A civ boundary value problem for the wave equation

Abstract

We consider the eigenvalue wave equation in 3 dimensions u_tt—Δu+pu=λ∣x∣^-2u, where u ε R, is a function of (x,y,z,t) ε R⁴, with t ≥ 0. Here r is the usual Euclidean norm r²=x²+y²+z². In the characteristic cone k = {(x,y,z,): 0<t<1,t<r<2–t} we impose boundary conditions along characteristic so that;tu(x,y,z,t)|_r=t =(1+t)u(x,y,z,1-t)|_r=1+t,0≤t≤1. We also asume that the conditions;u(x,y,z,1)|_r=1 = 0 = u(x,y,z,0),0≤r≤2, are satisfied. If we rewrite the wave equation in spherical coordinates and separate the variables we obtain three eigenvalue equations. The eigenvalues and eigenfunctions of these equations will provide the eigenvalues and eigenfunctions of the CIV boundary value problem for the wave equation in the characteristic cone k.

Keywords:

• Characteristics
• eigenvalues
• eigenfunctions
• Riemann function
• Bessel function
• Green's function
• Legendre polynomials

AMS:

• 35L05
• 35P99
• 35L20