PDE: Assignment Four

Deadline: April 20

Question 1. Consider the following initial-boundary value problem for hyperbolic equationin a two-dimensional bounded domain Ω

utt = a2∆u, u ∈ Ω, t > 0,

u(x, y, 0) = 0, ut(x, y, 0) = 0, (x, y) ∈ Ω,

homogenous boundary conditions, (x, y) ∈ ∂Ω, t > 0

Show that there is only trivial solution to the above problem (for Dirichlet, Neumann andRobin conditions).

Question 2. Using the method of separation of variables to solve the following problemutt = a2uxx, 0 < x < l, t > 0,

u(x, 0) = sinx, ut(x, 0) = cos x, 0 < x < l,

u(0, t) = 2t, u(l, t) = t2, t > 0.

Question 3. Find a solution of the following Cauchy problemutt = uxx, −∞ < x <∞, t > 0,

u(x, 0) = φ(x), ut(x, 0) = 0, 0 < x < l,

given that

φ(x) =

x3 + 5, x > 0,

5, x = 0,

− x2 + 5, x < 0.

Question 4. Find the solution ofutt − uxx = f(x, t), −∞ < x <∞, t > 0,

u(x, 0) = ut(x, 0) = 0, −∞ < x <∞.

[Hint: Use the Duhamel’s method which is similar to the method of variation of param-eters for ordinary differential equations. We attempt to write the solution in the form:

u(x, t) =

∫ t

0

v(x, t, τ)dτ,

1

where v(x, t, τ) satisfys the corresponding homogeneous equation. In addition, assumev(x, τ, τ) = 0 for all τ > 0. ]

Question 5. Find the solution ofutt − uxx = −xt, −∞ < x <∞, t > 0,

u(x, 0) = sin x, ut(x, 0) = x, −∞ < x <∞.

[Hint: Use Duhamel’s method (see Question 4)]

Question 6. Consider the Cauchy problemutt = uxx, −∞ < x <∞, t > 0,

u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < l,

where f(x) and g(x) are both functions that vanish for |x| > M with M being a positiveconstant. Show that u(t;x) = 0 whenever |x| > M + t (for positive t only).

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