# Fractional Calculus involving (p, q)-Mathieu Type Series

• 1 Department of Applied Sciences, Guru Kashi University, 151302, Bathinda, India
• 2 Department of Mathematics, Anand International College of Engineering, 303012, Jaipur, India
• 3 Department of Applied Sciences, Guru Kashi University, 151302, Bathinda, India
• 4 Department of Mathematics, Baba Farid College, 151001, Bathinda, India
Daljeet Kaur
, Praveen Agarwal
and Mehar Chand

## Abstract

Aim of the present paper is to establish fractional integral formulas by using fractional calculus operators involving the generalized (p, q)-Mathieu type series. Then, their composition formulas by using the integral transforms are introduced. Further, a new generalized form of the fractional kinetic equation involving the series is also developed. The solutions of fractional kinetic equations are presented in terms of the Mittag-Leffler function. The results established here are quite general in nature and capable of yielding both known and new results.

## 1 Introduction and Preliminaries

Fractional calculus is a very rapidly growing subject of mathematics which deals with the study of fractional order derivatives and integrals. Fractional calculus is an efficient tool to study many complex real world systems . It is demonstrated that fractional order representation of complex processes appearing in various fields of science, engineering and finance, provides a more realistic approach with memory effects to study these problems. (see e.g. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and ). Among the research work developing the theory of fractional calculus and presenting some applications, we have to point out some literature (see [16, 17, 18, 19, 20]). Kumar et al.  analyzed the fractional model of modified Kawahara equation by using newly introduced Caputo-Fabrizio fractional derivative. One also et al.  studied a heat transfer problem and presented a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Fabrizio fractional derivative. Recently, one et al.  presented a new fractional extension of regularized long wave equation by using Atangana-Baleano fractional operator. In et al.  one introduced a new numerical scheme for fractional Fitzhugh-Nagumo equation arising in transmission of new impulses. In et al.  one constituted a modified numerical scheme to study fractional model of Lienard’s equations. Hajipour et al.  in their work formulated a new scheme for class of fractional chaotic systems. Baleanu et al.  proposed a new formulation of the fractional control problems involving Mittag-Leffler non-singular kernel. In another work, Baleanu et al.  studied the motion of a Bead sliding on a wire in fractional analysis. Jajarmi et al.  analyzed a hyperchaotic financial system and its chaos control and synchronization by using fractional calculus.

For mathematical modeling of many complex problems appearing in various fields of science and engineering such as fluid dynamics, plasma physics, astrophysics, image processing, stochastic dynamical system, controlled thermonuclear fusion, nonlinear control theory, nonlinear biological systems, quantum physics and heat transfer problems, the fractional calculus operators involving various special functions have been used successfully. There is rich literature available revealing the notable development in fractional order derivatives and integrals (see, [1, 10, 11, 18, 19, 20, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]). Recently, Caputo and Fabrizio  introduced a new fractional derivative which is more suitable than the classical Caputo fractional derivative for many engineering and thermodynamical processes. Atangana  used a new fractional derivative to study the nature of Fisher’s reaction diffusion equation. Riemann and Caputo fractional derivative operators both have a singular kernel which cannot exactly represent the complete memory effect of the system. To overcome these limitations of the old derivatives, very recently Atangana and Baleanu  presented a new non-integer order derivative having a non-local, non-singular and Mittag-Leffler type kernel.

In recent years, many researchers have extensively studied the properties, applications and extensions of various fractional integral and differential operators involving the various special functions. (for detail see McBride , Kalla [44, 45], Kalla and Saxena [46, 47], Saigo [48, 49, 50], Saigo and Maeda , Kiryakova [32, 52],  etc).

For our present study, we recall the following pair of Saigo hypergeometric fractional integral operators.

For x > 0, λ, σ, ϑ ∈ ℂ and ℛ(λ) > 0, we have

$(I0,xλ,σ,ϑf(t))(x)=x−λ−σΓ(λ)∫0x(x−t)λ−1F21(λ+σ,−ϑ;λ;1−tx)f(t) dt$
and
$(Ix,∞λ,σ,ϑf(t))(x)=1Γ(λ)∫x∞(t−x)λ−1t−λ−σF21(λ+σ,−ϑ;λ;1−xt)f(t) dt$
where the 2F1(.), a special case of the generalized hypergeomteric function, is the Gauss hypergeometric function.

The operator $I0,xλ,σ,ϑ(.)$ contains the Riemann-Liouville $R0,xλ(.)$ fractional integral operators by means of the following relationships:

$(R0,xλf(t))(x)=(I0,xλ,−λ,ϑf(t))(x)=1Γ(λ)∫0x(x−t)λ−1f(t) dt$
$(Wx,∞λf(t))(x)=(Jx,∞λ,−λ,ϑf(t))(x)=1Γ(λ)∫x∞(t−x)λ−1f(t) dt$

It is noted that the operator (1.2) unifies the Erdêlyi-Kober fractional integral operators as follows:

$(E0,xλ,ϑf(t))(x)=(I0,xλ,0,ϑf(t))(x)=x−λ−ϑΓ(λ)∫0x(x−t)λ−1tηf(t) dt$
$(Kx,∞λ,ϑf(t))(x)=(Jx,∞λ,0,ϑf(t))(x)=xϑΓ(λ)∫x∞(t−x)λ−1t−λ−ϑf(t) dt$

The following lemmas proved in Kilbas and Sebastin  are useful to prove our main results.

Lemma 1

(Kilbas and Sebastian 2008) Let λ, σ, ϑ ∈ ℂ be such that ℛ(λ) > 0, ℛ(ρ) > max[0, ℛ(σϑ)], then

$(I0,xλ,σ,ϑtρ−1)(x)=Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)xρ−σ−1.$

Lemma 2

(Kilbas and Sebastian 2008) Let λ, σ, ϑ ∈ ℂ be such that ℛ(λ) > 0, ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], then

$(Ix,∞λ,σ,ϑtρ−1)(x)=Γ(σ−ρ+1)Γ(ϑ−ρ+1)Γ(1−ρ)Γ(λ+σ+ϑ−ρ+1)xρ−σ−1.$

The image formulas for special functions of one or more variables are very useful in the evaluation and solution of differential and integral equations. Motivating by the above discussion, we developed new fractional calculus formulas involving extended generalized Mathieu series.

