Fractional calculus and application of generalized Struve function

A new generalization of Struve function called generalized Galué type Struve function (GTSF) is defined and the integral operators involving Appell’s functions, or Horn’s function in the kernel is applied on it. The obtained results are expressed in terms of the Fox–Wright function. As an application of newly defined generalized GTSF, we aim at presenting solutions of certain general families of fractional kinetic equations associated with the Galué type generalization of Struve function. The generality of the GTSF will help to find several familiar and novel fractional kinetic equations. The obtained results are general in nature and it is useful to investigate many problems in applied mathematical science.

and the generalized Struve function of the first kind (Nisar et al. 2016b). Here, in this paper, we aim at presenting the integral transforms and the solutions of certain general families of fractional kinetic equations associated with newly defined Galué type generalization of Struve function. Galué (2003) introduced a generalization of the Bessel function of order p given by Baricz (2010) investigated Galué-type generalization of modified Bessel function as: The Struve function of order p given by is a particular solution of the non-homogeneous Bessel differential equation where Ŵ is the classical gamma function whose Euler's integral is given by (see, e.g., Srivastava and Choi 2012, Section 1.1): The Struve function and its more generalizations are found in many papers (Bhowmick 1962(Bhowmick , 1963Kanth 1981;Singh 1974;Nisar and Atangana 2016;Singh 1985Singh , 1988aSingh , b, 1989. The generalized Struve function given by Bhowmick (1962) and by Kanth (1981) Singh (1974) found another generalized form as The generalized Struve function of four parameters was given by Singh (1985) (also, see Nisar and Atangana 2016) as: where > 0, α > 0 and µ is an arbitrary parameter. Another generalization of Struve function by Yagmur (2014, 2013) is, Motivated from (1), (3) and (10), here we define the following generalized form of Struve function named as generalized Galué type Struve function (GTSF) as: where α > 0, ξ > 0 and µ is an arbitrary parameter and studied fractional integral representations of generalized GTSF.

Fractional integration of (11)
The following lemmas proved in Kilbas and Sebastian (2008) are needed to prove our main results.

Lemma 1 (Kilbas and Sebastian 2008) Let
Then ∃ the relation Lemma 2 (Kilbas and Sebastian 2008 The main results are given in the following theorem.
Theorem 1 Let a ∈ N, , σ , ϑ, ρ, l, b, c ∈ C, α > 0 and µ is an any arbitrary parameter be such that l . Nisar et al. SpringerPlus (2016) 5:910 Proof Notice that the condition given in Eq. (17) holds for 3 4 given in (20) and then interchanging the integration and summation, (11) and (12) together imply and hence by Lemma 1, In view of definition of Fox-Wright function (16) we obtain the desired result.
If we set α = a = 1, µ = 3 2 and ξ = 1 in Theorem 1 then we obtain the theorem 1 of Nisar et al. (2016a) as follows: (10) Theorem 2 Let a ∈ N, , σ , ϑ, ρ, l, b, c ∈ C, α > 0 and µ is an any arbitrary parameter be such that l Proof The Fox-Wright function 3 4 given in (22) is well-defined as it satisfy inequality (17) and changing the order of integration and summation, (13) and (16) together imply Now using Lemma 2 and the under the conditions mentioned in Theorem 2, we have Now (22) can be deduced from (23) by using (17), hence the proof.
The familiar Riemann-Liouville fractional integral operator (see, e.g., Miller and Ross 1993; Kilbas et al. 2006) defined by and the Laplace transform of Riemann-Liouville fractional integral operator ( Erdélyi et al. 1954;Srivastava and Saxena 2001) is where F (p) is the Laplace transform of f(t) is given by whenever the limit exist (as a finite number).

Kinetic equations
The standard kinetic equation is of the form, with N i (t = 0) = N 0 , which is the number of density of species i at time t = 0 and c i > 0 . The integration of (29) gives an alternate form as follows: where 0 D −1 t is the special case of the Riemann-Liouville integral operator and c is a constant. The fractional generalization of (30) is given by Haubold and Mathai (2000) as: where 0 D −υ t defined in (26). Recently, Saxena and Kalla (2008) considered the following equation and obtained the solution as: For more details about the solution of kinetic equations interesting readers can refer (Saxena and Kalla 2008;Nisar and Atangana 2016).

Solution of fractional kinetic equation involving (11)
In this section, we will discuss about the solution fractional kinetic equation involving newly defined function generalized GTSF to show the potential of newly defined function in application level.
Given the equation where e, t, v ∈ R + , a, b, c, l ∈ C and R(l) > −1.
Taking the Laplace transform of (34) and using (11) and (27) The following results are more general than (38) and they can derive parallel as above, so the details are omitted.

Conclusion
In this paper, we investigated the integral transforms of Galué type generalization of Struve function and the results expressed in terms of Fox-Wright function. By substituting the appropriate value for the parameters, we obtained some results existing in the literature as corollaries. The results derived in section "Application" of this paper are general in character and likely to find certain applications in the theory of fractional   Kumar et al. (2015) calculus and special functions. The solutions of certain general families of fractional kinetic equations involving generalized GTSF presented in section "Conclusion". The main results given in section "Solution of fractional kinetic equation involving (11)" are general enough to be specialized to yield many new and known solutions of the corresponding generalized fractional kinetic equations. For instance, if we put a = α = ξ = 1 and µ = 3 2 in (34), (39) and (41), then we get the Eqs. (15), (19) and (24) of Nisar et al. (2016b).