We study the general solution of equation □B,cku(x)=f(x), where □B,ck is the ultrahyperbolic Bessel operator iterated k-times and is defined by □B,ck=[(1/c2)(Bx1+Bx2+⋯+Bxp)−(Bxp+1+⋯+Bxp+q)]k,p+q=n, n is the dimension of ℝn+={x:x=(x1,x2,…,xn),x1>0,…,xn>0}, Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi), 2vi=2βi+1, βi>−1/2, xi>0 (i=1,2,…,n), f(x) is a given generalized function, u(x) is an unknown generalized function, k is a nonnegative integer, c is a positive constant, and x∈ℝn+.
1. Introduction
The n-dimensional ultrahyperbolic operator □k iterated k-times is defined by □k=(∂2∂x12+∂2∂x22+⋯+∂2∂xp2-∂2∂xp+12-∂2∂xp+22-⋯-∂2∂xp+q2)k,
where p+q=n,n is the dimension of space ℝn, and k is a nonnegative integer.
Consider the linear differential equation of the form □ku(x)=f(x),
where u(x) and f(x) are generalized functions and x=(x1,x2,…,xn)∈ℝn.
Gel'fand and Shilov [1] first introduced the fundamental solution of (1.2), which is a complicated form. Later, Trione [2] has shown that the generalized function R2k(x), defined by (2.8) with |v|=0, is a unique fundamental solution of (1.2) and Téllez [3] also proved that R2k(x) exists only in the case when p is odd with n odd or even and p+q=n. A wealth of some effective works on the fundamental solution of the n-dimensional classical ultrahyperbolic operator have, presented by Kananthai and Sritanratana [4–9].
In 2004, Yildirim et al. [10] have introduced the Bessel ultrahyperbolic operator iterated k-times with x∈ℝn+={x:x=(x1,x2,…,xn),x1>0,…,xn>0},□Bk=(Bx1+Bx2+⋯+Bxp-Bxp+1-⋯-Bxp+q)k,
where p+q=n,Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi),2vi=2βi+1,βi>-1/2 [11], k is a nonnegative integer, and n is the dimension of ℝn+. They also have studied the fundamental solution of Bessel ultrahyperbolic operator.
In 2007, Sarikaya and Yildirim [12] have studied the weak solution of the compound Bessel ultrahyperbolic equation and also studied the Bessel ultrahyperbolic heat equation [13].
In 2009, Saglam et al. [14] have developed the operator of (1.3), defined by (1.6), and it is called the ultrahyperbolic Bessel operator iterated k-times. They have also studied the product of the ultrahyperbolic Bessel operator related to elastic waves.
Next, Srisombat and Nonlaopon [15] have studied the weak solution of □B,cku(x)=f(x),
where u(x) and f(x) are some generalized functions. They have developed (1.4) into the form ∑k=0mCk□B,cku(x)=f(x),
which is called the compound ultrahyperbolic Bessel equation. In finding the solution of (1.5), they have used the properties of B-convolution for the generalized functions.
The purpose of this study is to find the general solution of equation □B,cku(x)=f(x), where □B,ck is the ultrahyperbolic Bessel operator iterated k-times and is defined by □B,ck=[1c2(Bx1+Bx2+⋯+Bxp)-(Bxp+1+⋯+Bxp+q)]kp+q=n, n is the dimension of ℝn+={x:x=(x1,x2,…,xn),x1>0,…,xn>0},Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi),2vi=2βi+1,βi>-1/2,xi>0(i=1,2,…,n),f(x) is a given generalized function, u(x) is an unknown generalized function, k is a nonnegative integer, c is a positive constant, and x∈ℝn+.
2. Preliminaries
Let Txy be the generalized shift operator acting on the function φ, according to the law [11, 16]: Txyφ(x)=Cv*∫0π⋯∫0πφ(x12+y12-2x1y1cosθ1,…,xn2+yn2-2xnyncosθn)×(∏i=1nsin2vi-1θi)dθ1⋯dθn,
where x,y∈ℝn+ and Cv*=∏i=1n(Γ(vi+1)/Γ(1/2)Γ(vi)). We remark that this shift operator is closely connected to the Bessel differential operator [11]: d2Udx2+2vxdUdx=d2Udy2+2vydUdy,U(x,0)=f(x),Uy(x,0)=0.
