Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator

26 Pages·0.22 MB·English

by  J. Barros-Neto

Bessel Integrals and Fundamental Solutions for
1
0
0 a Generalized Tricomi Operator
2

n
a
J J. Barros-Neto

8 Rutgers University, Hill Center

] 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019
A
C e-mail: [email protected]
.
h
Fernando Cardoso
t ∗
a
m Departmento de Matem´atica, Universidade Federal de Pernambuco
[ 50540-740 Recife, Pe, Brazil

1 e-mail: [email protected]
v
5
6
0
1
0 Abstract
1
0 ThemethodofpartialFouriertransformisusedtofindexplicitfor-
/
h mulas for two remarkable fundamental solutions for a generalized Tri-
at comi operator. These fundamental solutions reflect clearly the mixed
m type of the Tricomi operator. In proving these results, we establish
: explicit formulas for Fourier transforms of some functions involving
v
i Bessel functions.
X
r
a
1 Introduction
Consider the generalized Tricomi operator
∂2
(1.1) P = y∆+ ,
∂y2
∗
Partially supported by CNPq (Brazil)
1
∂2
where ∆ = n is the Laplace operator. Our aim is to find, via partial
j=1 ∂x2
P j
Fourier transform with respect to x = (x ,...,x ), fundamental solutions
1 n
relative to an arbitrary point (a,0) on the hyperplane y = 0 in Rn+1, that is,
distributions that are solutions to
Pu = δ(x a,y),
−
where δ(x a,y) is the Dirac measure concentrated at (a,0). Since P is
−
invariant under translationsparalleltothathyperplane, itsuffices toconsider
the case when the Dirac measure is concentrated at the origin.
If n = 1, then (1.1) is the classical Tricomi operator,
∂2 ∂2
(1.2) = y + .
T ∂x2 ∂y2
It is known that, for this operator, the equation 9x2 + 4y3 = 0 defines the
two characteristic curves that originate at the origin. They divide the plane
in two disjoint regions, namely,
D = (x,y) R2 : 9x2 +4y3 > 0 ,
+
{ ∈ }
the region “outside” the characteristics and
D = (x,y) R2 : 9x2 +4y3 < 0 ,
−
{ ∈ }
the region “inside” the characteristics. Note that D is entirely contained in
−
the hyperbolic region y < 0.
In the paper [2] it was shown the existence of the following two funda-
mental solutions relative to the origin:
Γ(1/6)
(9x2 +4y3)−1/6 in D
 −3 22/3π1/2Γ(2/3) +
(1.3) F+(x,y) =  ·


0 elsewhere



and
3Γ(4/3)
9x2 +4y3 −1/6 in D
 22/3π1/2Γ(5/6)| | −
(1.4) F (x,y) =
− 


0 elsewhere.



2
The support of F , the closure of D , is, except for the origin, entirely
− −
contained in the hyperbolic region (y < 0), while the support of F , the
+
closure of D , consists of the whole elliptic region (y > 0), the parabolic
+
region (x-axis), and extends to the hyperbolic region up to and included the
characteristic curves. Themethodusedin[2]toprovethese resultswas based
upon the property that solutions to the equation u = 0 are invariant under
the dilation d (x,y) = (t3x,t2y) in R2. T
t
In [1] we proved formulas (1.3) and (1.4) by using partial Fourier trans-
form with respect to the x variable. In this paper we extend the results of [1]
to the generalized Tricomi operator (1.1). Since the dimension of the space
variable x is now n > 1, technical difficulties in evaluating Fourier transforms
involving Bessel functions do occur. We circumvent them by calculating in-
tegrals of the type
∞
I (a,b) = e−ǫtt−λJ (at)J (bt)dt
ǫ µ ν
Z
0
(Section 3). As a result, we obtain for the operator (1.1) the following fun-
damental solutions:
3n−2 Γ(n2 − 13) (9 x 2 +4y3)13−n2 in Dn
 −22/3πn/2 Γ(2/3) | | +
F (x,y) =
+ 


