On the Bessel Heat Equation Related to the Bessel Diamond Operator

Abstract

In this article, we study the equation partial derivative/partial derivative t u(x, t) = c(2)lozenge(k)(B)u(x, t) with the initial condition u(x, 0) = f (x) for x is an element of R(n)(+). The operator lozenge(k)(B) is named to be Bessel diamond operator iterated k-times and is defined by lozenge(k)(B) = [(B(x1) + B(x2) + ... + B(xp))(2) -(B(xp+1) + ... + B(xp+q))(2)](k), where k is a positive integer, p + q = n, B(xi) = partial derivative(2)/partial derivative x(i)(2) + 2 upsilon(i)/x(i) partial derivative/partial derivative(xi), 2 upsilon(i) = 2 alpha(i) + 1, alpha(i) > -1/2, x(i) > 0, i = 1, 2,..., n, and n is the dimension of the R(n)(+), u(x, t) is an unknown function of the form (x, t) = (x(1),...,x(n), t) is an element of R(n)(+) x(0, infinity), f (x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)): 102-143, 1951; Yildirim, Ph.D. Thesis, 1995; Yildirim and Sarikaya, J. Inst. Math. Comput. Sci. 14(3): 217-224, 2001; Yildirim, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4): 375-387, 2004; Sarikaya, Ph.D. Thesis, 2007; Sarikaya and Yildirim, 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.