The convolution operator is determined by the Txy as follows: (f*φ)(y)=∫Rn+f(y)Txyφ(x)(∏i=1nyi2vi)dy.
The convolution (2.3) is known as a B-convolution. We note the following properties of the B-convolution and the generalized shift operator.
Txy·1=1.
Tx0·f(x)=f(x).
If f(x),g(x)∈C(ℝn+),g(x) is a bounded function all x>0, and
∫Rn+|f(x)|(∏i=1nxi2vi)dx<∞,
then
∫Rn+Txyf(x)g(y)(∏i=1nyi2vi)dy=∫Rn+f(y)Txyg(x)(∏i=1nyi2vi)dy.
From (c), we have the following equality forg(x)=1:
∫Rn+Txyf(x)(∏i=1nyi2vi)dy=∫Rn+f(y)(∏i=1nyi2vi)dy.
(f*g)(x)=(g*f)(x).
Definition 2.1.
Let x=(x1,x2,…,xn) be a point of the n-dimensional space ℝn+. Denote the nondegenerated quadratic form by
V=c2(x12+x22+⋯+xp2)-xp+12-xp+22-⋯-xp+q2,
where p+q=n. The interior of the forward cone is defined by Γ+={x=(x1,…,xn)∈ℝn+:xi>0,i=1,…,nandV>0}, where Γ¯+ designates its closure. For any complex number α, we define
Rα,cH(x)={V(α-n-2|v|)/2Kn(α),forx∈Γ+,0,forx∉Γ+,
where
Kn(α)=π(n+2|v|-1)/2Γ((2+α-n-2|v|)/2)Γ((1-α)/2)Γ(α)Γ((2+α-p-2|v|)/2)Γ((p+2|v|-α)/2).
The function Rα,cH(x) is introduced by [10, 12, 17, 18]. It is well known that Rα,cH(x) is an ordinary function if Re(α)≥n and is the distribution of α if Re(α)<n. Let suppRα,cH(x)⊂Γ¯+, where supp Rα,cH(x) denotes the support of Rα,cH(x).
By putting p=c=1 into (2.7), (2.8), and (2.9), and using the Legendre's duplication of Γ(z),Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
the formula (2.8) is reduced to
MαH(x)={V((α-n-2|v|)/2)Hn(α),forx∈Γ+,0,forx∉Γ+,
where V=x12-x22-⋯-xn2 and
Hn(α)=π(n+2|v|-1)/22α-1Γ(2+α-n-2|v|2)Γ(α2).
Note that the function MαH(x) is precisely the Bessel hyperbolic kernel of Marcel Riesz.
Lemma 2.2.
Given the equation
□B,cku(x)=δ(x),
where □B,ck is defined by (1.6) and x∈ℝn+, then we obtain u(x)=R2k,cH(x) as a fundamental solution of (2.13), where R2k,cH(x) is defined by (2.8).
The proof of this Lemma is given in [14].
Lemma 2.3.
The B-convolutions of tempered distributions.
(□B,ckδ)*u(x)=□B,cku(x), where u(x) is any tempered distribution.
Let R2k,cH(x) and R2m,cH(x) be defined by (2.8); then R2k,cH(x)*R2m,cH(x) exists and is a tempered distribution.
Let R2k,cH(x) and R2m,cH(x) be defined by (2.8); then R2k,cH(x)*R2m,cH(x)=R2k+2m,cH(x), where k and m are nonnegative integers.
The proof of this Lemma is given in [15].
Lemma 2.4.
Given that P is a hypersurface
Pδ(m)(P)+mPδ(m-1)(P)=0,
where δ(m) is the Dirac-delta distribution with m derivatives.
