Find the energy for problem

Reference no: EM133016917

PARTIAL DIFFERENTIAL EQUATIONS ASSIGNMENT

A) Let be the next cauchy problem for Klein-Gordon equation:

utt-uxx + u = 0 ; x ∈ R t > 0

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

Show that if u is solution of (1) then:
v(t, x, y) = eiyu(t, x)

is solution of:
vtt - vxx - vyy = 0 ; (x, y) ∈ R2, t > 0
B) Using part A) propose a method to find all solutions of problem (1).
C) Find the energy for problem (1) and show that it is a conserved quantity.
D) Show that the velocity of wave propagation for the klein-gordon equation is finite.
E) Consider an infinite cylinder C = R × S1. Propose a model (PDE) that describes the propagation of waves in the cylinder. Write the problem as a Cauchy problem.
F ) Find or propose a method to solve the problem proposed in part E).
G) Analyze the solutions of F ) from the point of view of:
a) Huygens principle. b)Conservation of energy. c)Velocity of waves propagation.

Reference no: EM133016917

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