On the Bessel diamond and the nonlinear Bessel diamond operator related to the Bessel wave equation Original Research Article

In this article, we study the solution of the equation View the MathML source♢Bku(x)=f(x) where u(x)u(x) is an unknown generalized function and ff is a generalized function, View the MathML source♢Bk is the Bessel diamond operator iterated kk times and is defined by View the MathML source♢Bk=[(Bx1+Bx2+⋯+Bxp)2−(Bxp+1+⋯+Bxp+q)2]k Turn MathJax on where View the MathML sourcep+q=n,Bxi=∂2∂xi2+2vixi∂∂xi, where 2vi=2αi+12vi=2αi+1, View the MathML sourceαi>−12 [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)], xi>0,i=1,2,…,n,kxi>0,i=1,2,…,n,k, is a nonnegative integer and nn is the dimension of the View the MathML sourceRn+. Firstly, it found that the solution u(x)u(x) depends on the conditions of pp and qq 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 View the MathML source♢Bku(x)=f(x,ΔBk−1□Bku(x)). It is found that the existence of the solution u(x)u(x) of such an equation depends on the condition of ff and View the MathML sourceΔBk−1□Bku(x) and moreover such a solution u(x)u(x) related to the Bessel wave equation depends on the conditions of pp, qq and kk.