The following familiar infinite series

$S(r)=∑n=1∞2n(n2+r2)2, (r∈ℝ+),$
is called a Mathieu series. It was introduced and studied by Émile Leonard Mathieu in his book  devoted to the elasticity of solid bodies. Bounds for this series are needed for the solution of boundary value problems for the biharmonic equations in a two dimensional rectangular domain, see [56, Eq. (54), p. 258]. Several interesting problems and solutions dealing with integral representations and bounds for the following generalization of the Mathieu series, which is so-called generalized Mathieu series with a fractional power can be found in [57, 58, 60]:
$Sμ(r)=∑n=1∞2n(n2+r2)μ+1, (μ>0,r>0).$

In , the authors derived the following new Laplace type integral representation series

$Sμ(r)=π2μ−12Γ(μ+1)∫0∞e−rtκμ(t)dt, (μ>32)$
$κμ(t)=tμ+12∑k=1∞Jμ+12(kt)kμ−12$
and Jμ(z) is the Bessel function. Motivated essentially by the works of Cerone and Lenard , Srivastava and Tomovski in  defined a family of generalized Mathieu series
$Sμ(α,β)(r;a)=Sμ(α,β)(r;{an}n=1∞)=∑n=1∞2anβ(anα+r2)μ, (α,β,μ,r>0),$
where it is tacitly assumed that the positive sequence
$a={an}={a1,a2,...}$
such that
$limn→∞an=∞$
is so chosen that the infinite series in definition (1.11) converges, that is, that the following auxiliary series
$∑n=1∞1anμα−β$
is convergent.

Definition 1

(see [63, Eq. (6.1), p. 256] ) The extended Beta function Bp;q(x; y) is defined by

$Bp,q(x,y)=∫01tx−1(1−t)y−1Ep,qdt,(x,y,p,q,∈ℂ;min{ℛ(x),ℛ(y)}>0, min{ℛ(p),ℛ(q))≥0}$
where Ep,q(t) is defined by
$Ep,q(t)=exp(−pt−q1−t)$
(p, q ∈ ℂ and min{ℛ(p), ℛ(q)} > 0).

In particular, Chaudhry et al. [64, p. 591, Eq. (1.7)], introduced the p–extension of Euler’s Beta function B(x, y) :

$Bp(x,y)=∫01tx−1(1−t)y−1e−pt(1−t)dt$
(ℛ(p) > 0) whose special case, when p = 0 ( or p = q = 0 in (1.12)), is the familiar Beta integral
$B(x,y)=∫01tx−1(1−t)y−1dt$
(ℛ(x), ℛ(y) > 0).

Recently, Mehrez and Tomovski  introduces the (p, q)-Mathieu-type power series in terms of the extended Beta function (1.12), which is defined as:

$Sμ,ϑ,τ,ξα,β(r;a;p,q;z)=∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μznn!(r,α,β,ν>0;ξ>τ>0;p,q∈ℂ;min{ℛ(p),ℛ(q)}≥0;|z|<1)$

In particular case when p = q; we define the p-Mathieu-type power series defined by

$Sμ,ϑ,τ,ξα,β(r;a;p;z)=∑n=1∞2anβ(ϑ)nBp(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μznn!(τ,α,β,ϑ,ξ,τ>0,p∈C,|z|≤1)$

The function $Sμ,ϑ,τ,ξα,β(τ;a;p;z)$ has many other special cases. If we set p = q = 0; we get

$Sμ,ϑ,τ,ξα,β(r;a;z)=∑n=1∞2anβ(ϑ)n(anα+r2)μ(ξ)nznn!$
(τ, α, β, ϑ, ξ > 0, |z| ≤ 1)

On the other hand, by letting τ = ω in (1.15) we obtain [66, Eq. 5, p. 974]:

$Sμ,ϑα,β(r;a;z)=∑n=1∞2anβ(ϑ)n(anα+r2)μznn!, (τ,α,β,ϑ>0,|z|≤1).$

The concept of the Hadamard product (or the convolution) of two analytic functions is very useful in our present study. It can help us to decompose a newly emerging function into two known functions. Let

$f(z):=∑n=0∞anzn,(|z|
and
$g(z):=∑n=0∞bnzn,(|z|
be two power series whose radii of convergence are denoted by Rf and Rg, respectively.

Then their Hadamard product is the power series defined by

$(f*g)(z):=∑n=0∞anbnzn=(g*f)(z)(|z|
where
$R=limn→∞|anbnan+1bn+1|=limn→∞|anan+1|.limn→∞|bnbn+1|=RfRg$

Therefore, in general, we have RRf.Rg [67, 68].

For various investigations involving the Hadamard product (or the convolution), the interested reader may refer to several recent papers on the subject (see, for example, [69, 70] and the references cited therein).

## 2 Fractional integration

In this section, we will establish some fractional integral formulas for the generalized (p, q)-Mathieu-type power series. Then their special cases also introduced here.

Theorem 1

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) > max[0, ℛ(σϑ)], then

$(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;t))(x) =xρ−σ−1Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)Sμ,ϑ,τ,ξα,β(r;a;p,q;x)*F22[ρ,ρ+ϑ−σρ−σ,ρ+λ+ϑ;x].$

Proof

For convenience, we denote the left-hand side of the result (2.1) by . Using (1.13), and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then

$ℐ=∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ1n!(I0,xλ,σ,ϑtn+ρ−1)(x),$
applying the result (1.7), the above equation (2.2) reduced to
$ℐ=∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ1n!Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n)xρ+n−σ−1,$
after simplification, we have
$ℐ=xρ−σ−1Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ ×(ρ)n(ρ+ϑ−σ)n(ρ−σ)n(ρ+λ+ϑ)nxnn!,$
further interpret the above equation with the view of of the function given in equation (1.13), we have
$ℐ=xρ−σ−1Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)Sμ,ϑ,τ,ξ;ρ−σ,ρ+λ+ϑα,β;ρ,ρ+ϑ−σ(r;a;p,q;x),$
employing the concept of the Hadamard product given in equation (1.19) in the above equation (2.5), required result is obtained.

