Fredholm type integral equation with special functions

Acta Univ. Sapientiae, Mathematica, 10, 1 (2018) 5–17 DOI: 10.2478/ausm-2018-0001

Fredholm type integral equation with special functions Frdric Ayant

Dinesh Kumar

Teacher in High School, France email: [email protected]

College of Agriculture, Sumerpur-Pali, Agriculture University of Jodhpur, India Department of Mathematics & Statistics, Jai Narain Vyas University, India email: dinesh [email protected]

Abstract. Recently Chaurasia and Gill [7], Chaurasia and Kumar [8] have solved the one-dimensional integral equation of Fredholm type involving the product of special functions. We solve an integral equation involving the product of a class of multivariable polynomials, the multivariable H-function defined by Srivastava and Panda [29, 30] and the multivariable I-function defined by Prasad [21] by the application of fractional calculus theory. The results obtained here are general in nature and capable of yielding a large number of results (known and new) scattered in the literature.

1

Introduction and preliminaries

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of any arbitrary real or complex order. The widely investigated subject of fractional calculus has gained importance and popularity during the past four decades or so, chiefly due to its demonstrated applications in 2010 Mathematics Subject Classification: 33C60, 26A33 Key words and phrases: multivariable H-function, multivariable I-function, class of multivariable polynomials, Fredholm type integral equation, Riemann-Liouville fractional integral, Weyl fractional integral

5

6

F. Ayant, D. Kumar

numerous seemingly diverse fields of science and engineering including turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, astrophysics (see, for details, [6, 5, 15, 26, 31] and for recent works, see also [18, 19]). Under various fractional calculus operators, the computations of image formulas for special functions of one or more variables are important from the point of view the solution of differential and integral equations (see, [1, 2, 3, 13, 12, 15, 22, 23, 24, 25]). In this paper, we use the fractional calculus, more precisely the Weyl fractional operator to resolve the one-dimensional integral equation of Fredholm type involving the product of special functions. The integral equations occur in many fields of physics, mechanics and applied mathematics. In the last several years a large number of Fredholm type integral equations involving various polynomials or special functions as kernels have been studied by many authors notably Chaurasia et al. [7, 8], Buchman [4], Higgins [10], Love [16, 17], Prabhakar and Kashyap [20] and others. In the present paper, we obtain the solutions of following Fredholm integral equation. Z∞

"

y 0

−α

t t # x x 0,λ 0 :(α 0 ,β 0 );··· ;(α(r) ,β(r) ) u1 , · · · , us × HA 0 ,C 0 :(M 0 ,N 0 );··· ; M(r) ,N(r) ) ( y y  0, η 0) (r) , η(r) 0 , · · · , γ(r)  (q : (g ); γ 0 ;··· ; q j 0 1,M 1,A 1,M(r)  0 (r) 0 0 (r) (r) (fj ) ; ξ , · · · , ξ 1,C 0 : (p , )1,N 0 ; · · · ; p , 1,N(r) 

M1 ,··· ,Ms SN 1 ,··· ,Ns

u1 (y)p  .   . ur (y)p 

(1)

(1)

(r)

(r)

,n ;··· ;m ,n 2 ;0,n3 ;··· ;0,nr :m × I0,n p2 ,q2 ,p3 ,q3 ;··· ;pr ,qr :p(1) ,q(1) ;··· ;p(r) ,q(r) q  z1 yx (1) (r)  a ; α 0 , α 00 ; · · · ; a ; α , · · · , α : 2j 2j rj rj  2j rj .  1,p2 1,pr  (1) (r) 0 , β 00 ; · · · ; brj ; βrj , · · · , βrj :  . q b2j ; β2j 2j 1,q2 1,qr zr yx  (1) (1) (r) (r) aj , αj ; · · · ; a , α j j (1) (r) 1,p 1,p  f(y)dy = g(x) (0 < x < ∞) (1) (1) (r) (r) bj , βj ; · · · ; bj , βj (1) (r) 1,q

(1)

1,q

The multivariable I-function defined by Prasad [21] is an extension of the multivariable H-function defined by Srivastava and Panda [29, 30]. It is defined

Fredholm type integral equation with special functions

7

in term of multiple Mellin-Barnes type integral: (1)

(1)

(r)

(r)

