In this paper, our aim is to finding the solutions of the fractional kinetic equation related with the p,q-Mathieu-type series through the procedure of Sumudu and Laplace transforms. The outcomes of fractional kinetic equations in terms of the Mittag-Leffler function are presented.
1. Introduction and Preliminaries
Fractional calculus (FC) can be a valuable mathematical method for considering the integrals and derivatives in a fractional order. The fractional calculus has been evolved and utilized in numerous engineering and analysis areas. In several disparate sectors, along with applied research, material science, mathematical physics, chemistry, and architecture, the theory of fractional differential equations and their implementations has played a key role. The complex conditions program at a basic stem of differential equations, which illustrates the amount of modification of a star’s chemical composition with each configuration in terms of generation and annihilation reaction levels. The expansion and sweeping statement of fractional kinetic equations related with different special functions were established (see [1–11] for details). Nowadays, several scholars are developing a simplified structure of the fractional kinetic equation involving the Mathieu-type series to make the dynamic state extremely relevant and acceptable in a few astrophysical problems.
The first Mathieu series was explored by Mathieu in his book Elasticity of Solid Bodies [12], which is represented as an infinite series of the following form:(1)Sϑ=∑ℓ=1∞2ℓℓ2+ϑ22,ϑ>0.
An integral representation of (1) is defined as (see [13])(2)Sϑ=1ϑ∫0∞xsinϑxex−1dx.
A few curiously special cases and their solutions deal with integral representations, their another account with a fractional image power characterized by Cerone and Lenard ([14], p. 2, Equation (16)), Milovanovic and Pogány ([15], p. 181):(3)Sμϑ=∑ℓ=1∞2ℓℓ2+ϑ2μ+1,μ>0,ϑ>0.
Inspired fundamentally by the works of Cerone and Lenard [14] (see also [16]), Srivastava and Tomovski established a generalized Mathieu series family in [17].(4)Sμα,βϑ,a=Sμα,βϑ,aℓℓ=1∞−∑ℓ=1∞2aℓβaℓα+ϑ2μ,α,β,μ>0,ϑ>0,where it is tacitly presumed that the positive sequence a=aℓ=a1,a2,… such that limℓ⟶∞aℓ=∞ and so, taken that, perhaps, the (4) infinite series converges, which is to say the preceding auxiliary series ∑ℓ=1∞1/aℓα,μ−β is convergent.
Within the continuation, in terms of the following power series, Tomovski and Mehrez [18] introduced a more generalized form of the series (4):(5)Sμ,vα,βϑ,a;z=Sμ,vα,βϑ,aℓℓ=1∞;z=∑ℓ=1∞2aℓβvℓaℓα+ϑ2μzℓℓ!,α,β,ϑ,a,μ>0,z≤1,where(6)vℓ=1,ℓ=0;v∈ℂ\0,vv+1…v+ℓ−1,v=ℓ∈ℕ;v∈ℂ.
Quite recently, Mehrez and Tomovski [19] found the more conventional version of the so-called p,q-Mathieu power series in the following version:(7)Sμ,v,τ,ωα,βϑ,a;p,q;z=∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τzℓaℓα+ϑ2μℬτ,ω−τℓ!,ϑ,a,v,μ,τ,ω,α,β∈ℜ+,z≤1,ℜq≥0,ℜp>0,where ℬσ,ρ;p,q is the p,q-extended beta function provided by Choi et al. [20],(8)ℬσ,ρ;p,q=ℬp,qσ,ρ=∫01xσ−11−xρ−1e−p/x−q/1−xdx,when minℜσ,ℜρ>0;minℜp,ℜq≥0. This p,q-Mathieu-type series contains, as limited cases, different aspects of the Mathieu-type series:
When p=q=0, then the generalized Mathieu-type power series is defined by(9)Sμ,v,τ,ωα,βϑ,a;z=Sμ,v,τ,ωα,βϑ;a;0,0;z=∑ℓ=1∞2aℓβvℓτℓzℓaℓα+ϑ2μωℓℓ!,ϑ,α,β,μ,a,v,τ,ω∈ℜ+,z≤1.
By setting τ=ω in (9), we obtain ([21], Equation (5), p. 974)(10)Sμ,vα,βϑ,a;z=∑ℓ=1∞2aℓβvℓzℓakα+ϑ2μℓ!,ϑ,α,β,μ,v,a∈ℜ+,z≤1.
