Last Name, First

EGR 599 ______LAST NAME, FIRST Problem set #8

1. Use the solution u(x,t) = F(x + ct) + G(x  ct) to find the function that satisfies the wave u equation and the initial conditions u(x,0) = 0, (x,0) =  2xexp(x2). t

2. Use the solution u(x,t) = F(x + ct) + G(x  ct) to find the function that satisfies the wave 1 u equation and the initial conditions u(x,0) = , (x,0) =  2xexp(x2). 1  x 2 t

3. Let f(x) denote the shape of a plucked string of length p with endpoints fastened at x = 0 and x = p, as shown. y

x 0 a p a) Obtain the sine series expansion of f(x). b) Let a = 1/3, p = 1, and h = 1/10 and plotting the resulting function f(x) with 2 and 20 terms partial sums.

4. Determine whether the given partial differential equation and boundary conditions are linear or nonlinear, homogeneous or nonhomogeneous, and the order of the PDE. a) uxx + uxy = 2u, ux(0, y) = 0. b) uxx  ut = f(x, t), ut(x, 0) = 2. c) utux + uxt = 2u, u(0, t) + ux(0, t) = 0.

1 5. Verify that the given function u = x 2  y 2  z 2

is a solution of the three dimensional Laplace equation uxx + uyy + uzz = 0

 2u  2u 6. Solve = c2 c = 1, L= 1 t 2 x 2

The boundary and initial conditions required for the solution of the wave equations are

B.C. : u(0,t) = 0 and u(L,t) = 0, for t  0 I.C. : u(x,0) = f(x) = sin x + 3sin 2x  sin 5x and u (x,0) = g(x) = 0, for 0  x  L t

 2u  2u 7. Solve = c2 c = 4, L= 1 t 2 x 2

The boundary and initial conditions required for the solution of the wave equations are

B.C. : u(0,t) = 0 and u(L,t) = 0, for t  0

 2x if 0  x  1/ 2 I.C. : u(x,0) = f(x) =  and 2(1  x) if 1/ 2  x  1 u (x,0) = g(x) = 0, for 0  x  L t

8. In what regions are the following PDEs elliptic, hyperbolic or parabolic?

 2u  2u (a) + 4 = 0, x 2 y 2  2u u (b) 7  3 = 0, xy y  2u  2u  2u u (c) x2 + 4y + + 2 = 0, x 2 xy y 2 x  2u  2u (d) 3y  x = 0, x 2 y 2  2u u (e) + 4 = 0, x 2 x  2u  2u  2u (f) x2y2 + 2xy + = 0. x 2 xy y 2

9. Use d’Alembert’s formula

1 1 xct u(x,t) = [f*(x  ct) + f*(x + ct)] + g * (s)ds 2 2c xct

 2u  2u to solve = c2 c = 1, L= 1 t 2 x 2

The boundary and initial conditions required for the solution of the wave equations are B.C. : u(0,t) = 0 and u(L,t) = 0, for t  0 u I.C. : u(x,0) = sin x + 3 sin 2x and (x,0) = sin x, for 0  x  L t