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

Authors : Aziz SAĞLAM ,  hüseyin YILDIRIM

Pages : 179-189

View : 24 | Download : 8

Publication Date : 2011-02-01

Article Type : Research Paper

Abstract :In this article, we study the equation ∂ ∂tuinsert ignore into journalissuearticles values(x, t); = c 2 ⊗ m,k B uinsert ignore into journalissuearticles values(x, t); with the initial condition uinsert ignore into journalissuearticles values(x, 0); = finsert ignore into journalissuearticles values(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 , uinsert ignore into journalissuearticles values(x, t); is an unknown function of the form insert ignore into journalissuearticles values(x, t); = insert ignore into journalissuearticles values(x1, . . . , xn, t); ∈ R + n ×insert ignore into journalissuearticles values(0, ∞);, finsert ignore into journalissuearticles values(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