Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 13, Number 9 (2017), pp. 6973–6978© Research India Publicationshttp://www.ripublication.com/gjpam.htm

A general voltera-type integral equation associated withan integral operator with the H -function in the kernel

P. Harjule, M. K. Bansal1 and R. Jain

Department of Mathematics,Malaviya National Institute of Technology,

Jaipur 302017, Rajasthan, India.

Abstract

In this paper, we solve a general Volterra-type fractional equation associated withan integral operator with the H -function in its Kernel. We make use of convolutiontechnique to solve the main equation. Since H -function occuring in the fractionaloperator herein is general in nature we can obtain a number of special cases fromour main findings, by reducing H -function to its many special cases. We recordhere two such special cases which involve generalized Riemann-Zeta function andpolylogarithmic function respectively. Results obtained by Srivastava and Bush-man [4, 5], Rashmi Jain [9] follow as special cases of our main findings.

AMS subject classification: 45E10, 33C60, 44A10.Keywords: Convolution Integral Equation; H -function; Laplace transform.

1. Introduction

The H -function occurring in the present work will be defined and represented here inthe following manner [3]

Hm,n

p,q [z] = Hm,n

p,q

⎡⎣z

∣∣∣∣∣∣(aj , αj ; Aj)1,n, (aj , αj )n+1,p

(bj , βj )1,m, (bj , βj ; Bj)m+1,q

⎤⎦

:= 1

2πω

∫L

�(ξ)zξdξ (1.1)

1Corresponding author: manish.math12@gmail.com

6974 P. Harjule, M. K. Bansal and R. Jain

where, ω = √−1, z ∈ C \ {0}, C being the set of complex numbers,

�(ξ) =

m∏j=1

�(bj − βjξ)n∏

j=1

{�(1 − aj + αjξ)

}Aj

q∏j=m+1

{�(1 − bj + βjξ)

}Bjp∏

j=n+1�(aj − αjξ)

(1.2)

The sufficient condition for the absolute convergence of the integral have been establishedby Bushman and Srivastava [8, p. 4708] The series representation for the H−Functionis as follows:

Hm,n

p,q

⎡⎣z

∣∣∣∣∣∣(aj , αj ; Aj)1,n, (aj , αj )n+1,p

(bj , βj )1,m, (bj , βj ; Bj)m+1,q

⎤⎦ =

∞∑t=0

m∑h=1

�st,hzs

t,h (1.3)

where,

�(st,h) =

m∏j=1,j �=h

�(bj − βjst,h)n∏

j=1

{�(1 − aj + αjst,h)

}Aj

q∏j=m+1

{�(1 − bj + βjst,h)

}Bjp∏

j=n+1�(aj − αjst,h)

(1.4)

2. An integral operator involving H -function

In our present investigation, we make use of the following fractional integral operatorwith H -function in its kernel

(H1,n;σ

a+;p,q;ρ ϕ)

(x) :=x∫

a

(x − t)ρ−1 H1,n

p,q[(x − t)σ ]ϕ(t)dt R(ρ) > 0 (2.1)

The following property of Laplace transform [1]

L(f (n)(x); s) = snF (s)

holds provided that f (i)(0) = 0, i = 0, 1, 2, . . . , n−1, n being a positive integer, where

L(f (x); s) =x∫

0

e−sxf (x)dx = F(s)

The well-known convolution theorem for Laplace transform

L

⎛⎝ x∫

0

f (x − u)g(u)du; s

⎞⎠ = L(f (x); s)L(g(x); s) (2.2)

A general voltera-type integral equation... 6975

holds provided that the various Laplace transforms occuring in (2.2) exists. For a = 0,by using the Convolution Theorem for the Laplace Transform, we find from the definition(2.1) that

L[(

H1,n;σ0+;p,q;ρ ϕ

)(x)

](s)

= L[xρ−1 H

1,n

p,q[xσ ]](s) · L[ϕ(x)](s)

= s−ρH1,n+1p+1,q

⎡⎣s−σ

∣∣∣∣∣∣(1 − ρ, σ ; 1), (aj , αj ; Aj)1,n, (aj , αj )n+1,p

(0, 1), (bj , βj ; Bj)2,q

⎤⎦ �(s) (2.3)

where

�(s, ρ, σ ) > 0

3. Main Result

A general Volterra-type integral equation associated with an integral operator with theH -function in its kernel is given by

(H1,n;σ

0+;p,q;ρ y)

(x) + a

�(η)

∫ x

0(x − t)η−1y(t)dt := g(x) (3.1)

