Selçuk J. Appl. Math. Selçuk Journal of Vol. 10. No. 1. pp. 85-93, 2009 Applied Mathematics
On The Product of The Ultra-Hyperbolic Bessel Operator Related to The Elastic Waves
Aziz Sa˘glam1, Hüseyin Yıldırım and Mehmet Zeki Sarıkaya
Department of Mathematics, Faculty of Science and Arts, Kocatepe University, Afyon, Turkey
e-mail: azizsaglam @ aku.edu.tr,hyildir@ aku.edu.tr,sarikaya@ aku.edu.tr
Received: October 23, 2008
Abstract. In this article, we study the solution of the equation¤1¤2() = where () is an unknown generalized function and is a Dirac-delta function, ¤
1and¤
2are the Ultra-Hyperbolic Bessel Operator iterated −times and
are defined by ¤ 1 = ∙ 1 2 1 ¡ 1+ 2+ + ) − (+1+ + + ¢¸ ¤ 1 = ∙ 1 2 2 ¡ 1+ 2+ + ) − (+1+ + + ¢¸ where + = = 2 2 +2 where 2= 2+ 1, −12 [6] 0
= 1 2 1 and 2is positive constant, is a nonnegative integer and is
the dimension of the R+
. Firstly, it is found that the solution () depends on
the conditions of and and moreover such a solution is related to the solution of the Ultra-Hyperbolic Bessel Operator iterated −times.
Key words: Dirac-delta distribution, Bessel Operator, Ultra-Hyperbolic Bessel Operator, Tempered Distribution.
2000 Mathematics Subject Classification. 46F10, 35J05, 31B10. 1.Introduction
In [4, 5, 6, 7], Kananthai proved elementary solution of the operator¤
where
¤
is the ultra-hyperbolic operator iterated −times, defined by
¤= ⎛ ⎝ X =1 2 2 − + X =+1 2 2 ⎞ ⎠ + = 1Corresponding author
the point ∈ R
Furthermore, he has showed that the elementary solution of the equation ¤ 1¤ 2() = where¤ 1 ¤ 2 is defined by ¤ 1 = ⎡ ⎣1 2 1 X =1 2 2 − + X =+1 2 2 ⎤ ⎦ and ¤ 2 = ⎡ ⎣1 2 2 X =1 2 2 − + X =+1 2 2 ⎤ ⎦
However, we know from [8, 10, 11], that the generalized function 2() defined by (11) is an elementary solution of the operator¤, that is ¤2() = where¤ is the Bessel ultra-hyperbolic operator iterated −times, defined by (1) ¤=¡1+ 2+ + − +1− − + ¢ + = where = 2 2 +2
the point = (1 2 ) ∈ R+ and is the
Dirac-delta distribution.
In this paper, we develop the operator of (1) to be (2) ¤1 = ∙ 1 2 1 ¡ 1+ 2+ + ¢ −¡+1+ + + ¢¸ and (3) ¤2= ∙ 1 2 2 ¡ 1+ 2+ + ¢ −¡+1+ + + ¢¸ and we study the elementary solution of the equation
(4) ¤1¤
2() =
We can obtain
(5) () = 2() ∗ 2()
as an elementary solution of (4), where the symbol ∗ denote the B-convolution
2() and 2() and is defined by (13) and (14) respectively with = 2
and = (1 2 ) ∈ R+ In particular if = 1 = 1 with 1= (time)
1 and 2 are velocity, then (3) becomes the elementary solution of the Elastic
Waves of fourth order. Moreover, in the case of Elastic equilibium ( = 0)
we obtain the elementary solution of the equation ∆2() = where ∆2 is the Laplace Bessel operator iterated 2 defined by
(6) ∆2 =¡2+ +
¢2
where = (2 ) ∈ R+−1
2. Preliminaries
Shift operator acts according to the law [3, 8] () = ∗ R 0 R 0 (p2 1+ 21− 211cos 1 p 2 + 2− 2cos ) ×( Q =1 sin2−1 )1 where ∈ R+ , ∗ = Q =1 Γ(+12) Γ(12)Γ()
We remark that this shift operator is closely connected with the Bessel differential operator = (1 2 )
[3].
The convolution operator determined by the is as follows. For that can
be absolutely integrable, we call
(7) ( ∗ ) () = Z R+ () () Y =1 2
to be the −convolution of and , where =
µQ =1 2−12Γ¡+1 2 ¢¶−1 Convolution (7) is known as a -convolution. We note the following properties of the
-convolution and the generalized shift operator: a.