,n ;··· ;m ,n 2 ;0,n3 ;··· ;0,nr :m I (z1 , · · · , zr ) = I0,n p2 ,q2 ,p3 ,q3 ;··· ;pr ,qr :p(1) ,q(1) ;··· ;p(r) ,q(r)  z1  . a ; α 0 , α 00 ;··· ; 2j 2j  2j  . 1,p2  0 , β 00 ;··· ;  . b2j ; β2j 2j 1,q2 zr (1) (r) (1) (1) arj ; αrj , · · · , αrj : aj , αj ;··· 1,pr 1,p(1) (1) (1) (1) (r) : bj , βj ;··· brj ; βrj , · · · , βrj (1)

=

1 (2πω)r

Z

1,qr

Z

···

; ;

1,q

ψ (s1 , · · · , sr )

L1

(2)

r Y

Lr

(r) (r) aj , αj

(r) (r) bj , βj



1,p(r)  1,q(r)

θi (ti ) zti i dt1 · · · dtr ,

i=1

the existence and convergence conditions for the defined integral (2), see Prasad [21]. The condition for absolute convergence of multiple Mellin-Barnes type contour (1) can be obtained by extension of the corresponding conditions for multivariable H-function: 1 |arg zi | < Ωi π 2 where n X

(i)

p X

(i)

Ωi =

(i) αk

k=1

k=n(i) +1

 +

−

−

(i) αk

n2 X

(i) α2k

−

p2 X k=n2 +1

q2 X

q3 X

k=1

(i)

k=1

(i)

(i)

(i) βk

q X

−

(i)

βk

k=1

k=m(i) +1



ns X

(i) α2k 

k=1

β2k +

+

(i) m X

+ ··· +

β3k + · · · +

k=1 qs X

(i) αsk

−

ps X

! (i) αsk

(3)

k=ns +1

! (i)

βsk

,

k=1

where i = 1, · · · , r. The complex numbers zi are not zero. We establish the asymptotic expansion in the following convenient form: |z1 |α1 , · · · , |zr |αr , max (|z1 | , · · · , |zr |) → 0 I (z1 , · · · , zr ) = O |z1 |β1 , · · · , |zr |βr , min (|z1 | , · · · , |zr |) → ∞ I (z1 , · · · , zr ) = O

8

F. Ayant, D. Kumar

(k) b , j = 1, · · · , m(k) ; where k = 1, · · · , r; = min < j(k) βj (k) a −1 0 and βk = max < j (k) , j = 1, · · · , n(k) . αk0

αj

The generalized class of multivariable polynomials defined by Srivastava [28], is given as X

[N1 /M1 ] M1 ,··· ,Ms SN 1 ,··· ,Ns

[y1 , · · · , ys ] =

X (−N1 )M K (−Ns )Ms Ks 1 1 ··· K1 ! Ks !

[Ns /Ms ]

···

K1 =0

(4)

Ks =0

A [N1 , K1 ; · · · ; Ns , Ks ] yK1 1 · · · yKs s , where M1 , · · · , Ms are arbitrary positive integers and the coefficients A[N1 , K1 ; · · · ; Ns , Ks ] are arbitrary constants, real or complex. The generalized polynomials of one variable defined by Srivastava [27], is given in the following manner: X (−N) MK A [N, K] xK , K!

[N/M]

SM N (x) =

(5)

K=0

where the coefficientsA [N, K] are arbitrary constants real or complex. The series representation of the multivariable H-function defined by Srivastava and Panda [29, 30] is given by Chaurasia and Olka [9] as 0,λ 0 :(α 0 ,β 0 );··· ;(α(r) ,β(r) ) H [u1 , · · · , ur ] = HA 0 ,C 0 :(M 0 ,N 0 );··· ; M(r) ,N(r) ( )  u1  . (gj ); γ 0 , · · · , γ(r) : q(1) , η(1) 1,M(1) ; · · · ; q(r) , η(r) 1,M(r)  1,A 0  . (fj ) ; ξ 0 , · · · , ξ(r) : p(1) , (1) 1,N(1) ; · · · ; p(r) , (r) 1,N(r) 1,C 0 ur Pr Qr 0 α(i) X ∞ i X i=1 ni φi u U i=1Q i (−) , = φ r 0 i i=1 mi ni ! 0

    (6)

mi =0 ni =0

where Qλ 0 φ= Q A

j=λ 0 +1

j=1 Γ

Γ gj −

Pr

(i) i=1 γj Ui

1 − gj + Q 0 , Pr (i) (i) C γ U Γ 1 − f + ξ U i j i i=1 j j=1 i=1 j

Pr

(7)


