"
### $
A GENERAL THEOREM ON LAPLACE TRANSFORM AND ITS APPLICATIONS
*M. I. Qureshi
1, Chaudhary Wali Mohd.
1, Kaleem A. Quraishi
2and Ram Pal
11
Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi-110025 (India)
2
Mathematics Section, Mewat Engineering College (Wakf), Palla, Nuh, Mewat-122107, Haryana (India)
E-mails: [email protected]; [email protected]; [email protected]; [email protected]
! " # $ ! %
---
ABSTRACT
I
n this paper we obtain a general theorem on Laplace transform in terms of Fox-Wright hypergeometric function. Laplace transforms of some composite functions are also obtained as special cases.2010 Mathematics Subject Classification: Primary 44A10; Secondary 44A20.
Keywords and Phrases: Generalized hypergeometric functions; Fox-Wright hypergeometric functions; Fox’s
H
-function; Pochhammer’s symbol; Gauss-Legendre multiplication theorem, Laplace transform.--- 1. INTRODUCTION, DEFINITIONS AND PRELIMINARIES:
In the usual notation, the Pochhammer’s symbol or shifted factorial or generalized factorial function
(
b
)
k is defined by∈
=
=
∈
−
+
+
+
=
Γ
+
Γ
=
}
,
3
,
2
,
1
{
,
1
if
;
!
0
if
;
1
}
,
3
,
2
,
1
{
if
);
1
(
)
2
)(
1
(
)
(
)
(
)
(
k
b
k
k
k
k
b
b
b
b
b
k
b
b
kwhere
b
is neither zero nor a negative integer and the notationΓ
stands for Gamma function.n n
n n
mn mn
m
m
m
m
m
m
+
+
+
−
=
1
2
1
)
(
λ
λ
λ
λ
λ
where
m
is a positive integer,n
is a non negative integer andλ
may be real or complex number.The convenient notation
∆
(
N
;
b
)
is used to denote the array ofN
number of parameters given by,
1
,
,
2
,
1
,
N
N
b
N
b
N
b
N
b
+
+
+
−
where
b
is a real or complex number andN
is a positive integer.The generalized Gaussian hypergeometric function A
F
B of one variable is defined by∞
=
=
0 1 2 2 1
2 1
2 1
!
)
(
)
(
)
(
)
(
)
(
)
(
;
...,
,
,
;
...,
,
,
k k k B k
k k A k k
B A B
A
k
b
b
b
z
a
a
a
z
b
b
b
a
a
a
F
(1.1)where denominator parameters
b
1,
b
2,
...,
b
B are neither zero nor negative integers andA
,
B
are non-negative integers.' ( ! )
* ( + + ,
If
A
≤
B
,
then series AF
B is always convergent for all finite values ofz
(real or complex). IfA
=
B
+
1
,
then series AF
B is convergent for|
z
|
<
1
.
In 1933, Wright (or Fox-Wright) defined a more interesting generalized hypergeometric function of one variable[12,pp.50-51(1.5.21), p.179(34iii),p.395(23);4,p.11(1.7.8)] in the following forms:
∞
=
Γ
+
Γ
+
Γ
+
+
Γ
+
Γ
+
Γ
=
Ψ
0 1 1 2 2
2 2 1 1
2 2 1 1
2 2 1 1
!
)
(
)
(
)
(
)
(
)
(
)
(
;
)
,
(
...,
),
,
(
),
,
(
;
)
,
(
...,
),
,
(
),
,
(
n q q
n p p
q q
p p q
p
n
nB
nB
nB
z
nA
nA
nA
z
B
B
B
A
A
A
β
β
β
α
α
α
β
β
β
α
α
α
(1.2)
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
=
z
B
B
B
A
A
A
q q
p p q
p q
p
;
)
,
(
...,
),
,
(
),
,
(
;
)
,
(
...,
),
,
(
),
,
(
)
(
)
(
)
(
)
(
)
(
)
(
2 2 1 1
2 2 1 1 *
2 1
2 1
β
β
β
α
α
α
β
β
β
α
α
α
−
−
−
−
−
−
−
=
+)
,
1
(
,
),
,
1
(
),
,
1
(
),
1
,
0
(
)
,
1
(
,
),
,
1
(
),
,
1
(
2 2 1
1
2 2 1
1 ,
1 1 ,
q q
p p p
q
p
B
B
B
A
A
A
z
H
β
β
β
α
α
α
(1.3)
∞
=
=
Ψ
0 1 2
2 1
2 2 1 1
2 2 1 1 *
!
