Certain Fractional Integral Operators and Generalized Struve s Function

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1 Volume 8 No. 9 8, 9-5 ISSN: -88 (printed version); ISSN: 4-95 (on-line version) url: ijpam.eu Certain Fractional Integral Operators and Generalized Struve s Function * Sunil Kumar Sharma and ** Ashok Singh Shekhawat * Department of Mathematics, Arya College of Engineering and Research Centre, Jaipur, Rajasthan, India. ** Department of Mathematics, Suresh Gyan Vihar University, Jaipur-7, Rajasthan, India. addresses: * sunil9447@gmail.com, ** csmaths4@yahoo.com Abstract Several interesting and useful extensions of some familiar functions such as Beta and Gauss hypergeometric functions and their properties have, recently, been investigated by many authors. Motivated mainly by those earlier works, we establish some integral formulas involving generalized Galu type Struve function (GTSF) by using certain general pair of fractional integrals operators involving Gauss hypergeometric function. We also investigated a composition formula of the pathway fractional integration operator with product of generalized GTSF. The obtained results are expressed in terms of the Fox-Wright function. Special cases of the main results are also pointed out in the form of corollaries. All the results derived here are of general nature and can yield a number of results in the theory of special functions. Mathematical Subject Classification: Primary C6, C, C5; Secondary 44A, 6A9, 6A Keywords and phrases: Generalized Struve function, Fox-Wright function, Fractional integral operators, Integral transform.. Introduction and Preliminaries The fractional calculus operators have been extensively used in describing and solving various integral equations, ordinary differential equations and partial differential equations in applied sciences such as fluid mechanics, rheology, diffusive transport, electrical networks, electromagnetic theory, probability theory, turbulence and fluid dynamics, stochastic dynamical system plasma physics and controlled thermonuclear function, image processing, non-linear biological system and astrophysics (see, for very recent works [6,7,8,9,]). Recent studies observed that the solutions of fractional order differential equations could model real life situations better, particularly in reaction diffusion type problems. Due to the potential applicability to wide variety of problems, fractional calculus is developed to large area of physics and engineering applications. Here in this paper, we aim at presenting the integral transform and the solution of certain fractional integral operators associated with newly defined Galu type generalization of Struve function. Galu [5] introduced a generalization of the Bessel function of order h given by +h h Γ + h +!, R, N,,, () 9

2 h h k ξ, > k ξ, >, > H λ ξ, >, τ > h,τ h,τ ξ h H h,b,c ξ, h, b, c C k, h, λ C Baricz [] investigated Galu -type generalization of modified Bessel function as: +h h + h +!, R, N () The Struve function of order h given by k ξ k+h+ Hh ξ, () k Γ k + Γ k + h + is a particular solution of the non-homogeneous Bessel differential equation 4 ξ h+ ξ y ξ + ξ y ξ + ξ h y ξ π Γ h+ (4) where Γ is the classical gamma function whose Euler s integral is given by Srivastava and Choi (see []), ( ) > (5) The Struve function and its more generalizations are found in many papers (see [, 4, 6, 7,,,, and ]). The generalized Struve function given by Bhowmick [4]: H λ ξ k+h+ (6) (7) k Γ λk + h + Γ k + (8) and by Kanth [6] (9) ξ k+h+ H λ,α k Γ λk + h + Γ αk + Singh [] found another generalized form as () k ξ k+h+ k Γ λk + τ + Γ k + The generalized Struve function of four parameters was given by Singh [] (also see [7]) ξ k+h+ H λ,α where >, > and τ is an arbitrary parameters. k Γ λk + h + Γ αk + τ Another generalization of Struve function by Orhan and Yagmur [8, 9] is c k ξ k+h+ b k Γ k + Γ k + h + + 4