Theorem 2

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0;ξ > τ > 0; p, q ∈ ℂ;min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], Then

$(Ix,∞λ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;1/t))(x)=xρ−σ−1Γ(σ−ρ+1)Γ(ϑ−ρ+1)Γ(1−ρ)Γ(λ+σ−ϑ−ρ) ×Sμ,ϑ,τ,ξα,β(r;a;p,q;1/x)*F22[σ−ρ+1,ϑ−ρ+11−ρ,λ+σ−ϑ−ρ;x].$

Proof

Proof is parallel to Theorem 1.

### 2.1 Special cases of fractional integral formulae

In this section we reduces our main findings to the special cases by assigning particular values to the parameters as follows:

Case 1.

If we choose p = q the findings in equations(2.1)and(2.6)reduces to the following the form:

Corollary 1

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0; p ∈ ℂ; ℛ(p) ≥ 0, such that ℛ(ρ) > max[0, ℛ(σϑ)], then

$(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p;t))(x) =xρ−σ−1Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)Sμ,ϑ,τ,ξα,β(r;a;p;x)*F22[ρ,ρ+ϑ−σρ−σ,ρ+λ+ϑ;x].$

Corollary 2

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0;ξ > τ > 0; p ∈ ℂ; ℛ(p) ≥ 0, such that ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], Then

$(Ix,∞λ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p;1/t))(x)=xρ−σ−1Γ(σ−ρ+1)Γ(ϑ−ρ+1)Γ(1−ρ)Γ(λ+σ−ϑ−ρ) ×Sμ,ϑ,τ,ξα,β(r;a;p;1/x)*F22[σ−ρ+1,ϑ−ρ+11−ρ,λ+σ−ϑ−ρ;x].$

Case 2.

If we choose p = q = 0 the findings in equations(2.1)and(2.6)reduces to the following the form:

Corollary 3

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0, such that ℛ(ρ) > max[0, ℛ(σϑ)], then

$(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;t))(x) =xρ−σ−1Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)Sμ,ϑ,τ,ξα,β(r;a;x)*F22[ρ,ρ+ϑ−σρ−σ,ρ+λ+ϑ;x].$

Corollary 4

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0, such that ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], Then

$(Ix,∞λ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;1/t))(x)=xρ−σ−1Γ(σ−ρ+1)Γ(ϑ−ρ+1)Γ(1−ρ)Γ(λ+σ−ϑ−ρ) ×Sμ,ϑ,τ,ξα,β(r;a;1/x)*F22[σ−ρ+1,ϑ−ρ+11−ρ,λ+σ−ϑ−ρ;x].$

Case 3.

If we choose p = q = 0 and τ = ξ, the findings in equations(2.1)and(2.6)reduces to the following the form:

Corollary 5

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0, such that ℛ(ρ) > max[0, ℛ(σϑ )], then

$(I0,xλ,σ,ϑtρ−1Sμ,ϑα,β(r;a;t))(x) =xρ−σ−1Γ(ρ)Γ(ρ+ϑ−σ)Γ(ρ−σ)Γ(ρ+λ+ϑ)Sμ,ϑα,β(r;a;x)*F22[ρ,ρ+ϑ−σρ−σ,ρ+λ+ϑ;x].$

Corollary 6

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0, such that ℛ(ρ) < 1 + min[(σ), ℛ(ϑ)], Then

$(Ix,∞λ,σ,ϑtρ−1Sμ,ϑα,β(r;a;1/t))(x)=xρ−σ−1Γ(σ−ρ+1)Γ(ϑ−ρ+1)Γ(1−ρ)Γ(λ+σ−ϑ−ρ) ×Sμ,ϑα,β(r;a;1/x)*F22[σ−ρ+1,ϑ−ρ+11−ρ,λ+σ−ϑ−ρ;x].$

## 3 Image Formulas Associated With Integral Transform

In this section, we establish certain theorems involving the results obtained in previous section associated with the integral transforms like, Beta transform, Laplace transform and Whittaker transform.

### 3.1 Beta Transform

The Beta transform of f(z) is defined as :

$B{f(z):a,b}=∫01za−1(1−z)b−1f(z)dz$

Theorem 3

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) > max[0, ℛ(σϑ)], then

$B{(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;t))(x):l,m}=Γ(m)xρ−σ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;x)*Ψ33[(ρ,1),(ρ+ϑ−σ,1),(l,1)(ρ−σ),(ρ+λ+ϑ,1),(l+m,1);1x].$

Proof

For convenience, we denote the left-hand side of the result (3.2) by . Using the definition of beta transform, the LHS of (3.1) becomes:

$ℬ=∫01zl−1(1−z)m−1(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;tz))(x)dz,$
further using (1.13) and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then
$ℬ=∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!(I0+λ,σ,ϑtn+ρ−1)(x)∫01zl+n−1(1−z)m−1dz$
applying the result (1.7), after simplification the above equation (3.4) reduced to
$ℬ=xρ−σ−1∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!×Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n)∫01zl+n−1(1−z)m−1dz,$
applying the definition of beta transform, the above equation (3.5) reduced to
$ℬ=xρ−σ−1∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n) ×Γ(l+n)Γ(m)Γ(l+m+n)$
after simplification, we have
$ℬ=xρ−σ−1∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!(ρ)n(ρ+ϑ−σ)n(ρ−σ)n(ρ+λ+ϑ)n ×(l)nΓ(m)(l+m)n$
further interpret the above equation with the view of of the function given in equation (3.7), we have
$ℬ=xρ−σ−1Γ(m)Sμ,ϑ,τ,ξ;ρ−σ,ρ+λ+ϑ,l+mα,β;ρ,ρ+ϑ−σ,l(r;a;p,q;x),$
employing the concept of the Hadamard product given in equation (1.13) in the above equation (3.8), required result is obtained.

Theorem 4

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0;ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], Then

$B{(Ix,∞λ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;z/t))(x):l,m}=Γ(m)xρ−σ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;x)*Ψ33[(ρ,1),(ϑ−ρ+1,1),(l,1)(ρ−σ,1),(λ+ϑ+ν,1),(l+m,1);1/x].$

Proof

The proof of this theorem is the same as that of Theorem 3.

