Convolution Integral Equation Involving Generalized Hypergeometric Function and H-function of Two Variables
Poonam Kumari and Yashwant Singh*
Department of Mathematics, Institute of Integrated & Honors Studies, Kurukshetra University, Kurukshetra, INDIA.
*Department of Mathematics,
Government College, Kaladera, Jaipur, Rajasthan, INDIA.
email: [email protected], [email protected].
(Received on: April 15, 2019) ABSTRACT
In the present paper, the authors have established a solution regarding convolution integral equation whose kernel is a generalized hypergeometric function
p F Q [.] and the -function of two variables. Some interesting special cases of main result have also been discussed.
2010 Mathematical subject classification: 33C99.
Keywords: Convolution Integral Equation, Laplace Transform, Convolution Theorem, H-function of Two Variables, Generalized Hypergeometric Function.
1. INTRODUCTION
If f(t) and g(t) are piecewise continuous function on , then the convolution integral of f(t) and g(t) is,
(𝑓 ∗ 𝑔)(𝑡) = ∫ 𝑓(𝑡 − 𝜏)𝑔(𝜏)𝑑𝜏
0𝑡A nice property of convolution integrals is.
(𝑓 ∗ 𝑔)(𝑡) = (𝑔 ∗ 𝑓)(𝑡) Or
∫ 𝑓(𝑡 − 𝜏)𝑔(𝜏)𝑑𝜏 =
0𝑡∫ 𝑔(𝑡 − 𝜏)𝑓(𝜏)𝑑𝜏
0𝑡H
[0, )
The following fact will allow us to take the inverse transforms of a product of transforms.
Generalized hypergeometric function is defined as:
(1.1) Where for brevity, (a
P)denotes the array of parameters a
1,…,a
Pwith similar interpretation for (b
Q) etc. . For further details one can refer Rainville
5.
The -function of two variables (Mittal and Gupta
3, p.172) using the following notation, which is due essentially to Srivastava and Panda
7( p.266, eq. (1.5) )is defined and represented as:
= (1.2)
Where
(1.3)
(1.4)
(1.5)
( ) ( ) ( )
L f g F s G s
1
{ ( ) ( )} ( )( ) L
F s G s f g t
1
0 1
; ; ; ,
!
P j n n P j
P Q P Q P Q Q
Q n
j n j
a a z
F a b z F z
b b n
H
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
( ; , ) :( , ) ,( , )
0, : , : ,
, : , : , ( ; , ) :( , ) ,( , )
[ , ]
j j j p j j p j j pj j j q j j q j j q
a A c e E
n m n m n
x x
y p q p q p q y b B d f F
H x y H H
1 2
2 3
2
1 ( , ) ( ) ( )
4
L L x y d d
1
1 1
1
1
1 1
(1 )
( , )
( ) (1 )
n
j j j
j
p q
j j j j j j
j n j
a A
a A b B
2 2
2 2
2 2
1 1
2
1 1
(1 ) ( )
( )
( ) (1 )
n m
j j j j
j j
p q
j j j j
j n j m
c d
c d
3 3
3 3
3 3
1 1
3
1 1
(1 ) ( )
( )
( ) (1 )
n m
j j j j
j j
p q
j j j j
j n j m
e E f F
e E f F
2. THE CONVOLUTION INTEGRAL EQUATION
The solution of the following convolution integral equation has been given:
(2.1)
Where Re(σ)>0, Re(η)>0 and (i)
(ii)
(iii)
(iv)
(V) (vi)
Solution: In order to solve (2.1), we first take Laplace transform of both sides of (2.1), we get
Changing the order of integration, we get
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
1 1
0
( ; , ) :( , ) :( , )
, : , : , ( )
, : , : , ( ) ( ; , ) :( , ) :( , )
( ) ; ; ( ) .