3 ,,,. Fractional Calculus of Generalized Struve Function Motivated from (), () and (), here Nisar et al. [4] defined the following generalized form of Struve function named as generalized Galu -type Struve function (GTSF) as : c k ξ k+h+ a h,b,c, α,µ τ ξ, N, h, b, c C () h b + k Γ αk + µ Γ ak + τ + where α >, τ > and µ is an parameter and studied fractional integral representations of generalized GTSF. The generalized hypergeometric series is defined by Raiville (see []):,, ;,, ; (,, ;,, ; ) (),,! Here p and q are positive integers or zero (interpreting an empty product as ), and we assume (for simplicity) that variable z, the numerator parameters,, and the denominator parameters,, take on complex values, provided that no zeros appear in the denominator of eq.(), that is ( C / ;,, ). () The special case of () is called (Gauss) hypergeometric series. where is the Pochhammer symbol defined for C by Srivastava and Choi (see [4] ): ( ) Γ λ + C/ (4) + + ( ) Γ λ The generalized hypergeometric Wright function defined for C,, C and real, R, (, ;,,.., ;,,, ) by the series,, ; Γ + (5 ), ; Γ +! where C is the set of all the complex numbers and Γ is the Euler gamma function (see [4]) and the function (5) was introduced by Wright [] and is known as generalized hypergeometric Wright function. Condition of existence (5) together with its representation in terms Mellin-Bernes integral and of the H-function were established in [5]. In particulars, is an entire function if there holds the condition > (6)

4 Recently fractional integral operators involving the various special functions have been established by many authors (see, e.g. [5-]).Here in this section, we shall establish some fractional integral formulas for newly defined Galu type generalization of Struve function. 4

5 h,,,,,,,,,,,,,,,,,,,,,, Γ Γ + + Γ Γ Γ For our purpose, we begin by recalling the following pairs of Saigo hypergeometric operators of fractional integrations. For >,, and >, we have,, F +, ; ;, (7) and Γ F +, ; ;, (8) Γ where F(. ) is Gamma hypergeometric series which is a special case of the generalized hypergeometric series F in (). The operator,,, (. ) contains both the Reimann-Liouville and Erdelyi-Kobar fractional integral operators, by means of the following relationships: R, (9) and,,, () whereas the operator (8) unifies the Weyl type and Erdelyi-Kobar fractional integral operators as follows:,,, () and Γ, () We use the following image formulas which are easy consequences of the operators (7) and (8) (see [4, 4]):,, Γ Γ +, >, + > () and Γ + Γ +, + >, + > Γ Γ (4) Applying () to the Saigo fractional integral operator (7), we obtain a fractional integral asserted by Theorem. Theorem.Let >, N,,,,, h,, C and is an arbitrary parameters be such that +,,,, R + h + > max, R, R >. Then the following fractional integral formula holds true,,, h,,,, ( ) h+ h + +,, h + + +,, (,) 4 h + 4 (5) h+ +,, h + +,, h ,, (, ) 4

6 ,,, h+,, h,,, h,, h + h+ Proof. The Fox-Wright function 4 given in (5) is well defined as it satisfies inequality (6). Applying (), to the Saigo fractional integral operator (7) and changing the order of integration and summation, which is valid under the condition of Theorem, we find that (h+ +),,,,, +h+ h,,, ( ) Γ + Γ + h + +, Forany,,,, clearly R h R + h + > max[, R( )].Now applying the known result () with replaced by + h +, we have,,, h,,,, ( ) (h+ +) Γ h Γ h h + Γ h Γ + + Γ + Γ h Γ +! (6) 4 which, in view of definition of Fox-Wright function (5), proves the required result (5). Interestingly, on setting, and, Theorem yields corollary. Corollary. Let,,, h,, C be h +,,,, R >, R + h + >, then we get H h+ h + +,, h + + +,, (,) 4 4 (7) h + +,, h + +,, h ,,(, ) where Hh,, ( ) is given in (). If we set and using the relation (), Theorem and Corollary yield the following results: Corollary. Let >, N, and the parameters,,, h,, C be such that R >, R >. Then the right-side Erd lyi-kober fractional integrals are given by,, h,,,, ( ) h + + +,, (,) h+ 4 +,, h ,, (, ) (8) Corollary 4.With all assumption and condition on parameters, as stated in Corollary with,,, h,, CR >, R >, the following result holds true: H ( ) 4