)
(
)
(
)
(
)
(
)
(
)
(
;
)
,
(
...,
),
,
(
),
,
(
;
)
,
(
...,
),
,
(
),
,
(
2 1
2 1
n nB nB q nB
n nA p nA nA
q q
p p q
p
n
z
z
B
B
B
A
A
A
q p
β
β
β
α
α
α
β
β
β
α
α
α
(1.4)
where the coefficients
A
1,
A
2,
,
A
p,
B
1,
B
2,
,
B
qare positive real numbers andq
p
β
β
β
α
α
α
1,
2,
,
,
1,
2,
,
are complex parameters.Convergence Conditions:
(i) The Fox
H
-function makes sense when
δ
≡
(
1
+
B
1+
B
2+
+
B
q)
−
(
A
1+
A
2+
+
A
p)
>
0
and0
<
|
z
|
<
∞
;
z
≠
0
(ii) The equality holds true only for suitably constrained values of
|
z
|
or appropriatelybounded values of
|
z
|
i.e.δ
=
0
and0
|
|
1 2 1 2.
2 1 2
1
q
p B
q B B A p A A
B
B
B
A
A
A
R
z
<
≡
− − −<
Suppose given function
F
(
t
)
is well defined for all values oft
>
0
,
then Laplace transform ofF
(
t
)
is given by)
(
d
)
(
)}
(
{
L
0
e
F
t
t
f
s
t
F
=
∞ −st=
(1.5)
(
)
d
(
),
0
2
1
)}
(
{
L
-1=
+ ∞=
>
∞
−
e
f
s
s
F
t
h
i
s
f
h ii h
st
π
(1.6)provided that above integrals exist and
s
is a real or complex number.α
α
α
s
x
x
e
sxd
(
)
01
Γ
=
∞ − −
(1.7)
(
(Re(
s
)
>
0
,
Re(
α
)
>
0
)
or
(Re(
s
)
=
0
,
0
<
Re(
α
)
<
1
)
)
)
(
d
0
1
s
x
x
e
x s=
sΓ
−
∞ − −
α
α
(1.8)
(
Re(
s
)
<
0
,
Re(
α
)
>
0
)
)
(
d
)
1
(
0
1
s
x
x
e
x−
s=
Γ
∞ − −
(1.9)
' ( ! )
* ( + + %
Hankel’s Contour Integral:
)
(
2
d
A
i
t
t
e
i h i h A tΓ
=
∞ + ∞ − −π
(1.10)
(
Re(
A
)
>
0
,
h
>
0
)
Fractional Derivative:
The fractional derivative of
t
P−1 with respect tot
,
λ
times is given by the following formula 1 1)
(
)
(
)
(
d
d
− − −−
Γ
Γ
=
λ λ λλ
P Pt
P
P
t
t
(1.11)where
λ
is arbitrary complex number. Any values ofλ
andP
leading to the result which do not make sense, are tacitly excluded.2. GENERAL THEOREM:
Since Pochhammer’s symbol is associated with Gamma function and Gamma function is undefined for zero and negative integers, therefore arguments, numerator and denominator parameters are adjusted in such a way that each term of following results is completely well defined and meaningful then without any loss of convergence, we have
×
+
−
Γ
+
−
+
Γ
+
Γ
=
+ − +)
1
(
)
1
(
)
1
(
;
)
(
;
)
(
d
d
L
1λ
µ
λ
µ
ν
µ
λ µ ν µ λ λ νs
xt
b
a
F
t
t
t
k B A B A+
−
+
−
+
+
Ψ
×
+ + kB A B A
s
x
k
b
b
b
k
k
a
a
a
;
)
,
1
(
),
1
,
(
...,
),
1
,
(
),
1
,
(
;
)
,
1
(
),
,
1
(
),
1
,
(
...,
),
1
,
(
),
1
,
(
2 1 2 1 * 1 2λ
µ
λ
µ
ν
µ
(2.1)(
Re(
ν
+
µ
−
λ
)
>
−
1
;
k
is
a
positive
real
number
)
Proof:
=
+ ∞ = r kr B r
r r A k B A B A
t
t
r
b
x
a
t
xt
b
a
F
t
t
t
λ µλ ν µ λ λ ν
d
d
!