0 elsewhere,



whose supportis theclosure oftheregionDn = (x,y) Rn+1 : 9 x 2+4y3 >
+ { ∈ | |
0 and
}
3nΓ(4)
3 9 x 2 +4y3 13−n2 in Dn
 22/3πn/2Γ(4 n) | | | | −
F−(x,y) =  3 − 2


0 elsewhere



supportedbytheclosureoftheregionDn = (x,y) Rn+1 : 9 x 2+4y3 < 0 .
− { ∈ | | }
These fundamental solutions clearly generalize formulas (1.3) and (1.4).
In addition, we also show that no matter the parity of the dimension n,
F (x,y) does not have support on the boundary of the region D in the
− −
hyperbolic region and hence, in contrast with what happens for strictly hy-
perbolic operators – like the wave operator – the Huyghens principle does
not hold for the generalized Tricomi operator.
3
It is well known [9] that, for the wave operator, the regularity of the
fundamental solutions degenerates as the dimension n increases: a locally
constant function, when n = 1; an absolutely continuous measure relative
to the Lebesgue measure, when n = 2; a measure carried by the surface of
the forward light-cone, when n = 3; and so on. However, for the generalized
Tricomi operator they always remain locally integrable.
The plan of this paper is as follows. In Section 2 we reduce, via partial
Fourier transform, the original problem to an equivalent one of finding fun-
damental solutions for a second order ordinary differential equation andshow
how these can be represented in terms of Airy functions or Bessel functions.
In Section 3 we obtain explicit formulas for Fourier transforms of the func-
tions ξ ±νJ ( ξ ), ξ νK ( ξ ), and ξ νN ( ξ ), ξ = (ξ ,...,ξ ), n 1. These
ν ν ν 1 n
| | | | | | | | | | | | ≥
formulas are used to construct the fundamental solution supported by Dn
−
(Section 4) and the fundamental solution supported by Dn (Section 5). In
+
theAppendixthereaderwillfindthedefinitions oftheBessel functionsJ (z),
ν
I (z), K (z), and N (z), and a list of properties that are needed throughout
ν ν ν
this work. We also recall the definition of hypergeometric functions and some
of their main properties.
The method of finding solutions, via partial Fourier transform, to the
equation
∂2u
(1.5) Pu = y∆u+ = f
∂y2
with, say, f ∞(Rn+1) is a natural one although not particularly new. In
∈ Cc
the monograph [5], R. J. P. Groothuizen exhibits a solution u in terms of
a Fourier integral operator whose symbol is obtained by the partial Fourier
transform analysis employed in this paper. He also obtains fundamental
solutions given by Fourier integral operators. Our results are more precise
since we give explicit formulas for the fundamental solutions and these are
tempered distributions. Consequently, we can obtain more transparent rep-
resentations for the solution u to the equation (1.5) as a convolution of f
with any of the fundamental solutions here described.
2 Preliminaries
Consider the more general problem of finding fundamental solutions for the
operator (1.1) relative to an arbitrary point (0,b) on the y-axis. That is, one
4
wishes to find distributional solutions to the equation
∂2
(2.1) y∆F + F = δ(x) δ(y b).
∂y2 ⊗ −
Partial Fourier transform with respect to x reduces this problem into finding
fundamental solutions to the ordinary differential equation
(2.2) F˜ y ξ 2F˜ = δ(y b),
yy
− | | −
where F˜ denotes the partial Fourier transform of F, ξ 2 = n ξ2, and
| | j=1 j
δ(y b) is the Dirac measure concentrated at b. P
−
One way of solving (2.2) (see [9]) consists in selecting two linearly inde-
pendent solutions U (ξ,y) and U (ξ,y) to the homogeneous equation
1 2
(2.3) u˜ y ξ 2u˜ = 0
yy
− | |
so that their Wronskian at y = b (which in the case under consideration is
the same as the Wronskian at y = 0) is equal to 1 and in defining
−
U (ξ,b)U (ξ,y) if y b
2 1
≥
(2.4) F˜(ξ,y;b) = 
 U (ξ,b)U (ξ,y) if y b.
1 2
≤