7 , h,,, h+, h,,, h,,,, h+ 4 h h,,, ( ) h + + h+ +, h+ h + + +,,(,) 4 (9) h+ h + +,, h ,, (, ) Further, if we replace by and make use relation (9), in theorem and Corollary, we obtain yet another corollaries providing Riemann-Liouville fractional integrals asserted by Corollaries 5 and 6. Corollary 5.Let >, >, N and the,, h,, C parameters satisfies R>, R >. Then the following fractional integral formula holds true R, h+ + h + +,,(,) h + 4 +,, h + + +,, (, ) Corollary 6. Let >,,, h,, C be h +,,,, R >, R + h + >, then we obtain R H h + +,, (,) 4 (7) h + +,, h + + +,, (, ) where Hh,, ( ) is given in (). Theorem 7. Let >, >, N,,,,, h,, C and is an arbitrary parameters be h such that,, >, R Then the, R h < + min R R following + fractional integral formula holds true,,, + h +,, + h +,, (,) h + 4 () +,, h +,, + + h + +,, (, ) Proof. Applying (), to the Saigo fractional integral operator (8) and changing the order of integration and summation, which is valid under the condition of Theorem 7, we find that (h+ +),,,,, h h Γ + Γ + + Now applying the known result (4) with replaced by h, we have 44

8 , h,,, (h+), h,, h+, h,,,, h,, 4 h h+ h+ Γ + h + + Γ + h + + h + Γ h + +,,, h Γ + + Γ + Γ + + h Γ +! 4 () In view of definition of Fox-Wright function (5),we obtain the desired result. If we set, and, we obtain a simple special case () asserted by the following Corollary. Corollary 8. Let,,,, h,, C be h +,,,, R >, R h < + min R, R. Then the following fractional integral formula holds true,, H + h +,, + h +,, (,) h 4 (4) where Hh,, ( ) is given in (). h + +,, h +,, + + h + +,,, If we set and using the relation (), Theorem 7 and Corollary 8, yield the following results. Corollary 8.Let >, N and the parameters,,, h,, C satisfying the inequalities R >, R >, then we have,, h + h +,, (,) 4 (5) h + +,, + h + +,, (, ) Corollary 9.With all assumption and condition on parameters, as stated in Corollary 9 with,,, h,, C and R >, R >,the following result holds true, H + h +,, (,) 4 (6) h + +, + h + +, Further, if we replace by and making use of relation (), in Theorem 7 and Corollary 8, we have obtain yet another Corollaries providing Weyl fractional integrals asserted by Corollaries () and (). 45

9 , h,,,, h,, +,, ( ) + h+ h+ ( ) (7) Corollary. Let >, N and the parameters,, h,, C, > satisfying the inequalities R >, R >.Then we have, h + h +,, (,) h+ 4 h + +,, h +,, (, ) Corollary. Let >,,,, h,, C be h +,,,, R >, R + h +. Then the following fractional integral formula holds true H h + h +,,(,) h+ 4 (8) where Hh,, ( ) is given in (). h + +,, h +,,,. Pathway Fractional Integration of the Generalized Struve s Function: Recently, Nair [7] introduced the pathway fractional integral operator by using the pathway idea of Mathai [8], developed further by Mathai and Haubold [] and defined as follows (c.f. [5]). Let,, C with R >, R +, and < be the pathway parameter. Then +,,. (9) where, is the set of Lebsegue measurable function defined on pathway model for scalar random variables is represented by the function ( p.d.f.):,. For real scalar, the following probability density ( ), provided that R,, R +, R +, >. Here c is the normalizing constant and is called the pathway parameter. For >, (9) can be written as follows:, (4) and + ( ) Provided that R,, R +, R +. ( ), (4) 46