�(ρ, σ , η) > 0; 0 ≤ n ≤ p

has the solution

y(x) =∫ x

0(x − t)l−σk−ρ−1

∞∑λ=0

Cλ(x − t)σλ

�(l − σk + σλ − ρ)Dl

t {g(t)}dt, (3.2)

where l is a positive integer such that Re(l − σk − ρ) > 0, where k denotes the least ν

for which C′v �= 0, where

C′v =

�(ρ + σν)n∏

j=1{�(1 − aj + αjν)}Aj

q∏j=2

{�(1 − bj + βjν)}Bj

p∏j=n+1

�(aj − αjν)ν!(−1)ν (3.3)

6976 P. Harjule, M. K. Bansal and R. Jain

g is prescribed such that g(u)(0) = 0 for 0 ≤ u ≤ l − 1Cλ are given by

Cλ = (−1)λ(C′k)

−λ−1det

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C′k+1 C

′k ... 0 ... 0

C′k+2 C

′k+1 ... ... ... 0

. .

. .

. .(Ck+ η−ρ

σ+ a

).

. .

. .

. .

C′k+λ C

′k+λ−1 ...

(Ck+ η−ρ

σ+ a

)... C

′k+1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.4)Proof. To solve Eq. (3.1), we first take the Laplace transform of its both sides with thehelp of (2.3), we get

s−ρH1,n+1p+1,q

⎡⎣s−σ

∣∣∣∣∣∣(1 − ρ, σ ; 1), (aj , αj ; Aj)1,n, (aj , αj )n+1,p

(0, 1), (bj , βj ; Bj)2,q

⎤⎦ Y (s)+aS−ηY (s) = G(s)

(3.5)Now, expressing H -function involved in the above equation in terms of series with thehelp of (1.3) we have

s−ρ

[ ∞∑ν=0

C′νs

−σν + as−η+ρ

]Y (s) = G(s) (3.6)

where C′ν is given by (3.4). The other details of the proof would run parallel to those

given already in [7] so we omit them here. �

4. Special Cases

If we take σ = 1, Aj(j = 1, 2, . . . , n) = Bj(j = 2, . . . q) = 1 and a = 0 in our mainresult we arrive at the result derived by [4, 5].

If we take a = 0 in the main result we arrive at the result obtained by [9]

1. If we reduce the H -function involved in (3.1) to the generalized Riemann Zetafunction, φ

((x − t)σ , µ, ξ

), [2, 6], we arrive at the following interesting result:

x∫0

(x− t)ρ−1φ((x − t)σ , µ, ξ

)y(t)dt + a

�(η)

x∫0

(x− t)η−1y(t)dt := g(x) (4.1)

A general voltera-type integral equation... 6977

has the solution given by

y(x) =x∫

0

(x − t)l−σk−ρ−1∞∑

λ=0

Cλ(x − t)σλ

�(l − σk + σλ − ρ)Dl

t {g(t)}dt (4.2)

provided that min {R(ρ), R(σ ), R(l − ρ − σ)} > 0, l is a positive integer and Cλ

is given by (3.4) where

C′ν = �(ρ + σν)

(ξ + ν)µ, ν = 0, 1, 2, .... (4.3)

Also g(u)(0) = 0 for 0 ≤ u ≤ l − 1.

2. If we reduce the H -function involved in (3.1) to the Polylogarithm function F(t, µ)

of order µ [2, p.30,p.315][6], we get the following interesting result:

x∫0

(x − t)ρ−1F((x − t)σ , µ

)y(t)dt + a

�(η)

x∫0

(x − t)η−1y(t)dt := g(x) (4.4)

has the solution given by

y(x) =x∫

0

(x − t)l−σk−ρ−1∞∑

λ=0

Cλ(x − t)σλ

�(l − σk + σλ − ρ)Dl

t {g(t)}dt (4.5)

provided that min {R(ρ), R(σ ), R(l − ρ − σ)} > 0, l is a positive integer and Cλ

is given by (3.4) where

C′ν = �(ρ + σ + σν)

(1 + ν)µ, ν = 0, 1, 2, ... (4.6)

Also g(u)(0) = 0 for 0 ≤ u ≤ l − 1.

References

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[2] A. Erdélyi, Higher Transcendental Functions, Vol. I, McGraw Hill Book Co., NewYork, (1953).

[3] A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feyn-man Integrals II. A generalization of the H-function, J. Phys. A: Math. Gen., 20,4119–4128, (1987).

6978 P. Harjule, M. K. Bansal and R. Jain

[4] H.M. Srivastava, and R.G. Bushman, Some convolution integral equations, Nederl.Akad. Wetensch. Proc. Ser. A77, Indag. Math., 36, 211–216, (1974).

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[8] R.G. Bushman, and H.M. Srivastava, The H -function associated with a certain classof Feynman integrals, J. Phys. A: Math. Gen., 23, 4707–4710, (1990).

[9] Rashmi Jain, Convolution Integral Equations Involving the H -function, Tamsui Ox-ford Journal of Mathematical Sciences, 23, 317–324, (2007).