On the operator ⊕k related to the wave equation and Laplacian

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Abstract

In this paper, we study the Green function of the operator ⊕k, iterated k-times and is defined byk=r=1p2x2r4j=p+1p+q2x2j4k,where p+q=n is the dimension of the space Cn, where C is a complex field, x=(x1,x2,…,xn)∈Cn and k is a nonnegative integer. At first we study the elementary solution or the Green function of the operator ⊕k and then such a solution is related to the solution of the wave equation and the Laplacian. We found that the relationships of such solutions depending on the conditions of p, q and k.

Introduction

The operator ⊕k can be expressed in the formk=r=1p2x2r2j=p+1p+q2x2j2kr=1p2x2r+ij=p+1p+q2x2jkr=1p2x2rij=p+1p+q2x2jk,where p+q=n is the dimension of Cn, i=−1 and k is a nonnegative integer. The operatorr=1p2x2r2j=p+1p+q2x2j2is first introduced by Kanathai [1] and named as the Diamond operator and denoted by♢=r=1p2x2r2j=p+1p+q2x2j2.

Let us denote the operatorsL1=∑r=1p2x2r+ij=p+1p+q2x2jandL2=∑r=1p2x2rij=p+1p+q2x2j.Thus (1.1) can be written byk=♢kLk1Lk2.Now, the operator ♢ can also be expressed in the form ♢=□△=□△ where □ is the ultra-hyperbolic operator defined by□=2x21+2x22+⋯+2x2p2x2p+12x2p+2−⋯−2x2p+q,where p+q=n and ▵ is the Laplacian defined by▵=2x21+2x22+⋯+2x2n.By putting p=1 and x1=t(time) in (1.6) then we obtain the wave operator□=2t2−∑j=1n−12x2j,and from (1.1) with q=0 and k=1, we obtain⊕=▵4p,wherep=2x21+2x22+⋯+2x2p.

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 x1=t(time).

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 writeu=x21+x22+⋯+x2p−x2p+1−x2p+2−⋯−x2p+q,p+q=n.Denote by Γ+={x∈Rn:x1>0andu>0} the interior of forward cone and Γ+ denote its closure.

For any complex number α, we define the functionRHα(u)=u(α−n)/2Kn(α)ifx∈Γ+,0ifx∉Γ+,where the constant Kn(α) is given by the formulaKn(α)=πn−1/2Γ2+α−n2Γ1−α2Γ(α)Γ2+α−p2Γp−α2.

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 equationkK(x)=δ,wherek 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 obtainK(x)=[RH2k(u)∗(−1)kRe2k(v)]∗(−1)k(−i)q/2S2k(w)∗(−1)k(i)q/2T2k(z)as an elementary solution of (3.1) where RH2k(u),Re2k(v),S2k(w) and T2k(z) are defined by , , , , respectively, with α=β=γ=ν=2k, k is a nonnegative integer.

Moreover, from (3.2) we obtain(−1)kRe−2k(v)∗[(−1)k(−i)q/2S−2k(w)]∗[(−1)k(i)q/2T−2k(z)]∗K(x)=RH

Acknowledgements

The authors would like to thank the Thailand Research Fund for financial support.

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