On the operator ⊕k related to the wave equation and Laplacian
Introduction
The operator ⊕k can be expressed in the formwhere p+q=n is the dimension of Cn, and k is a nonnegative integer. The operatoris first introduced by Kanathai [1] and named as the Diamond operator and denoted by
Let us denote the operatorsandThus (1.1) can be written byNow, the operator ♢ can also be expressed in the form ♢=□△=□△ where □ is the ultra-hyperbolic operator defined bywhere p+q=n and ▵ is the Laplacian defined byBy putting p=1 and in (1.6) then we obtain the wave operatorand from (1.1) with q=0 and k=1, we obtainwhere
In this work, we can find the elementary solution K(x) of the operator ⊕k, that is ⊕kK(x)=δ where δ is the Dirac-delta distribution. Moreover, we can find the relationship between K(x) and the elementary solution of the wave operator defined by (1.8) depending on the conditions of p, q and k of (1.1) with p=1, q=n−1, k=1 and .
Also, we found that K(x) relates to the elementary solution of the Laplacian defined by , depending on the conditions of q and k of (1.1) with q=0 and k=1.
Section snippets
Preliminary
Definition 2.1 Let x=(x1,x2,…,xn)∈Rn and writeDenote by the interior of forward cone and denote its closure. For any complex number α, we define the functionwhere the constant Kn(α) is given by the formula
The function RHα is first introduced by Nozaki [4, p. 72] and is called the ultra-hyperbolic kernel of Marcel Riesz. Now RαH(u) is an ordinary function if Re(α)⩾n and
Main results
Theorem 3.1 Given the equationwhere ⊕k is the operator iterated k-times defined by (1.1), δ is the Dirac-delta distribution, x=(x1,x2,…,xn)∈Rn and k is a nonnegative integer. Then we obtainas an elementary solution of (3.1) where and T2k(z) are defined by , , , , respectively, with α=β=γ=ν=2k, k is a nonnegative integer. Moreover, from (3.2) we obtain
Acknowledgements
The authors would like to thank the Thailand Research Fund for financial support.
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