### 3.2 Laplace Transform

The Beta transform of f(z) is defined as :

$L{f(z)}=∫0∞e−szf(z)dz$

Theorem 5

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0;ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) > max[0, ℛ(σϑ)], then

$L{zl−1(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;tz))(x)}=xρ−σ−1slSμ,ϑ,τ,ξα,β(r;a;p,q;xs)×Ψ32[(ρ,1),(ρ+ϑ−σ,1),(l,1)(ρ−σ,1),(ρ+λ+ϑ,1);xs].$

Proof

For convenience, we denote the left-hand side of the result (3.11) by . Then applying the Laplace, we have:

$ℒ=∫0∞e−szzl−1(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;tz))(x)dz$
further using (3.3) and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then
$ℒ=∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!(I0+λ,σ,ϑtn+ρ−1)(x)×∫0∞e−szzn+l−1dz$
applying the result (1.7), after simplification the above equation (3.13) reduced to
$ℒ=∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n)×Γ(n+l)sn+l,$
after simplification, we have
$ℬ=xρ−σ−1Γ(m)sl∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(xs)n(ρ)n(ρ+ϑ−σ)nn!(ρ−σ)n(ρ+λ+ϑ)n(l)n$
further interpret the above equation with the view of of the function given in equation (3.15), we have
$ℬ=xρ−σ−1Γ(m)slSμ,ϑ,τ,ξ;ρ−σ,ρ+λ+ϑ,α,β;ρ,ρ+ϑ−σ,l(r;a;p,q;xs),$
employing the concept of the Hadamard product given in equation (1.13) in the above equation (3.16), required result is obtained.

Theorem 6

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0;ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} 0, such that ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], Then

$L{zl−1(Ix,∞λ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;z/t))(x)}=xρ−σ−1slSμ,ϑ,τ,ξα,β(r;a;p,q;1sx)*Ψ32[(σ−ρ+1,1),(ϑ−ρ+1,1),(l,1)(1−ρ,1),(λ+σ+ϑ−ρ+1,1);1sx].$

Proof

The proof of this theorem would run parallel as those of Theorem 5.

### 3.3 Whittaker Transform

Theorem 7

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) > max[0, ℛ(σϑ)], then

$∫0∞zξ−1e−δz/2Wτ,ω(ηz){(I0,xλ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;tz))(x)}dz=xρ−σ−1ηξ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;xη)*Ψ43[(ρ,1),(ρ+ϑ−σ,1),(1/2+ω+ξ,1),(1/2−ω+ξ,1)(ρ−σ,1),(ρ+λ+ϑ,1),(1/2−τ+ξ,1);xη]$

Proof

For convenience, we denote the left-hand side of the result (3.25) by 𝒲. Then using the result from (2.3), after changing the order of integration and summation, we get:

$𝒲=xρ−σ−1∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n) ×∫0∞zn+ξ−1e−ηz/2Wτ,ω(ηz)dz,$
by substituting ηz = ς, (3.19) becomes:
$𝒲=xρ−σ−1∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μxnn!Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n) ×∫0∞ςn+ξ−1e−ς/2Wτ,ω(ς)dς.$

Now we use the following integral formula involving Whittaker function

$∫0∞tν−1e−t/2Wτ,ω(t)dt=Γ(1/2+ω+ν)Γ(1/2−ω+ν)Γ(1/2−τ+ν), (ℛ(ν±ω)>−12).$

Then we have

$𝒲=xρ−σ−1ηξ−1∑n=1∞2anβ(v)nBp,q(τ+n,ω−τ)B(τ,ω−τ)(anα+τ2)μn!Γ(ρ+n)Γ(ρ+ϑ−σ+n)Γ(ρ−σ+n)Γ(ρ+λ+ϑ+n) ×Γ(1/2+ω+ξ+n)Γ(1/2−ω+ξ+n)Γ(1/2−τ+ξ+n)(xη)n,$
after simplification, we have
$𝒲=xρ−σ−1ηξ−1∑n=1∞2anβ(v)nBp,q(τ+n,ω−τ)B(τ,ω−τ)(anα+τ2)μn!(ρ)n(ρ+ϑ−σ)n(ρ−σ)n(ρ+λ+ϑ)n ×(1/2+ω+ξ)n(1/2−ω+ξ)n(1/2−τ+ξ)n(xη)n,$
further interpret the above equation with the view of of the function given in equation (3.23), we have
$𝒲=xρ−σ−1ηξ−1Sμ,ϑ,τ,ξ;ρ−σ,ρ+λ+ϑ,1/2−τ+ξα,β;ρ,ρ+ϑ−σ,1/2+ω+ξ,1/2−ω+ξ(r;a;p,q;xη),$
employing the concept of the Hadamard product given in equation (1.13) in the above equation (3.24), required result is obtained.

Theorem 8

Let λ, σ, ϑ, ρ, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, such that ℛ(ρ) < 1 + min[ℛ(σ), ℛ(ϑ)], Then

$∫0∞zξ−1e−δz/2Wτ,ω(ηz){(Ix,∞λ,σ,ϑtρ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;z/t))(x)}dz=xρ−σ−1ηξ−1Sμ,ϑ,τ,ξα,β(r;a;p,q;xη)*Ψ43[(σ−ρ+1,1),(ϑ−ρ+1,1),(1/2+ω+ξ,1),(1/2−ω+ξ,1)(1−ρ,1),(ρ+λ+ϑ,1),(1/2−τ+ξ,1);xη]$

Proof

The proof of this theorem would run parallel as those of Theorem 7.

## 4 Fractional Kinetic Equations

The importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in physics, dynamical systems, control systems and engineering, to create the mathematical model of many physical phenomena. Especially, the kinetic equations describe the continuity of motion of substance. The extension and generalization of fractional kinetic equations involving many fractional operators were found in [72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85].

In view of the effectiveness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a further generalized form of the fractional kinetic equation involving generalized k-Mittag-Leffler function.

The fractional differential equation between rate of change of the reaction, the destruction rate and the production rate was established by Haubold and Mathai  given as follows:

$dNdt=−d(Nt)+p(Nt),$
where N = N(t) the rate of reaction, d = d(N) the rate of destruction, p = p(N) the rate of production and Nt denotes the function defined by Nt(t*) = N(tt*), t* > 0.