˜
It is a matter of verification that F(ξ,y;b) is a solution to (2.2).
Equivalently [4], one can obtain a fundamental solution to the equation
(2.2) by finding two linearly independent solutions to the homogeneous equa-
tion (2.3): u (ξ,y;b), defined for y > b, and u (ξ,y;b), defined for y < b, so
1 2
that
(i) the limit of u as y b+ equals the limit of u as y b
1 2
→ → −
and
(ii) the limit of u as y b+ minus the limit of u as y b is equal
1,y 2,y
to 1. The function F˜(ξ,y;b)→is now defined by → −
u (ξ,y;b) if y b
1
≥
˜
(2.5) F(ξ,y;b) = 
 u (ξ,y;b) if y b.
2
≤

In what follows we will use at our convenience either one of these two expre-
sions for F˜(ξ,y;b).
5
Returning to the original problem (2.1), if we can choose U (ξ,y) and
1
U (ξ,y), in formula (2.4), or u (ξ,y;b) and u (ξ,y;b), in formula (2.5), so
2 1 2
that F˜(ξ,y;b) is a tempered distribution with respect to ξ = (ξ ,...,ξ ),
1 n
then its inverse Fourier transform of F˜(ξ,y;b) defines a fundamental solution
to the operator (1.1).
We now proceed to find linearly independent solutions to the homoge-
neous equation (2.3). The change of variables z = ξ 2/3y transforms (2.3)
| |
into the classical Airy’s equation u′′ zu = 0. Two linearly independent
−
solutions to that equation are Ai(z) and Bi(z) respectively called the Airy
functions of the first and second kinds. It is known [6] that these two func-
tionscan berepresented interms of Bessel functions oforder 1/3as follows.
±
If argz < (2π/3), then
| |
Ai(z) = z1/2[I (2z3/2) I (2z3/2)] = 1(z)1/2K (2z3/2),
3 −1/3 3 − 1/3 3 π 3 1/3 3
(2.6) 
 Bi(z) = (z)1/2[I (2z3/2)+I (2z3/2)].
3 −1/3 3 1/3 3
If argz< (2π/3), then
| |
Ai( z) = z1/2[J (2z3/2)+J (2z3/2)],
− 3 −1/3 3 1/3 3
(2.7) 
 Bi( z) = (z)1/2[J (2z3/2) J (2z3/2)].
− 3 −1/3 3 − 1/3 3
In the Appendix I, where a brief review of Bessel functions is presented,
we show that the following relations hold:
3−2/3 3−4/3
(2.8) Ai(0) = , Ai′(0) =
Γ(2/3) −Γ(4/3)
and
3−1/6 3−5/6
(2.9) Bi(0) = , Bi′(0) = .
Γ(2/3) Γ(4/3)
As a consequence, the Wronskian of Ai(z) and Bi(z) evaluated at z = 0 is
(2.10) W(Ai(z),Bi(z)) = 1/π.
|z=0
Indeed, we have
3−2/3 3−1/6
(cid:12) Γ(2/3) Γ(2/3) (cid:12)
(cid:12) (cid:12) 2 3−3/2 1
W(Ai(z),Bi(z)) = (cid:12) (cid:12) = · = ,
|z=0 (cid:12) (cid:12) Γ(2/3)Γ(4/3) π
(cid:12) 3−4/3 3−5/6 (cid:12)
(cid:12) (cid:12)
(cid:12) −Γ(4/3) Γ(4/3) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6
because
2π
(2.11) Γ(2/3)Γ(4/3) = .
33/2
We now choose the following two linearly independent solutions to the
equation (2.3):
U (ξ,y) = √π ξ −1/3Ai( ξ 2/3y) and U (ξ,y) = √π ξ −1/3Bi( ξ 2/3y),
1 2
| | | | − | | | |
and note that, by virtue of (2.10), the Wronskian of U (ξ,y) and U (ξ,y) is
1 2
equal to 1. Next, according to (2.4), the distribution
−
π ξ −2/3Bi( ξ 2/3b)Ai( ξ 2/3y) if y b
− | | | | | | ≥
˜
(2.12) F(ξ,y;b) = 
 π ξ −2/3Ai( ξ 2/3b)Bi( ξ 2/3y) if y b
− | | | | | | ≤