1 = 1
b. 0
() = ()
c. If () () ∈ (R+
) , () is a bounded function for ∈ R+ and
Z R+ | ()| ( Y =1 2 ) ∞ then Z R+ () ()( Y =1 2 ) = Z R+ () ()( Y =1 2 )
d. From c., we have the following equality for () = 1, Z R+ () ( Y =1 2 ) = Z R+ ()( Y =1 2 ) e. ( ∗ )() = ( ∗ )()
Definition 1. The Fourier-Bessel transformation and its inverse transformation are defined as follows respectively ([8]-[12]):
(8) ( ) () = Z R+ () Ã Y =1 −1 2() 2 ! (9) ¡−1¢() = ( ) (−) = Ã Y =1 2−12Γ µ + 1 2 ¶!−1 where −1
2() is the normalized Bessel function which is the eigenfunction
of the Bessel differential operator. The following identity for Fourier-Bessel transformation can be found in [1, 2, 8, 10]:
() = 1
(10) ( ∗ )() = ()()
Definition 2. Let = (1 2 ) be a point of the -dimensional space
R+
Denote the nondegenerated quadratic form by = 21+ 22+ + 2−
2
+1− − 2+ + = . By Γ we designate the interior of the forward cone,
Γ+=
©
= (1 2 ) ∈ R+ : 0, = 1 and 0
ª
and, by Γ+, designate its closure. For any complex number define
(11) () = ⎧ ⎨ ⎩ −−2||2 () for ∈ Γ 0 for ∈ Γ where () is given by the formula
(12) () = +2||−12 Γ ³2+ −−2|| 2 ´ Γ¡1− 2 ¢ Γ () Γ³2+−−2||2 ´Γ³+2||−2 ´
The function () is introduced by [9, 10, 11, 12]. It is well known that ()
is an ordinary function if Re() ≥ and is distribution of if Re() Let supp
() ⊂ Γ+ where supp() denote the support of ()
From (11), we define the following functions
(13) () = ( −−2||2 () for ∈ Γ+ 0 for ∈ Γ+ (14) () = ( −−2||2 () for ∈ Γ+ 0 for ∈ Γ+ where = 2 1(21+ 22+ + 2) − ¡ 2 +1+ + 2+ ¢ and = 2 2(21+ 22+ + 2 ) − ¡ 2 +1+ + 2+ ¢
1and 2 are positive constants.
Lemma 1. By putting = 1 in (12), (13) and (14) and using the Legendre’s duplication of Γ(). Γ(2) = 22−1−12Γ()Γ( + 1
2) then the formula (13)
and (14) are reduced to
(15) () = ( −−2||2 () for ∈ Γ+ 0 for ∈ Γ+ (16) () = ( −−2||2 () for ∈ Γ+ 0 for ∈ Γ+ here = 2 121 − 22− 23− − 2, = 2221 − 22 − 23− − 2 and () = +2||−1 2 2−1Γ ³ 2+−−2|| 2 ´
Γ¡2¢ () and () are, pre-cisely, the Bessel hyperbolic kernel of Marcel Riesz.
Lemma 2. Let 0 + 2 ||, then ³ ||−´= 2+2||−22 Γ µ + 2 || − 2 ¶ h Γ³ 2 ´i−1 ||−−2|| holds the means of distrubition.
Proof. See [10]. Lemma 3. Let
(17) ∆ () =
be given for ∈ R+. Then () = −2() is an elementary solution of (17),
where 2() = 2+2||−42 Γ ³+2 ||−2 2 ´ Y =1 2−12Γ¡ +12¢ ||2−−2||
Proof. See [9, 10].
Lemma 4. Given the equation
(18) ¤1() = ∈ Γ+= © = (1 2 ) ∈ R+ : 0 = 1 and () 0 ª Then, () = 2() is defined by 2() = 2−−2|| 2 (2) is an elementary solution of (18), where,
(2) = +2||−12 Γ ³2+2 −−2|| 2 ´ Γ¡1−2 2 ¢ Γ (2) Γ³2+2−−2||2 ´Γ³+2||−22 ´
Proof. We can use the mathematical induction method, for = 1 (18) reduces to ¤1 () = µ 1 2 1 ¡ 1+ 2+ + ¢ −¡+1+ + + ¢¶ () = For this equation, considering Lemma 3 and Fourier Bessel transformation, we obtain ³ 1 2 1 ¡ 1+ 2+ + ¢ −¡+1+ + + ¢´ () = −¡2 1 ¡ 2 1+ 22+ + 2 ¢ −¡2 +1+ + 2+ ¢¢ () = 1
Again for this equation, using Lemma 2 and inverse Fourier Bessel transforma-tion, we get () = 2+2||−42 Γ ³+2 ||−2 2 ´ Y =1 2−12Γ¡+1 2 ¢ 2−−2|| 2 where () = 21(21+ 22+ + 2) −¡2+1+ + 2+¢
In view of the properties of Gamma function, the equation above can be rewrit-ten in the following form:
() = Γ³2+2−−2||2 ´Γ³+2||−22 ´ +2||−12 Γ ³4 −−2|| 2 ´ Γ (2) 2−−2||2
Thus, for = 1,
¤1 () = −−1 () ()
holds. Assume the statement is true for − 1 i.e. ¤−1
1 () = (−1)
−1−1
−1() ()
Then we must prove that it is also true for So we have ¤ 1 () = ¤1 ³ ¤−1 1 () ´ = (−1) −1 () (−1)−1−1−1() () = (−1)−1() () where −1 = .