The special case of (4.1) for spatial fluctuations and inhomogeneities in N(t) the quantities are neglected, that is the equation

$dNdt=−ciNi(t),$
with the initial condition that Ni(t = 0) = N0 is the number density of the species i at time t = 0 and ci > 0. If we remove the index i and integrate the standard kinetic equation (4.2), we have
$N(t)−N0=−c0Dt−1N(t)$
where $Dt−10$ is the special case of the Riemann-Liouville integral operator $Dt−ν0$ defined as
$Dt−ν0f(t)=1Γ(ν)∫0t(t−s)ν−1f(s)ds, (t>0,R(ν)>0)$

The fractional generalization of the standard kinetic equation(4.3) is given by Haubold and Mathai  as follows:

$N(t)−N0=−cνDt−10N(t)$
and obtained the solution of (4.5) as follows:
$N(t)=N0∑k=0∞(−1)kΓ(νk+1)(ct)νk$

Further, (Saxena and Kalla ) considered the the following fractional kinetic equation:

$N(t)−N0f(t)=−cνDt−ν0N(t), (ℜ(v)>0),$
where N(t) denotes the number density of a given species at time t, N0 = N(0) is the number density of that species at time t = 0, c is a constant and f (0, ∞).

By applying the Laplace transform to (4.7) (see ),

$L{N(t);p}=N0F(p)1+cνp−ν=N0(∑n=0∞(−cν)np−νn)F(p), (n∈N0,|cp|<1)$
where the Laplace transform  is given by
$F(p)=L{N(t);p}=∫0∞e−ptf(t)dt, (ℛ(p)>0).$

## 5 Solution of generalized fractional kinetic equations

In this section, we investigated the solutions of the generalized fractional kinetic equations by considering generalized (p, q)-Mathieu Type Series

Theorem 9

If a > 0, d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ;min{ℛ(p), ℛ(q)} ≥ 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p,q;dνtν)=−aνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(aνtν)nn! ×Γ(νn+1)Eν,νn+1(−aνtν).$

Proof

Laplace transform of Riemann-Liouville fractional integral operator is given by (Erdelyi et.al. , Srivastava and Saxena ):

$L{Dt−ν0f(t);p}=p−νF(p)$
where F(p) is defined in (4.9). Now, applying Laplace transform on (5.1) gives,
$L{N(t);p}=N0L{Sμ,ϑ,τ,ξα,β(r;a;p,q;dνtν);p}−aνL{Dt−ν0N(t);p}$
$i.e. N(p)=N0(∫0∞e−pt∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dνtν)nn!dt)−aνp−νN(p)$
interchanging the order of integration and summation in (5.5), we have
$N(p)+aνp−νN(p)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dν)nn!∫0∞e−pttνndt$
$=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dν)nn!Γ(νn+1)pνn+1$
$N(p)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dν)nn! ×Γ(νn+1){p−(νn+1)∑l=0∞[−(pa)−ν]l}$

Taking Laplace inverse of (5.8), and by using

$L−1{p−ν;t}=tν−1Γ(ν),(R(ν)>0)$
we have,
$L−1{N(p)}=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(aν)nn! ×Γ(νn+1)L−1{∑l=0∞(−1)laνlp−[ν(n+l)+1]}$
$i.e. N(t)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dν)nn!Γ(νn+1){∑l=0∞(−1)laνltν(n+l)Γ(ν(n+l)+1)}$
$=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dνtν)nn!Γ(νn+1){∑l=0∞(−1)l(aνtν)lΓ(ν(n+l)+1)}$

The equation (5.12) can be written as

$N(t)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μΓ(νn+1)(aνtν)nn!Eν,νn+1(−aνtν).$

Theorem 10

If d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p,q;dνtν)=−dνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dνtν)nn! ×Γ(νn+1)Eν,νn+1(−dνtν).$

Theorem 11

If d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0; p, q ∈ ℂ; min{ℛ(p), ℛ(q)} ≥ 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p,q;t)=−dνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)nBp,q(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(tν)nn! ×Γ(νn+1)Eν,νn+1(−dνtν).$

Proof

The proof of the Theorem 10 and Theorem 11 are same as that of Theorem 9, so we would like to skip here.

### 5.1 Special cases

Here we introduce some special cases of our results established in this section.

Case 4.

If p = q, then Theorem 9, Theorem 10 and Theorem 11 reduces to

Corollary 7

If a > 0, d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0; p ∈ ℂ;ℛ(p) ≥ 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p;dνtν)=−aνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)nBp(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(aνtν)nn!Γ(νn+1)Eν,νn+1(−aνtν).$

Corollary 8

If d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0; p ∈ ℂ; ℛ(p) ≥ 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p;dνtν)=−dνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)nBp(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(dνtν)nn!Γ(νn+1)Eν,νn+1(−dνtν).$

Corollary 9

If d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0; p ∈ ℂ;ℛ(p) ≥ 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p;t)=−dνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)nBp(τ+n,ξ−τ)B(τ,ξ−τ)(anα+r2)μ(tν)nn!Γ(νn+1)Eν,νn+1(−dνtν).$

Case 5.

If p = q = 0, then Theorem 9, Theorem 10 and Theorem 11 reduces to

Corollary 10

If a > 0, d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;dνtν)=−aνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)n(anα+r2)μ(ξ)n(aνtν)nn!Γ(νn+1)Eν,νn+1(−aνtν).$

Corollary 11

If d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p;dνtν)=−dνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)n(anα+r2)μ(ξ)n(dνtν)nn!Γ(νn+1)Eν,νn+1(−dνtν).$

Corollary 12

If d > 0, ν > 0, r, α, β, ϑ > 0; ξ > τ > 0, then the solution of the fractional kinetic equation

$N(t)−N0Sμ,ϑ,τ,ξα,β(r;a;p;t)=−dνDt−ν0N(t)$
is given by the following formula
$N(t)=N0∑n=1∞2anβ(ϑ)n(anα+r2)μ(ξ)n(tν)nn!Γ(νn+1)Eν,νn+1(−dνtν).$

## 6 Conclusion

In the present work, fractional integral formulae involving (p, q)-Mathieu Type series has established. The image formulae of our findings by employing integral transform has been also introduced. Further in this work we gave the solution of fractional kinetic equation in terms of Mittag-Leffler function. All the results are general in nature and give numerous results as their special cases.