Furthermore, in the special cases when v=z=1, we get the generalized Mathieu series (4).
2. Fractional Kinetic Equations
As of late, a startling interest has emerged in learning regarding the solution of fractional kinetic equations owing to their importance in astronomy and scientific material science. The kinetic equations of fractional order have been effectively utilized to decide certain physical wonders overseeing dissemination in permeable media and response and unwinding forms in complex frameworks. Subsequently, a large body of research into the application of these equations has been spread by publishing.
Haubold and Mathai [22] study the fractional differential equation between reaction rate ℑ=ℑt, destruction rate d=dℑ, and production rate p=pℑ as follows:(11)dℑdt=−dℑt+pℑt,where ℑt is the function represented by ℑtt∗=ℑt−t∗,t∗>0. Undermining the inhomogeneity in the number ℑt, (11) is given a special case as follows:(12)dℑidt=−ciℑit,where the primary condition ℑit=0=ℑ0 is the number of density of species i at time t=0. Neglecting index i and integrating, (12) becomes(13)ℑt−ℑ0=cD0t−1ℑt.
We keep in mind that the standard fractional integral operator is D0t−1.
In fact, the fractional sweep argument of the standard kinetic equation (13) is defined by Haubold and Mathai [22] inside the equation:(14)ℑt−ℑ0=cςD0t−ςℑt,where D0t−ς is the most common Riemann–Liouville (R–L) fractional integral operator. More details of R–L in [23] are defined as(15)D0t−ςfx=1Γς∫0tt−uς−1fudu,x>0,ℜς>0.
The solution for (14), fractional equation, is given by (see [22])(16)ℑt=ℑ0∑k=0∞−1kΓςk+1ctςk.
Further, Saxena et al. [24, 25] explored the generalized type solution of (14) in terms of a generalized Mittag-Leffler function (see [26–28] for details),(17)Eς,ℓz=∑k=0∞zkΓςk+ℓς,ℓ,z∈ℂ;ℜς>0;ℜℓ>0,and the function Eς,ℓ is now called the two-parameter Mittag-Leffler function (also known as the Wiman function). The extension of (17) is called three-parameter Mittag-Leffler function (or else Prabhakar’s function), and Garra and Garrappa [29] introduced this in terms of a series representation.(18)Eς,ℓεz=∑k=0∞εkςk+ℓk!zk,ς,ℓ,ε,z∈ℂ,ℜς>0.
For the effectiveness and significance of the fractional kinetic equations in specific astronomy issues, the authors establish a modern and encourage generalized form of the fractional kinetic equation pertaining to the p,q-Mathieu-type power series utilizing the strategy of Laplace transform. Furthermore, the findings obtained here are very capable of generating a large range of established and (presumably) novel outcomes.
3. Solution of Generalized Fractional Kinetic Equations
In this section, we obtain a fractional kinetic equation pertaining to the p,q-Mathieu-type power series using the Laplace transforms technique.
We recall the Laplace transform of fx as defined by Sneddon [30]:(19)ℱs=Lfx;s=∫0∞e−sxfxdx,ℜs>0.
Theorem 1.
If γ,ς,d>0; d≠γ; a,ϑ,α,β,v,μ,τ,ω∈ℜ+,ℜp>0,ℜq≥0,γt≤1, then the equation solution,(20)ℑt−ℑ0Sμ,v,τ,ωα,βϑ,a;p,q;γt=−dςD0t−ςℑt,holds the formula(21)ℑt=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγtℓaℓα+ϑ2μℬτ,ω−τEς,ℓ+1−dςtς.
Proof.
The Laplace transform of the R–L fractional integral operator given by the authors in [23] is as follows:(22)ℒD0t−ςft;s=s−ςℑs,where in (19), ℑs is defined. Then, using the Laplace transform on both sides of the (20) and using (7) and (22) order, we get(23)Lft;s=ℑ0LSμ,v,τ,ωα,βϑ,a;p,q;γt;s−dςLD0t−ςft;s,ℑs=ℑ0∫0∞e−st∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγtℓaℓα+ϑ2μℬτ,ω−τℓ!dt−dςs−ςℑs,(24)ℑs+dςs−ςℑs=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγℓaℓα+ϑ2μℬτ,ω−τℓ!∫0∞e−sttℓdt.
Under these conditions, calculating the integral in (24) term by term and using Ltℓ;s=s−ℓ+1Γℓ+1, we have(25)ℑs1+dsς=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγℓaℓα+ϑ2μℬτ,ω−τℓ!Γℓ+1sℓ+1.
