Final Exam Problems on Partial Differential Equations by Paul J. Atzberger, Exams of Mathematics

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Name:

Final Exam Practice Problems:

Professor: Paul J. Atzberger

Partial Differential Equations, 124A

March 11 th, 2007

Problem 1:

a) Use method of characteristics to find the general solution of { 4 ut(x, t) + 26ux(x, t) = 0, −∞ < x < ∞, t > 0 u(x, 0) = φ(x), −∞ < x < ∞, t = 0

b) Use method of characteristics to find the general solution of { 6 ut(x, t) + 3ux(x, t) = 0, −∞ < x < ∞, t > 0 u(x, 0) = φ(x), −∞ < x < ∞, t = 0

Problem 3: Classify each of the following PDE’s as Elliptic, Hyperbolic, or Parabolic:

(i) uxx − uxy − 2 uyy + 4ux + 6uy = u + 5

(ii) uyy + 2uxy + 3uxx + 3ux − 2 uy = x^3 + 1

(iii) uxy + 2uyy + uxx + πux + 2uyx = sin(xy)

(iv) uxy − 3 uyx + 3uyy + 3ux − 2 uy = x^5 +

x + 1

(v) −uyy − 2 uxy + 2uxx + 3ux + 4uxx − 2 uy = x^3 + 1

(vi) 2uyy − 2 πuxy + 3uxx + 3ux − 2 uy = e−^ sin(2π+

√2)

Problem 4: Suppose that u(x, t) satisfies the wave equation on R with a source term:   

utt(x, t) − c^2 uxx(x, t) = f (x, t), −∞ < x < ∞, t > 0 u(x, 0) = φ(x), −∞ < x < ∞, t = 0 ut(x, 0) = ψ(x), −∞ < x < ∞, t = 0

a) Give the solution of the wave equation when φ(x) = e−xx^2 , ψ(x) = e−x, and f (x, t) = 0 when c = 2.

b) Give the solution of the wave equation when φ(x) = 0, ψ(x) = e−x, and f (x, t) = e−x−t when c = 3.

c) Give the solution of the wave equation when φ(x) = sin(x)/x, ψ(x) = 0, and f (x, t) = sin(x) when c = 1.

Problem 6: Suppose that u(x, t) satisfies the wave equation on [0, ℓ] with Dirichlet boundary conditions:       

utt(x, t) = c^2 uxx(x, t), 0 < x < ℓ, t > 0 u(x, 0) = φ(x), 0 < x < ℓ, t = 0 ut(x, 0) = ψ(x), 0 < x < ℓ, t = 0 u(0, t) = 0, t > 0 u(ℓ, t) = 0, t > 0

a) Give the solution of the wave equation at time (x 0 , y 0 ) = (^14 , 2) with φ(x) = sin(πx) and ψ(x) = 0 when c = 1 and ℓ = 1.

b) Give the solution of the wave equation when (x 0 , y 0 ) = (^34 , 2) when φ(x) = 0 and ψ(x) = x − x^2 when c = 1 and ℓ = 1.

Problem 7: Suppose that u(x, t) satisfies the heat equation on R with a source term: { ut(x, t) − kuxx(x, t) = f (x, t), −∞ < x < ∞, t > 0 u(x, 0) = φ(x), −∞ < x < ∞, t = 0

a) Give the solution of the heat equation for φ(x) = e^2 −x, f (x, t) = 0 when k = 2.

b) Give the solution of the heat equation when φ(x) =

1 , if |x| < 1 0 , otherwise

, f (x, t) = 0 when

k = 1.

c) Give the solution of the heat equation when φ(x) = e−x, f (x, t) = e−x−t^ when k = 14.

Problem 9: Suppose that u(x, t) satisfies the Laplace equation on the disk x^2 + y^2 < a^2 with Dirichlet boundary conditions: { uxx(x, y) + uyy(x, y) = 0, x^2 + y^2 < a^2 u(x, y) = φ(x, y), x^2 + y^2 = a^2

a) Give the Poisson solution formula, expressing it in polar coordinates with x = r cos(θ), y = r sin(θ) and the Dirichlet boundary data φ(θ).

b) For the boundary data φ(θ) = 1/(sin^2 (θ) + 2) give the location of the maximum and minimum of the function u(x, y) on the disk. Also give the maximum and minimum values attained by u.

c) Let u(x, y) = x + y, show that u is a Harmonic Function. Using the Mean Value Theorem and Poisson’s Formula give the function φ(θ) which gives the value of u(0, 0) for a disk of radius a = 2. Write out the Mean Value Theorem for u(0, 0) in terms of polar coordinates.