( 1 + x 2) y 2xy + l (l + 1) y = 0 (b) x 2 y + xy + ( x 2 n 2) y = 0 2. (a) Find the Sturm-Liouville eigenfunctions {φ n } forfor the following Sturm-Liouville system y + λ 2 y = 0, y (0) = 0, y (1) + y (1) = 0 (b) Suppose f (x) is a continuous function of [0, 1]. Give an integral formula for the coefficients a n corresponding to the expansion f (x) = a n φ n (x) where the functions φ n (x) are the Sturm-Liouville eigenfunctions found in part (a). i=1

6 6 Homework Set 6 1. Show that a function f (z) = u (z) + iv (z) of a complex variable z = x + iy that satisfies the Cauchy- Riemann equations u x = v u, y y = v x also has the property that both its real part u (z) and its imaginary part v (z) satisfy Laplace s equation: i.e., u xx + u yy = 0 = v xx + v yy 2. Let g (x) be any piecewise continuous function on R. Show directly from the definition, that the mapping φ g : Cc R given by φ g (f) := f (x) g (x) dx defines a distribution. It will be easy to show that φ g defines a linear functional. The hard part will be to demonstrate that if φ g is continuous. For this purpose show that if {f n } n N Cc (R) converges uniformly to a function f (x) Cc (R), then lim φ g (f n ) = φ g (f) n By the way, uniform convergence means the following {f n } converges uniformly to f if for every ε > 0 there exists a natural number such that f n (x) f (x) < ε for all x R and all n > N. 3. Let ψ be any distribution. Verify that the functional ψ defined by ( ) df ψ (f) := ψ dx is a distribution. 4. Let u (x) = u (x, y) be a harmonic function on a planar domain D. Derive the representation formula u (x 0 ) = 1 [u (x) ( ln x x 0 ) ( u (x)) ln x x 0 ] n ds 2π D that expresses u (x 0 ) at an interior point x 0 as a certain integral of u (x) and its gradient over the boundary of D. 5. A Green s function G y (x) for the Laplace operator 2 and domain D and a point y D, is a function defined for all x in D such that G y (x) posseses continuous second derivatives and 2 G y (x) = 0; except at the point x = y. G y (x) = 0 for all x on the boundary D of D. The function 1 G y (x) + 4π x y is finite at y, has continuous second partial derivatives everywhere and is harmonic at y. Show that such a function is unique. (You can assume such a function always exists - this is, in fact, true.)

7 1. Homework Set 7 (a) Use the Taylor expansion formula f (x + x, y) = f (x, y) + f x (x, y) x f 2 x 2 (x, y) ( x) f 6 x 3 (x, y) ( x)3 + O (( x) 4) to derive the following approximations u u (x, t + t) u (x, t) (x, t) + O ( t) t t 2 u (x, t) u (x + x, t) 2u (x, t) + u (x x, t) x 2 ( x) 2 + O (( x) 2) (b) Let x i i x, t j j t and u i,j u (x i, t j ). Use the approximations developed in part (a) to develop a recursive formula yielding an approximate numerical solution for the following Heat Equation problem 2. Consider the problem u t a 2 u xx = f (x, t) u (x, 0) = h (x) u (0, t) = 0 u (L, t) = 0 u xx + u yy = 4 0 x 1, 0 y 1 u (0, y) = 0 u (1, y) = 0 u(x, 1) = 0 u (x, 0) = 0 on the unit square. Partition the square into four triangles by utilizing its diagonals and then use the Finite Element Method to find an approximate value for u ( 1 2, 1 2). 3. Use the Laplace transform method to solve (3a) (3b) (3c) (3d) u t ku xx = 0, 0 x L, t > 0 u (x, 0) = 1 + sin (πx/l) u (0, t) = 1 u (L, t) = 1 (See Example 3 on page 354 of the text.) 7