On The Product of The Ultra-Hyperbolic Bessel Operator Related to The Elastic Waves (original) (raw)

We study the solution of the equation □ B,c 1 k □ B,c 2 k u(x)=δ, where u(x) is an unknown function and δ is a Dirac-delta function, □ B,c 1 k and □ B,c 2 k are the ultra-hyperbolic Bessel operator iterated k-times and are defined by □ B,c 1 k =1 c 1 2 (B x 1 +B x 2 +⋯+B x p )-(B x p+1 +⋯+B x p+q ) k □ B,c 2 k =1 c 2 2 (B x 1 +B x 2 +⋯+B x p )-(B x p+1 +⋯+B x p+q ) k where p+q=n, B x i =∂ 2 ∂x i 2 +2v i x i ∂ ∂x i , where 2v i =2α i +1, α i >-1 2[6], x i >0, i=1,2,⋯,n, c 1 and c 2 is positive constant, k is a nonnegative integer and n is the dimension of the ℝ n + . Firstly, it is found that the solution u(x) depends on the conditions of p and q and moreover such a solution is related to the solution of the ultra-hyperbolic Bessel operator iterated k-times.