)]
[(
)]
[(
L
;
)
(
;
)
(
d
d
L
0+
−
+
Γ
+
+
Γ
=
∞ = − +0
[(
)]
!
(
1
)
)
1
(
)]
[(
L
r B r
r k r r A
r
k
r
b
t
r
k
x
a
t
λ
µ
µ
µ λν
{
ν µ λ}
λ
µ
µ
λ
µ
µ
+ + −∞ =
−
+
+
+
−
Γ
+
Γ
=
krr B r kr
r k r r A
t
r
b
x
a
L
)
1
(
!
)]
[(
)
1
(
)]
[(
)
1
(
)
1
(
0 ∞ = + − ++
−
+
−
+
+
+
−
Γ
+
−
+
Γ
+
Γ
=
0 1!
)
1
(
)]
[(
)
1
(
)
1
(
)]
[(
)
1
(
)
1
(
)
1
(
r kr r k r B r r k r k r Ar
s
b
x
a
s
µ
λ
λ
µ
ν
µ
λ
µ
λ
µ
ν
µ
λ µ νNow writing above series in hypergeometric notation (1.4) of Fox-Wright, we get the right hand side of (2.1).
3. SOME DEDUCTIONS OF (2.1):
If
k
is a positive integer, then (2.1) reduces tok B A B A
xt
b
a
F
t
t
t
;
)
(
;
)
(
d
d
L
λ µλ ν
+
−
∆
+
−
+
∆
+
∆
+
−
Γ
+
−
+
Γ
+
Γ
=
+ − + + + kk B A k B k A
s
k
x
k
b
k
k
a
F
s
(
),
(
;
1
)
;
;
)
1
;
(
),
1
;
(
),
(
)
1
(
)
1
(
)
1
(
21
µ
λ
λ
µ
ν
µ
λ
µ
λ
µ
ν
µ
λ µ' ( ! )
* ( + + "
(
Re(
ν
+
µ
−
λ
)
>
−
1
;
A
+
k
≤
B
+
1
)
In (3.1), setting
λ
=
µ
=
0
and then adjusting parameters and variables suitably, we get on simplification− k
B A B
A
xt
b
a
F
t
(
)
;
)
(
;
)
(
L
ν 1=
Γ
+∆
kk
B A B k A
s
k
x
b
k
a
F
s
)
(
;
)
(
;
)
;
(
),
(
)
(
ν
ν
ν (3.2)
(
Re(
ν
)
>
0
;
Re(
s
)
>
0
;
A
+
k
≤
B
+
1
)
∆
+ −
k k k
B A k B A k
k
t
k
k
b
a
F
t
λ
σ
σ
;
)
;
(
),
(
;
)
(
L
1=
Γ
+k
B A B A k
s
b
a
F
s
k
σ
λ
σ
(
)
;
;
)
(
,
1
)
(
1 (3.3)
(
Re(
σ
)
>
0
;
Re(
s
)
>
0
;
A
≤
B
)
4. APPLICATIONS:Making suitable adjustments of parameters and variables in (3.2) and (3.3), we obtain following results
|
)
Im(
|
)
Re(
,
2
;
;
2
3
;
2
3
,
2
2
)
2
(
}
)
sin(
L{
22
1 2
2
s
n
s
b
b
n
n
F
s
n
b
bt
t
n n−
>
−
>
+
+
+
Γ
=
+ (4.1)|
)
Im(
|
)
Re(
,
1
;
;
2
1
;
2
2
,
2
1
)
1
(
}
)
cos(
L{
22
1 2
1
n
s
b
s
b
n
n
F
s
n
bt
t
n n−
>
−
>
+
+
+
Γ
=
+ (4.2){
}
;
Re(
)
0
4
1
exp
2
)
sin(
L
2
3
−
>
=
s
s
s
t
π
(4.3)
=
−
s
F
s
t
t
4
1
;
2
3
;
1
1
)
sin(
L
1 1 (4.4)
=
−
s
s
t
t
4
1
exp
)
cos(
L
π
(4.5)
{
}
=
−
s
F
s
s
t
t
4
1
;
2
1
;
2
3
2
1
)
cos(
L
1 1π
(4.6)
{
}
1
1
)
(
L
2 1
+
−
=
s
s
t
J
(4.7)
{
}
=
−
s
s
t
J
t
4
1
exp
2
1
)
(
L
1 2 (4.8){
1}
=
2 2 1−
216
25
;
1
;
4
7
,
4
5
8
3
)
(
L
s
F
s
s
t
J
t
π
(4.9)
{
}
=
−
s
s
t
J
4
1
exp
1
)
(
' ( ! )
* ( + +
-)
exp(
2
4
1
exp
1
L
2
3
s
t
t
−
=
−
π
(4.11))
exp(
4
1
exp
1
L
s
s
t
t
−
=
−
π
(4.12)
{
}
1
1
)
(
L
2 0
+
=
s
t
J
(4.13)
{
}
1
1
)
(
L
2 0
−
=
s
t
I
(4.14)
2
)
2
exp(
L
+
=
−
s
t
t
π
(4.15)
1
1
}
)
(
L{erf
+
=
s
s
t
(4.16)
s
s
t
)
2
exp(
1
erfc
L
=
−
(4.17)3 3
2 0 2
1
2
27
;
3
5
,
3
4
;
L
s
t
F
t
+
=
−
−
−
−
(4.18)
=
−
−
−
=
−
−
−
s
J
s
s
F
s
t
F
1
1
2
;
1
;
1
;
1
,
1
;
L
0 2 0 1 0 (4.19)=
−
+
Ψ
− +
m n
m n
m