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É.L. Mathieu, Traité de Physique Mathé matique. VI–VII, Theory de l’Elasticite desCorps, (Part 2), Gauthier-Villars, Paris, 1980.

• 

K. Schroder, Das problem der eingespannten rechteckigen elastischen platte i.die biharmonische randwertaufgabe furdas rechteck, Math. Anal. 121 (1949) 247–326.

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• Export Citation
• 

P. Diananda, Some inequalities related to an inequality of mathieu, Math. Ann. 250 (1980) 95–98.

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• Export Citation
• 

G. V. M. ć, T. K. P.ány, New integral forms of generalized mathieu series and related applications, Applicable Analysis and Discrete Mathematics 7 (1) (2013) 180–192.

• Crossref
• Export Citation
• 

H. M. Srivastava, K. Mehrez,Ž. Tomovski, New inequalities for some generalized Mathieu type series and the Riemann Zeta function, Journal of Mathematical Inequalities 12 (1) (2018) 163–174.

• 

Ž. Tomovski, K. Trencevski, On an open problem of Bai-Ni Guo and Feng Qi, J. Inequal. Pure Appl. Math. 4 (2) (2003) 1–7.

• 

P. Cerone, C. T. Lenard, On integral forms of generalized Mathieu series, J. Inequal. Pure Appl. Math. 4 (5) (2003) 1–11.

• 

H. M. Srivastava,Ž. Tomovski, Some problems and solutions involving Mathieu’s series and its generalizations, JIPAM 5 (2) (2004) Article 45.

• 

H. M. Srivastava, R. K. Parmar, P. Chopra, A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions, Axioms 1 (2012) 238–258.

• Crossref
• Export Citation
• 

M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometric and conuent hypergeometric functions, Appl. Math. Comput. 159 (2) (2004) 589–602.

• 

K. Mehrez, Z. Tomovski, On a new (p,q)-Mathieu-type power series and its applications, Applicable Analysis and Discrete Mathematics 13 (1) (2019) 309–324. URL https://www.jstor.org/stable/26614261

• Crossref
• Export Citation
• 

Z. Tomovski, K. Mehrez, Some families of generalized Mathieu-type power series associated probability distributions and related functional inequalities involving complete monotonicity and log-convexity, Math. Inequal. Appl. 20 (2017) 973–986.

• 

V. Kiryakova, On two saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9 (2006) 159–176.

• 

T. Pohlen, The Hadamard Product and Universal Power Series: Ph.D. Thesis, Universitat Trier, Trier, Germany, 2009.

• 

H. Srivastava, R. Agarwal, S. Jain, Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions, Math. Method Appl. Sci. 40 (2017) 255–273.

• Crossref
• Export Citation
• 

H. Srivastava, R. Agarwal, S. Jain, A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas, Filomat 31 (2017) 125–140.

• Crossref
• Export Citation
• 

I. N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill, New Delhi, 1979.

• 

J. Choi, D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun. 20 (2015) 113–123.

• 

V. Chaurasia, S. C. Pandey, On the new computable solution of the generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions, Astrophys. Space Sci. 317 (2008) 213–219.

• Crossref
• Export Citation
• 

A. Chouhan, S. Sarswat, On solution of generalized kinetic equation of fractional order, Int. J. Math. Sci. Appl. 2 (2) (2012) 813–818.

• 

A. Chouhan, S. Purohit, S. Saraswat, An alternative method for solving generalized differential equations of fractional order, Kragujevac J. Math. 37 (2) (2013) 299–306.

• 

V. Gupta, B. Sharma, On the solutions of generalized fractional kinetic equations, Appl. Math. Sci. 5 (19) (2011) 899–910.

• 

A. Gupta, C. Parihar, On solutions of generalized kinetic equations of fractional order, Bol. Soc. Paran. Mat. 32 (1) (2014) 181–189.

• Crossref
• Export Citation
• 

H. Haubold, A. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci. 327 (2000) 53–63.

• 

D. Kumar, S. Purohit, A. Secer, A. Atangana, On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind, Mathematical Problems in Engineering (2015) 7. URL http://dx.doi.org/10.1155/2015/289387

• 

R. Saxena, A. Mathai, H. Haubold, On fractional kinetic equations, Astrophys. Space Sci. 282 (2002) 281–28.

• Crossref
• Export Citation
• 

R. K. Saxena, A. M. Mathai, H. J. Haubold, On generalized fractional kinetic equations, Physica A 344 (2004) 657–664.

• Crossref
• Export Citation
• 

R. K. Saxena, A. M. Mathai, Haubold, Solution of generalized fractional reaction-diffusion equations, Astrophys. Space Sci. 305 (2006) 305–313.

• Crossref
• Export Citation
• 

R. K. Saxena, S. L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput. 199 (2008) 504–511.

• 

A. Saichev, M. Zaslavsky, Fractional kinetic equations: solutions and applications, Caos 7 (1997) 753–764.

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G. M. Zaslavsky, Fractional kinetic equation for hamiltonian chaos, Physica D 76 (1994) 110–122.

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M. R. Spiegel, Theory and Problems of Laplace Transforms, Schaums Outline Series. McGraw-Hill, New York, 1965.

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A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, In: Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York-Toronto-London, 1954.

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H. M. Srivastava, R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001) 1–52.

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R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore-New Jersey-Hong Kong, 2000.

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R. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng. 32 (1) (2004) 1–104.

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H. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model. 45 (2017) 192–204.

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D. Benson, M. Meerschaert, J. Revielle, Fractional calculus in hydrologicmodeling: a numerical perspective, Adv. Water Resour 51 (2013) 479–497.

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M. Abdelkawy, M. Zaky, A. Bhrawy, D. Baleanu, Numerical Simulation Of Time Variable Fractional Order Mobile-Immobile Advection-Dispersion Model, Rom. Rep. Phys. 67 (3) (2015) 773–791.