The proof of this Lemma is given in [1].
Lemma 2.5.
Given the equation
□B,cku(x)=0,
where □B,ck is the ultrahyperbolic Bessel operator iterated k-times, as defined by (1.6), and x∈ℝn+, then
u(x)=[R2(k-1),cH(x)](m),
defined by (2.8) with m derivatives, as a solution of (2.15) with m=((n+2|v|-4)/2),n+2|v|≥4 and n is an even dimension.
Proof.
We first show that the generalized function δ(m)(c2r2-s2), where r2=x12+x22+⋯+xp2,s2=xp+12+xp+22+⋯+xp+q2,p+q=n, is a solution of□B,cu(x)=0,
and □B,c is defined by (1.6) with k=1 and x∈ℝn+. Now for 1≤i≤p, we have
∂∂xiδ(m)(c2r2-s2)=2c2xiδ(m+1)(c2r2-s2),∂2∂xi2δ(m)(c2r2-s2)=2c2δ(m+1)(c2r2-s2)+4c4xi2δ(m+2)(c2r2-s2).
Thus, we have
1c2∑i=1p[∂2∂xi2δ(m)(c2r2-s2)+2vixi∂∂xiδ(m)(c2r2-s2)]=2pδ(m+1)(c2r2-s2)+4c2r2δ(m+2)(c2r2-s2)+4|v′|δ(m+1)(c2r2-s2)=(2p+4|v′|)δ(m+1)(c2r2-s2)+4(c2r2-s2)δ(m+2)(c2r2-s2)+4s2δ(m+2)(c2r2-s2)=(2p+4|v′|)δ(m+1)(c2r2-s2)-4(m+2)δ(m+1)(c2r2-s2)+4s2δ(m+2)(c2r2-s2)=[2p+4|v′|-4(m+2)]δ(m+1)(c2r2-s2)+4s2δ(m+2)(c2r2-s2)
by applying Lemma 2.4 with P=c2r2-s2, where |v′|=v1+v2+⋯+vp.
Similarly, we have∑i=p+1p+q[∂2∂xi2δ(m)(c2r2-s2)+2vixi∂∂xiδ(m)(c2r2-s2)]=[-(2q+4|v′′|)+4(m+2)]δ(m+1)(c2r2-s2)+4c2r2δ(m+2)(c2r2-s2)
by applying Lemma 2.4 with P=c2r2-s2, where |v′′|=vp+1+vp+2+⋯+vp+q.
Thus, we have□B,cδ(m)(c2r2-s2)=1c2∑i=1p[∂2∂xi2+2vixi∂∂xi]δ(m)(c2r2-s2)-∑i=p+1p+q[∂2∂xi2+2vixi∂∂xi]δ(m)(c2r2-s2)=[2(p+q+2|v|)-8(m+2)]δ(m+1)(c2r2-s2)-4(c2r2-s2)δ(m+2)(c2r2-s2)=[2(n+2|v|)-8(m+2)]δ(m+1)(c2r2-s2)+4(m+2)δ(m+1)(c2r2-s2)=[2(n+2|v|)-4(m+2)]δ(m+1)(c2r2-s2)
by applying Lemma 2.4 with P=c2r2-s2, where |v|=|v′|+|v′′|.
If [2(n+2|v|)-4(m+2)]=0, we obtain□B,cδ(m)(c2r2-s2)=0.
That is, u(x)=δ(m)(c2r2-s2) is a solution of (2.15) with m=(n+2|v|-4)/2,n+2|v|≥4, and n is an even dimension. Now □B,cku(x)=0 can be written in the form
□B,c(□B,ck-1u(x))=0.