( ) ( )
j j j p j j p j j p
j j j q j j q j j q
x
u
P Q P Q
a A c e E
o n m n m n t x t
p q p q p q t x t b B d f F
t x t F g h at x t
H f t dt g x
1 2 1 2
1 1 1 1
0
p p q q
j j j j
j j j j
R
1 2 1 2
1 1 1 1
0
p p q q
j j j j
j j j j
S A E B F
1 1 2 2 2 2
1 1 1 1 2 1 1 2 1
0
p q m q n p
j j j j j j
j n j j j m j j n
U
3 3 3 3
1 1
1 1 1 1 3 1 1 3 1
0
m q n p
p q
j j j j j j
j n j j j m j j n
V A B F F E E
| arg | 1
x 2 U 1
| arg | y 2 V
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
1 1
0 0
( ; , ) :( , ) :( , )
0, : , : , ( )
, : , : , ( ) ( ; , ) :( , ) :( , )
0
{ ( ) ; ; ( )
( ) ( )
j j j p j j p j j p
j j j q j j q j j q
x
px u
P Q R S
a A c e E
n m n m n t x t px
p q p q p q t x t b B d f F
e t x t F g h at x t
H f t dt dx e g x dx
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
( ; , ) :( , ) :( , )
0, : , : , 1
, : , : , 1 ( ; , ) :( , ) :( , )
1 1
0 0
( ) .
{ ( ) ( ) ( )
j j j p j j p j j p
j j j q j j q j j q
a A c e E
n m n m n
p q p q p q b B d f F
x
px ur r
f r H
e t x t f t dt dx g p
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
( ; , ) :( , ) :( , )
0, : , : , 1
, : , : , 1 ( ; , ) :( , ) :( , )
1 1
0
( ) .
( ){ ( ) ( )
j j j p j j p j j p
j j j q j j q j j q
a A c e E
n m n m n
p q p q p q b B d f F
ur px r
t
f r H
t f t e x t dx dt g p
Putting (x-t)=u, we obtain
Or
(2.2) Where and denote the Laplace transform of f(t) and g(t), respectively, and
(2.3) And . (2.4)
A series expansion for can be obtain as a special case of series expansion of H[x,y] ([6],eq.(6.2.1),p.84)and, since this specialization leads to a power series, the series representation for the reciprocal can be formed. To do this we note that
(2.5)
Where (2.6) Where are given by (1.3), (1.4), (1.5) respectively.
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
( ; , ) :( , ) :( , )
0, : , : , 1
, : , : , 1 ( ; , ) :( , ) :( , )
1 ( ) 1
0 0
( ) .
( ){ ( )
j j j p j j p j j p
j j j q j j q j j q
a A c e E
n m n m n
p q p q p q b B d f F
ur p u t r
f r H
t f t e u du dt g p
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
( ; , ) :( , ) :( , )
0, : , : , 1
, : , : , 1 ( ; , ) :( , ) :( , )
1 1
0 0
( ) .
( ){ ( )
j j j p j j p j j p
j j j q j j q j j q
a A c e E
n m n m n
p q p q p q b B d f F
pt ur pu r
f r H
e t f t e u du dt g p
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
1
1
( ; , ) :( , ) :( , )
0, : , : , 1
, : , : , 1 ( ; , ) :( , ) :( , )
1
( ) .
( )
( 1) ( ) ( )
j j j p j j p j j p
j j j q j j q j j q
ur
ur
a A c e E
n m n m n
p q p q p q b B d f F
ur
r
f r H
d r
f p g p
dp p
1
1
1 1 1
( )
2,
{( 1)
ur ur( )} ( )
ur
r
f r H p p d
f p g p
p dp
1
1
1
1 1
2
{ ( )} ( 1) ( )
( ) ,
ur
ur
ur
d
rf p p g p
f r H p p dp
( )
f p g p ( )
1 1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1 1,1 1,2 1,3
(1 ,1,1),( ; , ) :( , ) :( , )
0, : , : ,
1 1
2
,
1, : , : , ( ; , ) :(j ,j )j :(p ,j )j p j j pj j j q j j q j j q
r a A c e E
n m n m n p
p q p q p q p b B d f F
H p p H
1
0
1
( ) !
p
j r r j
q r
j r j
g a
f r r
h
1 1
2
,
H p
p
1 1
2 ,
, 0
( )
, ( )
! !