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J. Zhao, L. Zheng, X. Chen, X. Zhang, F. Liu, Unsteady marangoni convection heat transfer of fractional maxwell fluid with cattaneo heat flux, Appl. Math. Model. 44 (2017) 497–507.

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B. Moghaddam, J. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Comput. Math. Appl. 73 (6) (2017) 1262–1269.

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C. Sin, L. Zheng, J. Sin, F. Liu, L. Liu, Unsteady flow of viscoelastic fluid with the fractional K-BKZ model between two parallel plates, Appl. Math. Model. 47 (2017) 114–127.

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A. Razminia, D. Baleanu, V. Majd, Conditional optimization problems: fractional order case, J. Optim. Theory App. 156 (1) (2013) 45–55.

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I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198, Academic Press, New York, London, Sydney, Tokyo and Toronto, 1999.

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D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: models and numerical methods, N. Jersey, London, Singapore: World Scientific, Berlin, 2012.

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L. Huang, D. Baleanu, G. Wu, S. Zeng, A new application of the fractional logistic map, Rom J Phys. 61 (7–8) (2016) 1172–1179.

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D. Baleanu, S. D. Purohit, J. C. Prajapati, Integral inequalities involving generalized Erdélyi-Kober fractional integral operators, Open Mathematics 14 (1) (2016) 89–99. doi:10.1515/math-2016-0007.

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R. Nigmatullin, D. Baleanu, New relationships connecting a class of fractal objects and fractional integrals in space, Fractional Calculus and Applied Analysis 16 (4) (2013) 911–936.

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P. Agarwal, M. Chand, G. Singh, Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Engineering journal 55 (4) (2016) 3053–3059.

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P. Agarwal, S. K. Ntouyas, S. Jain, M. Chand, G. Singh, Fractional kinetic equations involving generalized k-Bessel function via Sumudu transform, Alexandria Engineering journal doi:10.1016/j.aej.2017.03.046.

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Z. Hammouch, T. Mekkaoui, P. Agarwal, Optical solitons for the calogero-bogoyavlenskii-schiff equation in (2 + 1) dimensions with time-fractional conformable derivative, The European Physical Journal Plus 133:248. doi:https://doi.org/10.1140/epjp/i2018-12096-8.

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M. Chand, Z. Hammouch, J. K. Asamoah, D. Baleanu, Certain fractional integrals and solutions of fractional kinetic equations involving the product of s-function, In: Ta? K., Baleanu D., Machado J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity 24 (2019) 213–244. doi:https://doi.org/10.1007/978-3-319-90972-1_14.

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M. Chand, P. Agarwal, Z. Hammouch, Certain sequences involving product of k-Bessel function, International Journal of Applied and Computational Mathematics 4:101. doi:https://doi.org/10.1007/s40819-018-0532-8.

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D. Kumar, J. Singh, D. Baleanu, Modified kawahara equation within a fractional derivative with non-singular kernel, Thermal Science doi:10.2298/TSCI160826008K.

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D. Kumar, J. Singh, D. Baleanu, A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Therm. Sci. doi:10.2298/TSCI170129096K.

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D. Kumar, J. Singh, D. Baleanu, S. Baleanu, Analysis of regularized long-wave equation associated with a new fractional operator with mittag-leffler type kernel, Physica A 492 (2018) 155–167.

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D. Kumar, J. Singh, D. Baleanu, New numerical algorithm for fractional fitzhugh-nagumo equation arising in transmission of nerve impulses, Nonlinear Dynamics 91 (2018) 307–317.

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D. Kumar, R. Agarwal, J. Singh, A modified numerical scheme and convergence analysis for fractional model of lienard’s equation, Journal of Computational and Applied Mathematics doi:10.1016/j.cam.2017.03.011.

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M. Hajipou, A. Jajarmi, D. Baleanu, An efficient nonstandard finite difference scheme for a class of fractional chaotic systems, Journal of Computational and Nonlinear Dynamics 13 (2) (2017) 9 pages. doi:10.1115/1.4038444.

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D. Baleanu, A. Jajarmi, M. Hajipour, A new formulation of the fractional optimal control problems involving mittagleffler nonsingular kernel, Journal of Optimization Theory and Applications 175 (3) (2017) 718–737.

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D. Baleanu, A. Jajarmi, J. Asad, T. Blaszczyk, The motion of a bead sliding on a wire in fractional sense, Acta Physica Polonica A 131 (6) (2017) 1561–1564.

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A. Jajarmi, M. Hajipour, D. Baleanu, New aspects of the adaptive synchronization and hyperchaos suppression of a financial model, Chaos, Solitons and Fractals 99 (2017) 285–296.

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K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, USA, 1993.

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D. Baleanu, Z. B. Guvenc, J. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer Dordrecht Heidelberg, London, New York, 2010.

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L. Carlitz, Generating functions, Fibonacci Quart. 7 (1969) 359–393.

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P. Agarwal, Q. Al-Mdallal, Y. J. Cho, S. Jain, Fractional differential equations for the generalized Mittag-Leffler function, Advances in difference equations 58. doi:10.1186/s13662-018-1500-7.

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P. Agarwal, S. Jain, T. Mansour, Further extended caputo fractional derivative operator and its applications, Russian Journal of Mathematical physics 24 (4) (2017) 415–425.

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M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015) 73–85.

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A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction diffusion equation, Applied Mathematics and Computation 273 (2016) 948–956.

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A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel, theory and application to heat transfer model, Thermal Science 20 (2016) 763–769.

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A. McBride, Fractional powers of a class of ordinary differential operators, Proc. London Math. Soc. (III) 45 (1982) 519–546.

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S. Kalla, Integral operators involving Fox’s H-function I, Acta Mexicana Cienc. Tecn. 3 (1969) 117–122.

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S. Kalla, Integral operators involving Fox’s H-function II, Acta Mexicana Cienc. Tecn. 7 (1969) 72–79.

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S. Kalla, R. Saxena, Integral operators involving hypergeometric functions, Math. Z. 108 (1969) 231–234.

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S. Kalla, R. Saxena, Integral operators involving hypergeometric functions ii, Univ. Nac. Tucuman, Rev. Ser. A 24 (1974) 31–36.

