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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.151-154
Some Fractional Derivative of H-Function in one Variable Neelam Pandey and Jai Prakash Patel Department of Mathematics, Govt. Model Science College, Rewa, M.P., INDIA. (Received on: March 1, 2014) ABSTRACT In this paper,we derive the some Fractional derivative of Hfunction in one variable defined by
(3) and (4) . The results
established are of general character and include some known results. Keywords : Fractional Calculus, H-function , Hypergeometric function.
INTRODUCTION Fractional calculus is a field of Applied mathematics that deals with derivatives and integrals of arbitrary orders during the left some decades Fractional Calculus has been applied to almost every field of science and technology .Many applications of fractional can be found in turbulence and fluid dynamics,Stochastic dynamical system,Plasam Physics and Controlled Thermonuclear fusion,Non linear control theory,Image Processing,Non linear biological system,Astrophysics etc. The fractional derivative,extension of the
equation as well as in other contents. The Gauss hypergeometric function is defined4. (a) k (b) k z k (c ) k k! k =1 ∞
2 F1[a, b; c; z ] =
∑
(1)
| z |< 1, c ≠,1,2,...
The Pochhammers symbol defined in term of the Gamma function by (λ ) n =
Γ (λ + n ) n =0 = {1,λ ( λ when +1)( λ + 2)( λ + 3)....( λ + n −1), Γ (λ ) when n∈ N for λ ≠ 0,1,2,....
(2)
Binomial expression7
d n f ( x) familiar derivative to no integral dx n
x x ∞ ( x + ξ ) λ = ξ λ ∑r = 0( λr )( ) r ; ( ) < 1 ξ ξ
values of n,is of immense utility in finding the solution of ordinary, partial and integral
Dxµ ( x λ ) =
Γ( λ + µ ) λ − µ x ; Re(λ ) > −1 Γ(λ )
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
(3)
152
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (2), 151-154 (2014)
µ λ −µ z D∞ z
µ
is
Γ (λ + µ ) − µ − λ z , Γ (λ )
= (−1) µ
(4)
arbitrary
(a1 , α 1 ), (a2 , α 2 ), (a3 , α 3 ),......, (a p , α p )
Oldham & Sapnier2 and Shrivastav & Goyal10 the fractional derivative of a function f(x) of complex order γ (or th
alternatively, a − γ order) integrals of f(x)is defined as
α
Where (a) ι is a square root of -1 and (a j , α j ) represent the set of p-order pairs
∫
(5) where m is a positive integer. For simply we use D
−γ t
=0 D When the lower terminal limit α → 0 (5) reduce to the Riemann-Liouville representation used for differ integral of arbitrary order which α → ∞ gives the differ integral of arbitrary order in weyl sense. The Leibnitz rule for the n th derivative of a Product can be generalized to a derivative of arbitrary order (see Ross and Northover6) ∞
∑(
γ m ).α
θ (s) =
Π mj=1Γ (b j − β j s ) Π nj =1Γ (1 − a j + α j s ) Π qj = m +1Γ (1 − b j + β j s ) Π pj = n +1Γ( a j − α j s )
,
(c) m,n,p,and q are integers satisfying 1 ≤ m ≤ q,0 ≤ n ≤ p, α j (1 ≤ j ≤ p ), β j (1 ≤ j ≤ q) are positive numbers and a j (1 ≤ j ≤ p ), b j (1 ≤ m ≤ q ) are complex numbers. MAIN RESULTS
−γ t
γ α D x [ f ( x ) g ( x )] =
(b)
fractional
t 1 ( t − x ) −γ −1 f ( x ) dx Re ( γ ) < 0, Γ ( −γ ) α dm Dtγ − m f ( t ); 0 ≤ Re ( γ ) < m dx m α
Dt−γ ( f ( x )) = {
and similarly for (bq , β q ).
Dxm f ( x)α
m= 0
The fractional derivative to be evaluate here is D xγ {( x (x
m1
m1
+ a ) λ (b − x σ
+ a ) 1 (b − x
Dxγ − m g ( x), ∀α
H [ zx
m
∑
= a λ b −δ x −γ
− ρ1
(8)
m2 −δ1 ) ]}
∞
r1 , r2 = 0
ρ
(6)
m2 −δ )
(
m
x 1 r1 x 2 r2 ) ( ) a b r1 ! r2 !
