EGR 509 Advanced Differential Equations ______

EGR 509 Advanced Differential Equations ______

LAST NAME, FIRST

Problem set #4

1. Find the Fourier series of f(x)

f(x) = - (x - p/2) if 0 £ x £ p and f(x) = (x + p/2) if -p £ x £ 0 .

Ans: f(x) = cos[(2k + 1)x]

2. Find the sine series expansion of x(x - 1) on the interval 0 < x < 1.

Ans: x(x - 1) = sin[(2k + 1)px]

3. Find the sine series expansion of sin px on the interval 0 < x < 1.

Ans: f(x) = sin px

4. Find the sine series expansion of (sin px)(cos px) on the interval 0 < x < 1.

Ans: f(x) = sin 2px

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

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.

Ans: f(x) = sin(np)sin(npx)

6. 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.

Ans:

linear / nonlinear / homogeneous / nonhomogeneous / order
uxx + uxy = 2u
ux(0, y) = 0
uxx - ut = f(x, t)
ut(x, 0) = 2
utux + uxt = 2u
u(0, t) + ux(0, t) = 0. / X
X
X
X
X / X / X
X
X
X / X
X / 2
2
2

7. Verify that the given function u =

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

8. Solve = c2 c = 1, L= 1

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 ³ 0

I.C. : u(x,0) = f(x) = sin px + 3sin 2px - sin 5px and

(x,0) = g(x) = 0, for 0 £ x £ L

Ans: u(x, t) = sin(px)cos(pt) + 3sin(2px)cos(2pt) - sin(5px)cos(5pt)

9. Solve = c2 c = 4, L= 1

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 ³ 0

I.C. : u(x,0) = f(x) = and

(x,0) = g(x) = 0, for 0 £ x £ L

Ans: u(x, t) = sin[(2k + 1)px]cos[4(2k + 1)pt]

10. Solve + =

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

B.C. : u(0,t) = 0 and u(p,t) = 0, for t ³ 0

I.C. : u(x,0) = sin x and (x,0) = g(x) = 0 for 0 £ x £ p

Ans: u(x, t) = e-t/2[cos t + sin t]sin x

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

(a) + 4 = 0, Elliptic everywhere

(b) 7 - 3= 0, Hyperbolic everywhere

(c) x2 + 4y + + 2 = 0,

|x| > 2|y| : Elliptic, |x| < 2|y| Hyperbolic.

(d) 3y - x = 0,

Elliptic: Second and Fourth quadrants

Hyperbolic: First and Third quadrants

(e) + 4 = 0,

Parabolic everywhere

(f) x2y2 + 2xy + = 0.

Parabolic everywhere