is a fundamental solution to the ordinary differential equation (2.2). A fun-
damental solution F(x,y;b) to the generalized Tricomi operator (1.1) and
relative to the point (0,b) would then be the inverse Fourier transform of
F˜(ξ,y;b) whenever that Fourier transform exists.
We do not know how to obtain an explicit formula (or formulas) for the
˜
inverse Fourier transform of F(ξ,y;b) when b = 0, a problem that merits to
6
be investigated. We conjecture that when b < 0, that is, the point (0,b) is in
the hyperbolic region, there exists two fundamental solutions that converge,
as b 0, to the fundamental solutions F (x,y) and F (x,y) defined, re-
+ −
→
spectively, by the formulas (5.6) and (4.2). The two fundamental solutions
described in the monograph [5] do not seem to satisfy these requirements.
However, when b = 0,we will show inthe following sections how to obtain
from formula (2.12), as well as formulas similar to it, explicit expressions for
fundamental solution to (1.1).
The case n = 1. For sake of completeness and in order to motivate our
work in the forthcoming sections we briefly present the results of the paper
[1], for the Tricomi operator (1.2), that is the case when n = 1. If b = 0 and
we take into account the values of Ai(0) and Bi(0) as given by (2.8) and
(2.9), then formula (2.12) simplifies as follows:
π ξ −2/3
| | Ai( ξ 2/3y) if y 0
 −31/6Γ(2/3) | | ≥
(2.13) F˜(ξ,y) = 


 π ξ −2/3
| | Bi( ξ 2/3y) if y 0,
 −32/3Γ(2/3) | | ≤




7
where, for simplicity, we wrote F˜(ξ,y) for F˜(ξ,y;0). Now the inverse Fourier
transform of both expressions on the right-hand side of (2.13) can be explic-
itly calculated. To see this, first consider the representations of Ai(z) and
Bi(z) in terms of Bessel functions as given by formulas (2.6) and (2.7):
1 ξ 2/3y 1/2 2
Ai( ξ 2/3y) = | | K ( ξ y3/2)
1/3
| | π (cid:18) 3 (cid:19) 3| |
and
ξ 2/3( y) 1/2 2 2
Bi( ξ 2/3y) = | | − [J ( ξ ( y)3/2) J ( ξ ( y)3/2)].
−1/3 1/3
| | (cid:18) 3 (cid:19) 3| | − − 3| | −
Next by introducing the change of variables s = (2/3)y3/2, whenever y 0,
≥
and t = (2/3)( y)3/2, whenever y 0, rewrite formula (2.13) as
− ≤
s
α ( )1/3K (s ξ ) if y 0
1/3
· ξ | | ≥