From (2.17), we have
□B,ck-1u(x)=δ(m)(c2r2-s2)
with m=(n+2|v|-4)/2,n+2|v|≥4, and n being an even dimension. By Lemma 2.3(a), we can write (2.24) in the from
□B,ck-1δ*u(x)=δ(m)(c2r2-s2).B-convolving both sides of the above equation with the function R2(k-1),cH(x), we obtain
R2(k-1),cH(x)*□B,ck-1δ*u(x)=R2(k-1),cH(x)*δ(m)(c2r2-s2),□B,ck-1[R2(k-1),cH(x)]*u(x)=[R2(k-1),cH(x)](m),δ*u(x)=u(x)=[R2(k-1),cH(x)](m),
by Lemma 2.2.
It follows that u(x)=[R2(k-1),cH(x)](m) is a solution of (2.15) with m=(n+2|v|-4)/2,n+2|v|≥4 and n is an even dimension.
The generalized function δ(m)(c2r2-s2) mentioned in Lemma 2.5 has been also studied on the aspect of multiplicative product, distributional product and applications, for more details, see [19–23].
3. Main ResultTheorem 3.1.
Given the equation
□B,cku(x)=f(x),
where □B,ck is the ultrahyperbolic Bessel operator iterated k-times and is defined by (1.6), f(x) is a generalized function, u(x) is an unknown generalized function, x∈ℝn+, and n is an even, then (3.1) has the general solution
u(x)=[R2(k-1),cH(x)](m)+R2k,cH(x)*f(x),
where [R2k,cH(x)](m) is a function defined by (2.8) with m derivatives.
Proof.
B-convolving both sides of (3.1) with R2k,cH(x), we obtainR2k,cH(x)*(□B,cku(x))=R2k,cH(x)*f(x).
By Lemma 2.2, we have
□B,ck(R2k,cH(x))*u(x)=δ*u(x)=R2k,cH(x)*f(x).
So, we obtain that
u(x)=R2k,cH(x)*f(x)
is the solution of (3.1).
For a homogeneous equation □B,cku(x)=0, we have a solutionu(x)=[R2(k-1),cH(x)](m)
by Lemma 2.5. Thus the general solution of (3.1) is
u(x)=[R2(k-1),cH(x)](m)+R2k,cH(x)*f(x).
This completes the proof.
By putting c=1, (3.1) becomes the Bessel ultrahyperbolic equation□Bkw(x)=f(x),
where □Bk is the Bessel ultrahyperbolic operator iterated k-times, and is defined by (1.3), f(x) is a generalized function and w(x) is an unknown generalized function. From (3.5) we have that w(x)=R2kH(x)*f(x)
is a solution of (3.8), where R2kH(x)=R2k,1H(x) defined by (2.8).
From (3.2), we obtain that the general solution of the Bessel ultrahyperbolic equation isw(x)=[R2(k-1)H(x)](m)+R2kH(x)*f(x).
Moreover, if we put k=1,p=1 and x1=t(times), then (3.8) is reduced to the Bessel wave equation □Bw(x)=(Bt-∑i=2nBxi)w(x)=f(x),
where □B=Bt-∑i=2nBxi
is the Bessel wave operator and Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi).
Thus, we obtain w(x)=M2(x)*f(x) as a solution of the Bessel wave equation, since R2H(x) becomes M2H(x), where M2H(x) is the Bessel ultrahyperbolic kernel of Marcel Riesz, and is defined by (2.11) with α=2. And from (3.2), we obtain the general solution of Bessel wave equation as w(x)=δ(m)(x)+M2H(x)*f(x),
where δ(m)(x) is a solution of (Bt-∑i=2nBxi)w(x)=0.
Now we put V=t2-x22-x32-⋯-xn2 and s2=x22+x32+⋯+xn2. By [24], we obtain that w(x,t)=δ(m)(t2-s2)
is the solution of (3.14) with the initial conditions w(x,0)=0 and ∂w(x,0)/∂t=(-1)m2πm+1δ(x) at t=0 and x=(x2,x3,…,xn)∈ℝn-1+.
Acknowledgments
The authors would like to thank an anonymous referee who provided very useful comments and suggestions. This work is supported by the Commission on Higher Education, the Thailand Research Fund, and Khon Kaen University (contract number MRG5380118), and the Centre of Excellence in Mathematics, Thailand.
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