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M. Saigo, A remark on integral operators involving the gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11(2) (1978) 135–143.

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M. Saigo, A certain boundary value problem for the Euler-Darboux equation II, Math. Japonica 25 (2) (1980) 211–220.

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M. Saigo, N. Maeda, More generalization of fractional calculus, Transform Methods and Special Functions, Bulgarian Acad. Sci., Sofia, Varna, Bulgaria, 1996.

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D. Baleanu, D. Kumar, S. Purohit, Generalized fractional integrals of product of two h-functions and a general class of polynomials, International Journal of Computer Mathematics doi:10.1080/00207160.2015.1045886.

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A. Kilbas, N. Sebastian, Generalized fractional integration of bessel function of the first kind, Int Transf Spec Funct 19 (2008) 869–883.

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• Export Citation
• 

É.L. Mathieu, Traité de Physique Mathé matique. VI–VII, Theory de l’Elasticite desCorps, (Part 2), Gauthier-Villars, Paris, 1980.

• 

K. Schroder, Das problem der eingespannten rechteckigen elastischen platte i.die biharmonische randwertaufgabe furdas rechteck, Math. Anal. 121 (1949) 247–326.

• Crossref
• Export Citation
• 

P. Diananda, Some inequalities related to an inequality of mathieu, Math. Ann. 250 (1980) 95–98.

• Crossref
• Export Citation
• 

G. V. M. ć, T. K. P.ány, New integral forms of generalized mathieu series and related applications, Applicable Analysis and Discrete Mathematics 7 (1) (2013) 180–192.

• Crossref
• Export Citation
• 

H. M. Srivastava, K. Mehrez,Ž. Tomovski, New inequalities for some generalized Mathieu type series and the Riemann Zeta function, Journal of Mathematical Inequalities 12 (1) (2018) 163–174.

• 

Ž. Tomovski, K. Trencevski, On an open problem of Bai-Ni Guo and Feng Qi, J. Inequal. Pure Appl. Math. 4 (2) (2003) 1–7.

• 

P. Cerone, C. T. Lenard, On integral forms of generalized Mathieu series, J. Inequal. Pure Appl. Math. 4 (5) (2003) 1–11.

• 

H. M. Srivastava,Ž. Tomovski, Some problems and solutions involving Mathieu’s series and its generalizations, JIPAM 5 (2) (2004) Article 45.

• 

H. M. Srivastava, R. K. Parmar, P. Chopra, A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions, Axioms 1 (2012) 238–258.

• Crossref
• Export Citation
• 

M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometric and conuent hypergeometric functions, Appl. Math. Comput. 159 (2) (2004) 589–602.

• 

K. Mehrez, Z. Tomovski, On a new (p,q)-Mathieu-type power series and its applications, Applicable Analysis and Discrete Mathematics 13 (1) (2019) 309–324. URL https://www.jstor.org/stable/26614261

• Crossref
• Export Citation
• 

Z. Tomovski, K. Mehrez, Some families of generalized Mathieu-type power series associated probability distributions and related functional inequalities involving complete monotonicity and log-convexity, Math. Inequal. Appl. 20 (2017) 973–986.

• 

V. Kiryakova, On two saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9 (2006) 159–176.

• 

T. Pohlen, The Hadamard Product and Universal Power Series: Ph.D. Thesis, Universitat Trier, Trier, Germany, 2009.

• 

H. Srivastava, R. Agarwal, S. Jain, Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions, Math. Method Appl. Sci. 40 (2017) 255–273.

• Crossref
• Export Citation
• 

H. Srivastava, R. Agarwal, S. Jain, A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas, Filomat 31 (2017) 125–140.

• Crossref
• Export Citation
• 

I. N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill, New Delhi, 1979.

• 

J. Choi, D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun. 20 (2015) 113–123.

• 

V. Chaurasia, S. C. Pandey, On the new computable solution of the generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions, Astrophys. Space Sci. 317 (2008) 213–219.

• Crossref
• Export Citation
• 

A. Chouhan, S. Sarswat, On solution of generalized kinetic equation of fractional order, Int. J. Math. Sci. Appl. 2 (2) (2012) 813–818.

• 

A. Chouhan, S. Purohit, S. Saraswat, An alternative method for solving generalized differential equations of fractional order, Kragujevac J. Math. 37 (2) (2013) 299–306.

• 

V. Gupta, B. Sharma, On the solutions of generalized fractional kinetic equations, Appl. Math. Sci. 5 (19) (2011) 899–910.

• 

A. Gupta, C. Parihar, On solutions of generalized kinetic equations of fractional order, Bol. Soc. Paran. Mat. 32 (1) (2014) 181–189.

• Crossref
• Export Citation
• 

H. Haubold, A. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci. 327 (2000) 53–63.

• 

D. Kumar, S. Purohit, A. Secer, A. Atangana, On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind, Mathematical Problems in Engineering (2015) 7. URL http://dx.doi.org/10.1155/2015/289387

• 

R. Saxena, A. Mathai, H. Haubold, On fractional kinetic equations, Astrophys. Space Sci. 282 (2002) 281–28.

• Crossref
• Export Citation
• 

R. K. Saxena, A. M. Mathai, H. J. Haubold, On generalized fractional kinetic equations, Physica A 344 (2004) 657–664.

• Crossref
• Export Citation
• 

R. K. Saxena, A. M. Mathai, Haubold, Solution of generalized fractional reaction-diffusion equations, Astrophys. Space Sci. 305 (2006) 305–313.

• Crossref
• Export Citation
• 

R. K. Saxena, S. L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput. 199 (2008) 504–511.

• 

A. Saichev, M. Zaslavsky, Fractional kinetic equations: solutions and applications, Caos 7 (1997) 753–764.

• 

G. M. Zaslavsky, Fractional kinetic equation for hamiltonian chaos, Physica D 76 (1994) 110–122.

• Crossref
• Export Citation
• 

M. R. Spiegel, Theory and Problems of Laplace Transforms, Schaums Outline Series. McGraw-Hill, New York, 1965.

• 

A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, In: Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York-Toronto-London, 1954.

• 

H. M. Srivastava, R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001) 1–52.

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