σ
+1, n + 2 1 1 −δ Hm p + 3, q + 3 [ zx a b / ( − λ ,σ 1 )(1−δ − r2 , s )( a j ,α j )1, p (1−γ + r1m1 + r2 m2 , ρ1 ) (1+ r1m1 + r2 m2 , ρ1 )( b j , β j )1, q )( − λ + m1 ,σ 1 )(1−δ ,δ1 )
According to a standard notation of one variable which is introduced by Fox,C.7 will be represented as follows-
m1 m γ + a) λ (b − x 2 ) −δ x D∞ {( x m1 −σ 1 m2 −δ1
(x
+ a)
(b − x
)
H [ zx
]
ρ1
(9)
]}
( a ,α )
H [ x] = H pm,,qn [ x] = H pm,,qn [ x/ (b p, β p) ] q
( a ,α )
= H pm,,qn [ x/ (b j , β j )1, p ] = j
j 1, q
q
1 θ ( s ) x s ds 2πι l
∫
m
(7)
= a λ b − δ x − γ ( − 1) γ
∞
∑ r1 , r2 = 0
(
m
x 1 r1 x 2 r2 ) ( ) a b r1 ! r2 !
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (2), 151-154 (2014) ρ
+2,n+1 1 Hm p +3,q+3 [ zx a
REFERENCES
−σ1 δ1 b /
(1−δ −r2 , s )(a j ,α j )1, p (λ −r1,σ1)(−m1r1−m2r2 , ρ1 (1+λ,σ1)(b j , β j )1,q )(γ −m1r1−m2r2 , ρ1)(−δ ,σ1) ]
To Prove of (8) , we first express the H-function occurring on the L.H.S. of equation (8) in terms of Mellin Barnes type of contour integral given by equation(7) and collected the power of x, ( x
m1
+ a) and
m
(b − x 2 ) and appply binomial expansion, we get a λ b −δ
1 2πι
∞ Π mj=1Γ(b j − β j s )Π nj =1Γ(1 − a j + α j s ) . q p −ι∞ Π j = m +1Γ(1 − b j + β j s )Π j = n +1Γ( a j − α j s ) r1, r2 = 0
∫
+ι∞
∑
Γ(1 − (−λ ) + σ 1s )Γ (1 − δ − δ 1s ) . Γ(1 − (λ ) − r1 + σ 1s )(Γ(1 − δ − r2 − δ 1s )) 1 r x1 r ( ) 1( ) 2 m r +m r −ρ S a b .D xγ {x 1 1 2 2 1 } r1! r2 !
Further using the result(4) the above equation becomes R.H.S. of (8). m
a λ b −δ x − γ
∞
∑
(
m
x 1 r1 x 2 r2 ) ( ) ρ 1 σ 1 −δ +1, n + 2 a b Hm p + 3, q + 3 [ zx a b / r1!r2 !
r1 , r2 = 0 ( − λ ,σ 1 )(1−δ − r2 , s )( a j ,α j )1, p (1−γ + r1m1 + r2 m2 , ρ1 ) (1+ r1m1 + r2 m2 , ρ1 )( b j , β j )1, q )( − λ + m1 ,σ 1 )(1−δ ,δ1 )
]
On the same parallel lines,second formula given by equation (9) can be easily obtained by using the results (4 ) and (5) respectively.
153
1. Erdelyi, A. Higher Transdental Function, Vol.I, MacGraw-Hill, New York, Toronto and London (1953) . 2. Fox,C. The G and H-functions as symmetrical Fourier kernals, Amer. Math. Soc. 98 p.395 − 429, (1961) . 3. Oldham, K.B. and Spanier, J. The Fractional Calculus, Academic Press, NewYork/ London (1974). 4. Rainville, E. D. Special Function, NewYork, The MacMillan Company (1960). 5. Sharma, C.K. and Singh, Indrajeet: Fractional derivatives of the Lauricella Function and multivariable h-function, Jananabha 21 p.165 − 170. (1991). 6. Ross, B. Fractional calculus and its applications, Lecture Notes in Math.Vol. 457 Spring-verlag, NewYork (1975). 7. Ross, B. and Northover, F.H. A use for a derivative of complex order in the fractional calculus, Indian J.Pure App. Math. 9 , p.400 − 406. (1978). 8. Srivastava, H.M., Gupta, K.C. and Goyal, S.P. The H-function of one and two variable with application, South Asian Publishers, New Delhi-Madras (1986). 9. Srivastava, H.M., Singh Chandel, R.C. and Vishwakarma,P.K, (1994) :Fractinal derivatives of certain generalized hypergeometric functions of several variables, J. Math. Anal. Appl. 185 P.560-573 (1994). 10. Srivastava, H.M. and Goyal, S.P. Fractional derivatives of the H-function of severable (1985).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
154
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (2), 151-154 (2014)
11. Srivastava, H.M. and Manocha, H.L. A Treatise on generating functions, Halsted press, (Ellis Horwood limited,
Chichester) John Wiley and sonâ&#x20AC;&#x2122;s New York, Chicester, Brisbane and Toronto (1984).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)