ON THE GENERALIZED BESSEL HEAT EQUATION RELATED TO THE GENERALIZED BESSEL DIAMOND OPERATOR

Abstract

In this article, we study the equation
∂
∂tu(x, t) = c
2 ⊗
m,k
B u(x, t)
with the initial condition u(x, 0) = f(x) for x ∈ R
+
n . Here the operator
⊗
m,k
B is called the Generalized Bessel Diamond Operator, iterated k
times, and is defined by
⊗
m,k
B =
Bx1 + Bx2 + · · · + Bxp
m
−

Bxp+1 + · · · + Bxp+q
mk
,
where k and m are positive integers, p + q = n, Bxi =
∂
2
∂x2
i
+
2vi
xi
∂
∂xi
,
2vi = 2αi + 1, αi > −
1
2
, xi > 0, i = 1, 2, . . . , n, n being the dimension
of the space R
+
n , u(x, t) is an unknown function of the form (x, t) =
(x1, . . . , xn, t) ∈ R
+
n ×(0, ∞), f(x) is a given generalized function and c
a constant. We obtain the solution of this equation, which is related to
the spectrum and the kernel, the so called generalized Bessel diamond
heat kernel. Moreover, the generalized Bessel diamond heat kernel is
shown to have interesting properties and to be related to the kernel of
an extension of the heat equatio

Keywords

• : Heat kern,, Dirac-delta distributi,, Bessel Diamond Operat

ON THE GENERALIZED BESSEL HEAT EQUATION RELATED TO THE GENERALIZED BESSEL DIAMOND OPERATOR

Abstract

References

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