9

and Q (i) (i) (i) (i) (i) β U U + η − Γ 1 − q Γ p i i j=1,j6=mi j j j j j=1 , Q (i) φi = Q (i) (i) (i) (i) (i) M N U U − η + Γ q Γ 1 − p (i) (i) i i j j j j j=β +1 j=α +1 Qα(i)

(8)

(i)

for i = 1, · · · , r and Ui =

pmi +ni0

following conditions:

(i)

mi

i = 1, · · · , r, which is valid under the

,

i h (i) (i) (i) mi pj + pi 6= j [pmi + ni ] , for j = mi , mi = 1, · · · , α(i) ; pi , ni0 = 0, 1, 2, · · · ; ui 6= 0 0

0

Σi =

A X j=1

(i) γj

−

C X

B X

j=1

+

D X (i)

(i)

(i) ξj

(i) ηj

j=1

−

(i)

j < 0, ∀i ∈ {1, · · · , r}.

(9)

j=1

Let = denote the space of all functions f which are defined on R+ and satisfy (i) f ∈ C∞ (R+ ) (ii) limx→∞ [xγ fr (x)] = 0 for all non-negative integersγ and r. (iii) f(x) = 0(1) as x → 0. For correspondence to the space of good functions defined on the whole real line (−∞, ∞). The Riemann-Liouville fractional integral (of order µ) is defined by D−µ {f(x)} = 0 D−µ x {f(x)} Zx 1 (x − ω)µ−1 f (ω) dω = Γ (µ) 0

( 0, f ∈ =) ,

(10)

where Dµ {f(x)} = φ(x) is understood to mean that φ is a locally integrable solution of f(x) = D−µ {φ(x)}, implying that Dµ is the inverse of the fractional operator D−µ . The Weyl fractional (of order h) is defined by Laurent [14] as following: W −h {f(x)} = x D−h ∞ {f(x)} = where 0 and f ∈ =.

1 Γ (h)

Z∞ x

(ξ − x)h−1 f(ξ)dξ,

(11)

10

F. Ayant, D. Kumar

2

Solution of the Fredholm integral equation (1)

For convenience, we shall use these following notations: Ur = p2 , q2 ; p3 , q3 ; · · · ; pr−1 , qr−1 ; Vr = 0, n2 ; 0, n3 ; · · · ; 0, nr−1 , (12) Wr = p(1) , q(1) ; · · · ; p(r) , q(r) ; Xr = m(1) , n(1) ; · · · ; m(r) , n(r) , (13)

(1)

(2)

A = a2k ; α2k , α2k

1,p2

(1)

(2)

(r−1)

; · · · ; a(r−1)k ; α(r−1)k , α(r−1)k , · · · , α(r−1)k

, 1,pr−1

(14)

(1)

(2)

(r)

A = ark ; αrk , αrk , · · · , αrk

: 1,pr

(1)

(1)

ak , αk

1,p(1)

(r)

(r)

; · · · ; ak , αk

1,p(r)

,

(15)

(1)

(2)

B = b2k ; β2k , β2k

1,q2

(r−1)

(2)

(1)

; · · · ; b(r−1)k ; β(r−1)k , β(r−1)k , · · · , β(r−1)k

, 1,qr−1

(16)

(1)

(2)

(r)

B = brk ; βrk , βrk , · · · , βrk

: 1,qr

(1)

(1)

bk , βk

1,q(1)

(r)

(r)

; · · · ; bk , βk

1,q(r)

,

(17) (−N1 )M1 K1 (−Ns )Ms Ks ··· A [N1 , K1 ; · · · ; Ns , Ks ] , K1 ! Ks ! U = α 0 , β 0 ; · · · ; α(r) , β(r) ; V = M 0 , N 0 ; · · · ; M(r) , N(r) , h i (r) (r) (1) (1) , ; · · · ; q , η C = (gj ) ; γ 0 , · · · , γ(r) : q , η (r) (1) 0 1,M 1,M h i 1,A ; · · · ; p(r) , (r) . D = (fj ) ; ξ 0 , · · · , ξ(r) : p(1) , (1) (1) (r) 0

as =

1,C

1,N

1,N

(18) (19) (20) (21)