M N M N
M N
r M N
H p p C p
M N
,
( , )
2( ) ( )
3C
M N M N M N
2 3
( , ), ( ), ( )
From the well -known rearrangement property (Rainville
5, p.56)
(2.7) We can rewrite H
2as a single (power) series in the form
(2.8)
Where (2.9) If k denotes the least value of for which h
≠0, then
(2.10) So that if we let the coefficients H
be determined by the relation
(2.11) Then (2.2) becomes
= (2.12)
Consequently, on taking the inverse Laplace transform of (2.12) and applying its convolution theorem, we obtain the following:
Theorem: If Re(σ)>0, Re(η)>0, g
r(0)=0 for 0≤r≤, an integer, Re(-k-σ-ηr)>0, then under suitable restrictions on the parameters of the H-functions of two variables occurring in (2.1) [obtainable easily from the set of conditions (i) to (vi) mentioned with (2.1)]the solution to the convolution integral equation (2.1) is given by
or
(2.13) Where
(2.14)
, 0 0 0
( , ) ( , )
M
M N M N
F M N f M N N
1 1
2
0
,
H p p h p
,0
( 1) ( )
!
h r C
1 1
2
0
,
k k n nn
H p p p h p
1
0 0
n k n n
h p H p
1
1 1
0
( ) ( 1) ( ) 1
( )
ur
ur r k
ur
d f p p g p H p
dp f r
1 ( )
0
( ) 1 ( )
( )
ur k r
p H p p g p
f r
1 1
1 1
0
( 1)
ur( ) ( 1)
ur( ) ( ) ( )
t
ur k r
t
f t
t x V t x g x dx
1 1
0
( ) ( ) ( ) ( )
ur
t
k r
t
f t t x
V t x g
x dx
0
( ) 1
( ) ( )
V x H x
f r k r
The coefficients H
being defined by the recurrences
H
kH
0=1, and for >0 by (2.15) And the power series coefficients h
being given by (2.9).
3. SPECIAL CASES
(i) If we take p
1=q
1=0 in (2.1), the H-function of two variables reduces to the product of two single Fox’s H-function (Fox
2) as:
(3.1)
Where g
(r)(0)=0 for 0≤r≤, an integer and Re(σ)>0, Re(η)>0, has its solution given by
(3.2)
Where Re(-k-σ-ηr)>0 and
(3.3) The H
determined by (2.15), h
given by (2.9), and the coefficients in (2.6) reduces to the form:
(ii) If we put r=0 in (2.1), we get the result due to Buschman, Koul and Gupta
1in some slight different form:
Where g
(r)(0)=0 for 0≤r≤, an integer and Re(σ)>0, Re(η)>0, has its solution given by
(3.4)
Where Re(-k-σ)>0 and
(3.5) The H
determined by (2.15), h
being given by (2.9).
0
k
0 H h
2 22 1,1, 22
3 1,3
3 3 1, 3
( , )
,
1 1
, ( , )
0
( , )
,
, ( , )
( ) ; ; ( ) ( )
( ) ( ) ( )
j j p
j j q
j j p
j j q
x
o n c u
P Q P Q p q d
e E o n
p q f F
t x t F g h at x t H t x t
H t x t f t dt g x
1 1
0
( ) ( ) ( ) ( )
ur
t
k r
t
f t t x
V t x g
x dx
0
( ) 1
( ) ( )
x H x
f r k r
, 2
( ) ( )
3C
1, 1, 1,
1 2 2 3 3 1 2 3
1 1 2 2 3 3 1,1 1,2 1,3
( ; , ) :( , ) :( , )
, : , : ,
1 1 ( )
, : , : , ( ) ( ; , ) :( , ) :( , )
0
( )
j j j p j j p j j p( ) ( )
j j j q j j q j j q
x
a A c e E
o n m n m n x t
p q p q p q x t b B d f F
t
x t
H
f t dt g x
1 1
0
( ) ( ) ( ) ( )
t
t
f t t x
k V t x g
x dx
0
( ) 1
( ) ( )
x H x
f r k