By the condition = 1, we have
= (−1)−1() ()
1 = (−1)() ()
(−1)− = ()
Finally, applying Lemma 2 in (−1)−() = () and considering the
na-ture of the Gamma function, we deduce
() = Γ³2+2−−2||2 ´Γ³+2||−22 ´ +2||−12 Γ ³2+2 −−2|| 2 ´ Γ (2) 2−−2||2
which is the desired equality. Hence, the proof is completed.
Remark 1. It can be easily seen from Lemma 4 that the elemantary solution of¤ 2() = is 2(). Here, 2() = Γ³2+2−−2||2 ´Γ³+2||−22 ´ +2||−12 Γ ³2+2 −−2|| 2 ´ Γ (2) 2−−2||2 where = 2 2(21+ 22+ + 2) − ¡ 2 +1+ + 2+ ¢ . Lemma 5. (The Convolution
() ∗ ()). The function () and ()
are tempered distributions. The convolution
() ∗ () exists and also a
tempered distribution. Proof. See [13].
3. Main Results
Theorem 1. Given the equation
(19) ¤1¤2() =
where¤1 and¤2 are defined by (2) and (3) respectively, is the Dirac-delta distrubution, = (1 2 ) ∈ R+ Then
(20) () =
2() ∗ 2()
is an elementary solution of (19), where 2() and 2() are defined by (13) and (14) respectively, with = 2 Moreover, in particular if = 1with 1=
and 16= 2then (20) becomes () = 2()∗2() as an elementary solution
of the following Elastic Wave equation à 1 2 1 − X =2 !à 1 2 2 − X =2 ! () =
where 2() and 2() are defined by (15) and (16) respectively. If elastic equilibrium (
= 0) holds, then (19) becomes ∆
2
() = where ∆2 is
defined by (6) and we obtain () =
4() is an elementary solution of such
equation and 2() is defined by
(21) 2() = || 2−−2|| 2 (2) where || = (2 1+ 22+ + 2)12 (2) = Y =1 2−12Γ(+12)Γ() 2+2||−4Γ(+2||−2 2 ) and is a complex parameter.
Proof. B-convolving both sides of (19) by
2() we obtain 2() ∗ ¤1¤ 2() = 2() ∗ = 2() ¤ 1 2() ∗ ¤2() = ∗ ¤ 2 = 2()
or ¤2() = 2() by Lemma 4. B-convolving both sides of the equation again by
2() and Lemma 4, we obtain () = 2() ∗ 2() since 2() ∗
2() = 2() ∗ 2() exists by Lemma 5.
Thus () = 2() ∗ 2() is an elementary solution of (19). In particular, if = 1 with 1 = and 1 6= 2 the function () reduces to () defined
by (15) and () reduces to () defined by (16). Thus the equation (20)
becomes () =
2() ∗ 2() as the elementary solution of the Elastic
Wave Equation. Moreover if elastic equilibrium (
() = 4(), where = (2 ) ∈ R+−1 is an elementary solution of the
Laplace equation ∆2() = see [9, 10], where 4() is defined by (21) and ∆2 is defined by (6).
References
1. Kipriyanov I.A., Doklady Acad. Nauk USSR, 158, No:2, p.274-278 (1964). 2. Kipriyanov I.A., Tr. Math. Im. V.A. Steklova Akad. Nauk SSSR, Vol. No:89, p.130-213(1967).
3. Levitan B. M. , Expansion in Fourier Series and Integrals with Bessel Functions, Uspeki Mat., Nauka (N.S) 6, No: 2(42), pp.102-143 1951(in Russian).
4. Kananthai A., On the solution of−dimensional diamond operator, Applied Math-ematics and computation Vol 88, Elsever Sicience, Nev York, pp. 27-37(1997). 5. Kananthai A., On The Product of The Ultra-Hyperbolic Operator Related to The Elastic Waves , Compution Technology, pp: 88-91, Vol. 2 No. 6, 1999.
6. Kananthai A., On the diamond operator related to the wave equation, Nonlinear Analysis, 47(2),1373-1382(2001).
7. Kananthai A., On the nonlinear diamond operator related to the wave equation, Nonlinear Analysis:Real World Applications 3 465-470(2002).
8. Yıldırım H., Riesz Potentials Generated by a Generalized shift operator, Ph. D. Thesis, Ankara University,1995.
9. Yıldırım H., Sarıkaya M. Z. and Ozturk S., The solutions of the -dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution, Proc. Indian Acad. Sci. (Math. Sci.) Vol 114, No.4 , pp.375-387, (2004).
10. Sarıkaya M. Z., On the elemantary solution of the Bessel Diamond operator, Ph. D. Thesis Afyon Kocatepe University, 2007.
11. Sarıkaya M. Z. and Yıldırım H., On the Bessel diamond and the nonlinear Bessel diamond operator related to the Bessel wave equation, Nonlinear Analysis 68, 430-442 (2008).
12. Sarıkaya M. Z. and Yıldırım H., On the weak solutions of the compound Bessel ultra-hyperbolic equation, Applied Mathematics and Computation 189, 910-917 (2007). 13. Aguire Tellez M., Trione S.E., The distributional convolution products of Marcel Riesz’s ultra-hyperbolic Kernel, Revista de la Union Matematica Argentina, 39, 1995.