10 h,p,c,τ h,p,c,τ α,λ Γ + h+ ( ) +h+ h+ ( h p + + If we set and and replacing by in (9), then we have the following relationship: +,, Γ + ( ) (4) where + is the left sided Riemann-Liouville fractional integral operators []. For more detail on the pathway model and its particular cases, the interested reader may refer to the recent works (see [, 6, 7, and 8]). It is observed that the pathway fractional integral operator (9) can lead to other interesting examples of fractional calculus operators regarding some probability density function and applications in statistics. Our main result in this section is based on the following assertion giving a composition formula of the pathway fractional integrations operators (9) with a power function (see Nair [7, Lemma ]). Lemma. Let C, R>, C and <. If R > and R >, then we have + Γ Γ + +,, (4) ( ) Γ + + Now we are ready to present our result which is composition formula of the pathway fractional integration operator (9) with a product of generalized Struve function () asserted by the following theorem. Theorem 4. Let<, the parameters,,, h,, C, R >, >, R > and R >, then there holds the following formula +,, l h,p,c, α,λ τ ξ + +h+ ( + h +,), (,) h + 4 (44) +, l, + + h +,, λ, Proof. Applying () to (9) and changing the order of integration and summation, we find c k ( +h+) +,, l ξ,, + +h+, k Γ αk + λ Γ lk + τ + Now we applying Lemma to use (4) with replaced by+ + h + and obtain + +h+ Γ + Γ + h + + Γ + +,, l α,λ ξ ) +h+ h p + k Γ αk + λ τ Γ lk + + (45) 4 Γ + + h + +! 47

11 h,p,c, h+ Γ + +h+ Hence, in view of (5), the last expression of (45) is seen to correspond with our desired result (44). This completes the proof. Interestingly, on setting α l, λ and, Theorem 4 yields corollary 5. Corollary 5. Let <, the parameters,,, h,,, R>, >, R > and R >, then the following fractional integral formula holds true +,, H ξ + +h+ ( + h +,), (,) (46) 4 ( ) h + +,, + + h +,,, 4. Conclusion. In this paper, we investigated the integral transforms of Galu type generalization of Struve function and the result expressed in terms of Fox-Wright function. Recently, fractional operators theory was recognized to be good tool for modelling, complex problems, kinetic equations, fractional reaction, fractional diffusion equations, and so forth. In this work, we investigated and studied Saigo hypergeometric operators of fractional operators and pathway integral operator of fractional operators are associated with Galu type generalization of Struve function (GTSF). Besides some interesting cases of Saigo hypergeometric operators, the Riemann-Liouville, the Erd lyi-kober and Weyl type integral operators are also discussed. We obtained the results expressed in terms of Fox- Wright function, through these fractional operators. Results derived in this paper are very significant and may find applications in the solution fractional order differential equations that are arising in certain areas of turbulence, propagation of seismic waves, diffusion process, fractional kinetic equation etc. By substituting the appropriate value for the parameters, we obtained some results existing in the literature as Corollaries. References [] E.M.Wright, The asymptotic expansion of the generalized hypergeometric functions, J.London Math. Soc., Vol. (95), [] S.G.Samko, A.A.Kilbas and O.I.Marchev, Fractional integrals and derivatives, Theory and applications, Gordon Breach, Yverdon et al. (99). [] A.M.Mathai, H.J.Haubold, On generalized distribution and pathways, Phys. Lett. A 7 (8), 9-. [4] A. Erdlyi, W.Magnus, F.Oberhettinger and F.G. Tricom, Higher Transcendental Functions, Vol. I. McGraw-Hill, New York-Toronto-London (95). [5] D.Baleanu and P.Agarwal, A composition formula of the pathway integral transform operators, Note Mat. Vol. 4() (4), [6] J.Choi, P.Agarwal and Shilpi Jain, Certain fractional integral operators and extended generalized Gauss hypergeometric functions, KYUNPOOK Math. J., Vol. 55 (5),

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