Using 1+d/Sς−1 geometric series expansion for d<s, we have(26)ℑs=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγℓaℓα+ϑ2μℬτ,ω−τsℓ+1∑m=0∞1mm!−dSςm.
Taking inverse Laplace transform on both sides of (26) and using L−1s−ς;t=tς−1/Γς for ℜς>0, we obtain(27)ℑt=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγtℓaℓα+ϑ2μℬτ,ω−τ∑m=0∞−1mdtmςΓmς+ℓ+1.
Interpreting the result in (27) in the view of (17), the necessary result is (21).
Corollary 1.
If γ,d,ς>0; d≠γ; a,ϑ,α,β,μ,τ,v,ω∈ℜ+,γt≤1, then equation(28)ℑt−ℑ0Sμ,v,τ,ωα,βϑ,a;γt=−dςD0t−ςℑt,has the solution(29)ℑt=ℑ0∑ℓ=1∞2aℓβvℓτℓγtℓaℓα+ϑ2μωℓEς,ℓ+1−dςtς.
Corollary 2.
If γ,d,ς>0; γ≠d; ϑ,a,α,β,μ,v∈ℜ+,γt≤1, then equation(30)ℑt−ℑ0Sμ,vα,βϑ,a;γt=−dςD0t−ςℑt,has the solution(31)ℑt=ℑ0∑ℓ=1∞2aℓβvℓγtℓaℓα+ϑ2μEς,ℓ+1−dςtς.
Theorem 2.
If d,η,ς>0; ϑ,α,β,μ,v,τ,ω,a∈ℜ+,ℜp>0,ℜq≥0,ηt≤1, then equation(32)ℑt−ℑ0Sμ,v,τ,ωα,βϑ,a;p,q;ηtς=−∑k=1nnkd−ςkD0t−ςkℑt,has the solution(33)ℑt=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τΓℓς+1ηtςℓaℓα+ϑ2μℬτ,ω−τℓ!Eς,ℓς+1n−dςtς.
Proof.
Now, using the Laplace transform on both sides of (32) and using (7) and (22) lead to(34)Lℑt;s=ℑ0LSμ,v,τ,ωα,βϑ,a;p,q;ηtς;s−L∑k=1nnkd−ςkD0t−ςkℑt;s,which upon solving for ℑs yields(35)ℑs=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τηℓaℓα+ϑ2μℬτ,ω−τℓ!Γℓς+1sℓς+11+dsς−n.
Employing the binomial formula 1−x−δ=∑k=0∞δk/k!xk, which converges for x<1, we have(36)ℑs=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τΓℓς+1ηℓaℓα+ϑ2μℬτ,ω−τℓ!sℓς+1∑k=0∞nkk!−dςsςk.
Taking inverse Laplace transform on both sides of (36), we obtain(37)ℑt=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τΓℓς+1ηtςℓaℓα+ϑ2μℬτ,ω−τℓ!×∑k=0∞−1knkdtςkΓςk+ℓς+1.
Interpreting the result (37) in the view of (18), we get the necessary result (33).
Corollary 3.
If d,ς,η>0; a,ϑ,α,β,μ,v,τ,ω∈ℜ+,ηt≤1, then equation(38)ℑt−ℑ0Sμ,v,τ,ωα,βϑ,a;ηtς=−∑k=1nnkd−ςkD0t−ςkℑt,is given by(39)ℑt=ℑ0∑ℓ=1∞2aℓβvℓτℓΓℓς+1ηtςℓaℓα+ϑ2μωℓℓ!Eς,ℓς+1n−dςtς.
Corollary 4.
If d,ς,η>0; a,ϑ,α,β,μ,v∈ℜ+,ηt≤1, then equation(40)ℑt−ℑ0Sμ,vα,βϑ,a;ηtς=−∑k=1nnkd−ςkD0t−ςkℑt,gives a solution(41)ℑt=ℑ0∑ℓ=1∞2aℓβvℓΓℓς+1ηtςℓaℓα+ϑ2μℓ!Eς,ℓς+1n−dςtς.
Theorem 3.
If ς,d>0; a,ϑ,α,β,v,μ,ω,τ∈ℜ+,ℜp>0,ℜq≥0,t≤1, then the solution of the equation(42)ℑt−ℑ0Sμ,v,τ,ωα,βϑ,a;p,q;dςtς=−dςD0t−ςℑt,holds the formula(43)ℑt=ℑ0∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τΓςℓ+1dtℓaℓα+ϑ2μℬτ,ω−τℓ!Eς,ςℓ+1−dςtς.
