On the Bessel Heat Equation Related to the Bessel Diamond Operator

In this article, we study the equation

$frac{partial }{partial t}u(x,t)=c^{2}Diamond _{B}^{k}u(x,t)$

with the initial condition u(x,0)=f(x) for x∈? _n ⁺ . The operator ? _B ^k is named to be Bessel diamond operator iterated k-times and is defined by

$Diamond _{B}^{k}=bigl[(B_{x_{1}}+B_{x_{2}}+cdots +B_{x_{p}})^{2}-(B_{x_{p+1}}+cdots +B_{x_{p+q}})^{2}bigr]^{k},$

where k is a positive integer, p+q=n, (B_{x_{i}}=frac{partial ^{2}}{partial x_{i}^{2}}+frac{2v_{i}}{x_{i}}frac{partial }{partial x_{i}},) 2v _i =2α _i +1,(;alpha _{i}>-frac{1}{2}), x _i >0, i=1,2,…,n, and n is the dimension of the ? _n ⁺ , u(x,t) is an unknown function of the form (x,t)=(x ₁,…,x _n ,t)∈? _n ⁺ ×(0,∞), f(x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)):102–143, 1951; Y?ld?r?m, Ph.D. Thesis, 1995; Y?ld?r?m and Sar?kaya, J. Inst. Math. Comput. Sci. 14(3):217–224, 2001; Y?ld?r?m, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4):375–387, 2004; Sar?kaya, Ph.D. Thesis, 2007; Sar?kaya and Y?ld?r?m, Nonlinear Anal. 68:430–442, 2008, and Appl. Math. Comput. 189:910–917, 2007). We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel diamond heat kernel. Moreover, such Bessel diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.