On the Bessel diamond and the nonlinear Bessel diamond operator related to the Bessel wave equation

摘要

In this article, we study the solution of the equation lozenge(k)(B)u(x)=f(x) where u(x) is an unknown generalized function and f is a generalized function, lozenge(k)(B) is the Bessel diamond operator iterated k times and is defined by lozenge(k)(B)=[(B-x1+B-x2+...+B-xp)2-(Bxp+1+...+Bxp+q)(2)](k) where p+q=n, B-xi=partial derivative(2)/partial derivative(2)(xi) + 2 upsilon(i)/x(i) partial derivative/partial derivative x(i), where 2 upsilon(i)=2 alpha(i)+1, alpha(i)>-1/2 [B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Uspekhi Matematicheskikh Nauk (N.S.) 6 2 (42) (1951) 102-143 (in Russian)], x(i)> 0, i=1, 2, ..., n, k, is a nonnegative integer and n is the dimension of the R-n(+).Firstly, it 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 Laplace Bessel equation and the Bessel wave equation. Finally, we study the solution of the nonlinear equation lozenge(k)(B)u(x)=f(x, Delta(k-1)(B)square(k)(B)u(x)). It is found that the existence of the solution u(x) of such an equation depends on the condition of f and Delta(k-1)(B)square(k)(B)u(x) and moreover such a solution u(x) related to the Bessel wave equation depends on the conditions of p, q and k. (c) 2006 Elsevier Ltd. All rights reserved.