s
b
s
b
t
b
m
n
m
t
1
sin
;
)
2
,
(
),
2
,
2
(
;
)
1
,
1
(
L
2 22 1 1
(4.20)
=
−
Ψ
−
m n
m n
s
b
s
t
b
m
n
t
1
cos
;
)
2
,
(
),
2
,
1
(
;
)
1
,
1
(
L
1 1 2 2 2 (4.21)=
−
+
Ψ
−− +
m n
m n
m
s
b
s
b
t
b
m
n
m
t
1 2 2 2 21
tan
1;
)
2
,
(
),
2
,
2
(
;
)
2
,
1
(
),
1
,
1
(
L
(4.22)+
=
−
+
Ψ
− +
m n
m n
m
s
b
s
b
t
b
m
n
m
t
1
ln
1
;
)
,
(
),
1
,
2
(
;
)
1
,
1
(
),
1
,
1
(
L
1 2 2 (4.23)|
)
Im(
|
)
Re(
,
tan
)
sin(
L
1s
b
s
b
t
bt
>
=
−(4.24)
=
s
t
t
2
1
erf
)
sin(
L
π
(4.25)0
)
Re(
;
4
1
exp
2
}
)
(
L{sinh
2
3
>
=
s
s
s
t
π
(4.26){
}
=
s
s
t
I
4
1
exp
1
)
(
L
0 (4.27)' ( ! )
* ( + + )
REFERENCES:
1. Ditkin, V. A. and Prudnikov, A. P.; Integral Transforms and Operational Calculus, Pergamon Press, Oxford, London, Frankfurt, 1965.
2. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.; Table of Integral Transforms, Vol. I(Bateman Manuscript Project), McGraw-Hill Book Co. Inc., New York, Toronto and London, 1954.
3. Mathai, A. M. and Saxena, R. K.; Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lectures Notes in Mathematics No. 348, Springer-Verlag, Berlin, Heidelberg and New York, 1973.
4. Mathai, A. M. and Saxena, R. K.; The H-function with Applications in Statistics and Other Disciplines, John Wiley and Sons (Halsted Press), New York, 1978.
5. Marichev, O. I.; Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Halsted Press (Ellis Horwood, Chichester, U.K) John Wiley and Sons, New York, Brisbane, Chichester and Toronto, 1983.
6. Oberhettinger, F. and Badii, L.; Tables of Laplace Transforms, Springer-Verlag, New York, Heidelberg, Berlin, 1973.
7. Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I.; Integrals and Series, Vol. 3: More Special Functions, Gordon and Breach Science Publishers, New York, 1990.
8. Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I.; Integrals and Series, Vol. 4: Direct Laplace Transform, Gordon and Breach Science Publishers, New York, 1992.
9. Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I.; Integrals and Series, Vol. 5: Inverse Laplace Transform, Gordon and Breach Science Publishers, New York, 1992.
10. Spiegel, M. R.; Theory and Problem of Laplace Transforms, Schaum’s Outline Series, 1986.
11. Srivastava, H. M., Gupta, K. C. and Goyal, S. P.; The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.
12. Srivastava, H. M. and Manocha, H. L.; A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester, U.K.) John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
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