We have the following formula: Lemma 1 

p  v1 yx t t #  −α M ,··· ,Ms  0 ;U x x 1  W β−α  H0,λ A 0 ,C 0 ;V  y SN1 ,··· ,Ns u1 y , · · · , us y : p vr yx q   z1 yx A; A   Vr ;0,nr ;Xr   × IUr :pr ,qr ;Wr  : :   q B; B zr x "

y

 C  :   D

Fredholm type integral equation with special functions X

[N1 /M1 ]

=y

−β

···

K1 =0

as uK1 1    

z1 zr

· · · uKs s

q x y

: q x y

x y

[Ns /Ms ] α(i) X X

∞ X

Qr

0

i i=1 ni φi vU i=1Q i (−) r 0 i i=1 mi ni !

φ

Ks =0 mi =0 ni0 =0 t Psi=1 Ki +p Pri=1 Ui

Pr

11

Vr ;0,nr +1;Xr × IU r :pr +1,qr +1;Wr

(22) 

P P A; (1 − β − t s Ki − p r Ui ; q, · · · , q) , A  i=1 i=1  P P B; (1 − α − t s Ki − p r Ui ; q, · · · , q) , B  , i=1 i=1

where φ1 , φi and as are defined respectively by (7), (8) and (18). Also, provided that (a) 0, (i) (i) (c) |arg zi |
0, under the same conditions that (22).


3

13

Particular cases

We obtain the similar formula with the class of polynomials of one variable defined by Srivastava [27]. We have Corollary 1 Pr Qr Z∞ [N/M] α(i) ∞ 0 i X X X i=1 ni (−N) φi vU MK i=1Q −β i (−) y φ r i n 0! K! 0 i=1 mi i 0 K= 0 mi =0 ni =0

tK+p Pri=1 Ui x Vr ;0,nr +1;Xr u K × IU A[N, K] r :pr +1,qr +1;Wr y q  z1 yx P  A; (1 − β − tK − p r Ui ; q, · · · , q) , A i=1  P  : q B; (1 − α − tK − p r Ui ; q, · · · , q) , B i=1 zr yx p   v1 yx C t ! Z∞   0 ;U x  : :  u y−α SM = H0,λ 0 ,C 0 ;V  N A  y p 0 D vr yx q   z1 yx A; A   r ;Xr  :  × Dβ−α [f(x)] dy, : IVUrr;0,n :pr ,qr ;Wr   q x B; B zr

   f(y) dy  (26)

y

under the same conditions and notations that (22). If the multivariable I-function reduces to the multivariable H-function defined by Srivastava and Panda [29, 30], we obtain the following result: Corollary 2 Pr Qr Z∞ [N1 /M1 ] [Ns /Ms ] α(i) ∞ 0 i X X X X i=1 ni φi vU i=1Q −β i (−) y ··· φ r 0 i 0 i=1 mi ni ! 0 K1 =0

Ks =0 mi =0 ni =0

t Psi=1 Ki +p Pri=1 Ui x r +1;Xr uK1 1 · · · uKs s × Hp0,n as r +1,qr +1;Wr y q  z1 yx P P  (1 − β − t s Ki − p r Ui ; q, · · · , q) , A i=1  P Pi=1  : q (1 − α − t s Ki − p r Ui ; q, · · · , q) , B i=1 i=1 zr yx

   f(y) dy 

14

F. Ayant, D. Kumar

p  x # v t t Z∞ 1 y  0 ;U x x M1 ,··· ,Ms  u1 = y−α SN , · · · , us H0,λ A 0 ,C 0 ;V  1 ,··· ,Ns : p y y 0 vr yx q   z1 yx A   β−α r ;Xr   [f(x)] dy, × H0,n pr ,qr ;Wr  : q :  D x B zr "

 C  :   D

y

(27) under the same conditions and notations that (22) with Ur = Vr = A = B = 0.

4

Conclusion

The equation (1) is of general character. By suitably specializing the various parameters of the multivariable I-function, the multivariable H-function and the class of multivariable polynomials, our results can be reduce to a large number of integral equations involving various polynomials or special functions of one and several variables occur in many fields of physics, mechanics and applied mathematics.

Acknowledgments I am grateful to the National Board of Higher Mathematics (NBHM) India, for granting a Post-Doctoral Fellowship (sanction no.2/40(37)/2014/R&DII/14131).

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Received: August 9, 2017