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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 > 0B) 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.
An expert in partial differential equations to solve an assignment related to advance wave equation. You have time to solve the assignment until Monday
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