(2.14) F˜(ξ,y) =  | |

 t
β ( )1/3[J (t ξ ) J (t ξ )] if y 0
−1/3 1/3
· ξ | | − | | ≤
 | |

where α and β are constants given by
1 π
(2.15) α = and β = .
−21/331/3Γ(2/3) −21/335/6Γ(2/3)
Theorem 3.1 of [1] proves that F(x,y), the inverse Fourier transform of
F˜(ξ,y), is then
3 1
F = F F ,
+ −
2 − 2
a linear combination of the two fundamental solutions F and F respec-
+ −
tively defined by formulas (1.3) and (1.4). To prove this theorem, one re-
lies on known formulas (see [4], [7]) for the inverse Fourier transforms of the
functions ξ −1/3J ( ξ ), ξ −1/3J ( ξ ),and ξ −1/3K ( ξ ),formulas that
1/3 −1/3 1/3
| | | | | | | | | | | |
need be generalized to the case n > 1.
3 Fourier transforms involving Bessel func-
tions
As we have indicated in the previous section, we are going to need ex-
plicit formulas for the inverse Fourier transforms of the following functions:
8
ξ ±νJ ( ξ ), ξ νK ( ξ ),and ξ νN ( ξ ),where ξ = (ξ ,...,ξ ), n 1. Since,
ν ν ν 1 n
| | | | | | | | | | | | ≥
when n = 1, explicit formulas for the Fourier transforms are known and can
be found, for example, in [4] and [7], we concentrate on the case n > 1.
1. The inverse Fourier transform of ξ νK ( ξ ).
ν
| | | |
Assume that Re ν < 1/2 and define
| |
1
(3.1) −1[ ξ νK ( ξ )](x) = eihx,ξi ξ νK ( ξ )dξ.
F | | ν | | (2π)n ZR | | ν | |
n
Theassumptionontherealpartofν securesconvergenceoftheintegralatthe
origin. On the other hand, in view of the asymptotic behavior of K ( ξ ) for
ν
| |
largevalues of ξ ,the integral converges absolutely. By introducing spherical
| |
coordinates, rewrite the integral on the right-hand of (3.1) as
∞
rn+ν−1K (r) eihrx,ωidω dr.
ν
Z {Z }
0 Sn−1
Since
n
(2π)2
(3.2) ZSn−1 eihrx,ωidω = |rx|n2−1Jn2−1(r|x|),
it follows that
x 1−n2 ∞ n
(3.3) F−1[|ξ|νKν(|ξ|)](x) = |(2|π)n2 Z0 r2+νJn2−1(r|x|)Kν(r)dr.
We now quote the following result found in Watson’s treatise [11]:
Lemma 3.1. If Re(µ+1) > Re ν and Re b > Im a then
| | | |
∞ (2a)µ(2b)νΓ(µ+ν +1)
(3.4) tµ+ν+1J (at)K (bt)dt = .
µ ν
Z (a2 +b2)µ+ν+1
0
From this lemma, we obtain the following result
Theorem 3.1. If Re ν < 1/2 then
| |
2ν−1Γ(n +ν)
(3.5) −1[ ξ νK ( ξ )](x) = 2 1+ x 2 −n2−ν .
F | | ν | | πn2 | |
(cid:0) (cid:1)
9
Proof. The integral in (3.3) is the same as the integral in (3.4), where
µ = n/2 1, a = x , and b = 1. Clearly the conditions of the lemma are
− | |
2
satisfied and Theorem 3.1 follows at once.
We remark that, when n = 1, (3.5) becomes
2ν−1Γ(1 +ν) 1
−1[ ξ νK ( ξ )](x) = 2 1+ x 2 −2−ν ,
F | | ν | | π21 | |
(cid:0) (cid:1)
a formula found in [4] and [7].
2. Inverse Fourier transforms of ξ νJ ( ξ ) and ξ −νJ ( ξ ).
ν ν
| | | | | | | |
As before, assume that Re ν < 1/2. Formally, the inverse Fourier trans-
| |
form of ξ νJ ( ξ ) is
ν
| | | |
1
−1[ ξ νJ ( ξ )](x) = eihx,ξi ξ νJ ( ξ )dξ.
ν ν
F | | | | (2π)n ZR | | | |
n
In general, the integral diverges at and so we introduce a converging
∞
factor e−ǫ|ξ| and take a limit as ǫ 0. Since ξ νJ ( ξ ) is locally integrable,
ν
it defines a tempered distribution→and the lim| i|t exi|st|s in ′(R ), the space
n
S
of tempered distributions on R . Thus the precise meaning of the inverse
n
Fourier transform is
1
(3.6) −1[ ξ νJ ( ξ )](x) = lim eihx,ξi−ǫ|ξ| ξ νJ ( ξ )dξ.
F | | ν | | (2π)n ǫ→0ZR | | ν | |
n
By introducing spherical coordinates, one can see that the integral on the
right-hand can be written as
∞
e−ǫrrn+ν−1J (r) eihrx,ωidω dr.
ν
Z {Z }
0 Sn−1
Since
n
(2π)2
(3.7) ZSn−1 eihrx,ωidω = |rx|n2−1Jn2−1(r|x|),
it follows that
x 1−n2 ∞ n
(3.8) F−1[|ξ|νJν(|ξ|)](x) = |(2|π)n2 lǫi→m0Z0 e−ǫrr2+νJn2−1(r|x|)Jν(r)dr.
10