Proof.
The thorough proof of Theorem 3 is similar to that of Theorem 1, so we omit the details.
Corollary 5.
If ς,d>0; a,ϑ,α,β,v,μ,ω,τ∈ℜ+,t≤1, then equation(44)ℑt−ℑ0Sμ,v,τ,ωα,βϑ,a;dςtς=−dςD0t−ςℑt,has the solution(45)ℑt=ℑ0∑ℓ=1∞2aℓβvℓτℓΓςℓ+1dtℓaℓα+ϑ2μωℓℓ!Eς,ςℓ+1−dςtς.
Corollary 6.
If d,ς>0;α,β,μ,v,ϑ,a∈ℜ+,t≤1, then equation(46)ℑt−ℑ0Sμ,vα,βϑ,a;dςtς=−dςD0t−ςℑt,has the solution(47)ℑt=ℑ0∑ℓ=1∞2aℓβvℓΓςℓ+1dtℓaℓα+ϑ2μℓ!Eς,ςℓ+1−dςtς.
4. Examples
Details of the Mathieu-type series and their implementations can be contained in the monographs by different authors [21, 31]. The integral transform, named the Sumudu Transform, promotes the method of fathoming differential and integral equations inside time. It turns out that the Sumudu transform has exceptional and useful properties and that it is important in knowing the issues of research and regulation of the management of active situations. Here, we use the concept of the Sumudu transform given by Watugala [32] as follows:(48)Gω=Sfz;ω=∫0∞e−zfωzdz;ω∈−ε1,ε2,where the exponentially bound function class in the T is as follows:(49)T=fz∃M,ε1,ε2>0,fz<Mez/εj,t∈−1j×0,∞.
In addition, the Sumudu transform given in (48) can be calculated directly from the Fourier integral. The Sumudu transformation tends to be the theoretical dual transformation of Laplace. It is interesting to equate the Sumudu transform (48) with the well-known Laplace transform (see, for example, [33]):(50)ℱp=Lfz=∫0∞e−pzfzdz,ℜp>0.
Equation (15) can be described in the following form by using the Sumudu transformation theorem [34–36]:(51)SD0z−ςfz=Szς−1Γς⋅Sfz=uςGu.
It is simple to see that the function fz=zδ by using the Sumudu transform is given as(52)Sfz=∫0∞e−zyzδdz=uδΓ1+δ,ℜδ>−1.
The interested readers should search [37–41] for more subtle elements almost transforming the Sumudu and its properties as opposed to the Laplace transform.
Because of the significance of the abovementioned observation, in this section, we evaluate the solutions of generalized fractional kinetic equations by applying the Sumudu transform using the same analytical method as in Theorems 1, 2, and 3, presented in examples 1, 2, and 3.
Example 1.
If γ,d,ς,δ>0; d≠γ; a,ϑ,α,β,v,μ,τ,ω∈ℜ+,ℜp>0,ℜq≥0,γt≤1, then the solution of the equation(53)ℑt−ℑ0tδ−1Sμ,v,τ,ωα,βϑ,a;p,q;γtς=−dςD0t−ςℑt,holds the formula(54)ℑt=ℑ0tδ−2∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τγtςℓaℓα+ϑ2μℬτ,ω−τEς,ςℓ+δ−1−dςtς.
Example 2.
If d,η,ς>0; ϑ,α,β,μ,v,τ,ω,a∈ℜ+,ℜp>0,ℜq≥0,ηt≤1, then equation(55)ℑt−ℑ0tδ−1Sμ,v,τ,ωα,βϑ,a;p,q;ηtς=−∑k=1nnkd−ςkD0t−ςkℑt,has the solution(56)ℑt=ℑ0tδ−2∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τΓℓς+1ηtςℓaℓα+ϑ2μℬτ,ω−τℓ!Eς,ℓς+δ−1n−dςtς.
Example 3.
If ς,d>0, a,ϑ,α,β,v,μ,ω,τ∈ℜ+,ℜp>0,ℜq≥0,t≤1, then the solution of the equation(57)ℑt−ℑ0tδ−1Sμ,v,τ,ωα,βϑ,a;p,q;dςtς=−dςD0t−ςℑt,holds the formula(58)ℑt=ℑ0tδ−2∑ℓ=1∞2aℓβvℓℬp,qτ+ℓ,ω−τΓςℓ+1dtℓaℓα+ϑ2μℬτ,ω−τℓ!Eς,ςℓ+δ−1−dςtς.
