with the initial condition

u(x, 0) = f(x)

for x Î Rⁿ - the n-dimensional Euclidean space. The operator à is named the Diamond operator defined by

p + q = n is the dimension of the space Rⁿ, u(x, t) is an unknown function for (x, t) = (x₁, x₂, …, x_n, t) Î Rⁿ ´ (0, ¥), f(x) is the given generalized function and c is a positive constant.

We obtain the solution of such equation which is related to the spectrum and the kernel which is so called the Diamond heat kernel. Moreover, such a Diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.