5. Concluding Remarks
It is not troublesome to get a few who encourage closely fractional kinetic equations and their solutions as those displayed here by Theorem 1, 2, and 3 and its Corollaries. It is popular to support that a variety of other special cases of our results can also be obtained as shown in Section 4, if we take p=q=0,τ=ω, and v=z=1, and we can obtain nine different findings. We leave those to the interested reader as an exercise. Moreover, if we set p=q=0 and τ=ω in our main results, then we arrive at [4]. In this article, we considered the traditional kinetic equation as a recent fractional generalization and proposed their solutions. Besides, in view of near connections of the p,q-Mathieu-type series and p,q-Mittag-Leffler with other special functions, it does not seem difficult to construct different known and unused fractional kinetic equations. Hence, the examined which comes about in this paper would, at once, grant numerous outcomes about including assorted special functions happening within the issues of astronomy, scientific mathematical science, and engineering.
Data Availability
The data can be obtained from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
ChoiJ.KumarD.Solutions of generalized fractional kinetic equations involving Aleph functions2015201113123HabenomH.SutharD. L.GebeyehuM.Application of Laplace transform on fractional kinetic equation pertaining to the generalized Galué type Struve function201920198507403910.1155/2019/50740392-s2.0-85067788231KachhiaK. B.PrajapatiJ. C.On generalized fractional kinetic equations involving generalized Lommel-Wright functions20165532953295710.1016/j.aej.2016.04.0382-s2.0-84969247990KhanO.KhanN.BaleanuD.NisarK. S.Computable solution of fractional kinetic equations using Mathieu-type series20192019123410.1186/s13662-019-2167-42-s2.0-85067351475KiryakovaV.A brief story about the operators of the generalized fractional calculus2008112203220KumarD.ChoiJ.SrivastavaH. M.Solution of a general family of fractional kinetic equations associated with the mittag-leffler function2018233455471KumarD.PurohitS. D.SecerA.AtanganaA.On generalized fractional kinetic equations involving generalized Bessel function of the first kind20152015728938710.1155/2015/2893872-s2.0-84924257414NisarK. S.PurohitS. D.MondalS. R.Generalized fractional kinetic equations involving generalized Struve function of the first kind201628216717110.1016/j.jksus.2015.08.0052-s2.0-84940974972ShafeeA.GalueL.KallaS.Solution of certain fractional kinetic equations201627543745310.12732/ijam.v27i5.2SutharD. L.HabenomH.NisarK. S.Solutions of fractional kinetic equation and the generalized Galué type Struve function20192271167118410.1080/09720502.2019.1706841SutharD. L.KumarD.HabenomH.Solutions of fractional kinetic equation associated with the generalized multiindex Bessel function via Laplace-transform2019201910.1007/s12591-019-00504-9MathieuE. L.1980Paris, FranceGauthier-VillarsPogányT. K.SrivastavaH. M.TomovskiŽoSome families of Mathieu a-series and alternating Mathieu a-series200617316910810.1016/j.amc.2005.02.0442-s2.0-32244445657CeroneP.LenardC. T.On integral forms of generalized Mathieu series200345111MilovanovicG.PogányT.New integral forms of generalized Mathieu series and related applications20137118019210.2298/aadm121227028m2-s2.0-84874952112NisarK. S.SutharD. L.BohraM.PurohitS. D.Generalized fractional integral operators pertaining to the product of Srivastava’s polynomials and generalized Mathieu series20197220610.3390/math70202062-s2.0-85069972851SrivastavaH. M.TomovskiŽ.Some problems and solutions involving Mathieu’s series and its generalizations20045213TomovskiŽoMehrezK.Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity2017220497398610.7153/mia-2017-20-612-s2.0-85031704106MehrezK.TomovskiZ.On a new (p, q)-Mathieu-type power series and its applications201913130932410.2298/aadm190427005m2-s2.0-85065520894ChoiJ.RathieA. K.ParmarR. K.Extension of extended beta, hypergeometric and confluent hypergeometric functions201436233936710.5831/hmj.2014.36.2.357TomovskiŽoMehrezK.Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity201720497398610.7153/mia-2017-20-612-s2.0-85031704106HauboldH. J.MathaiA. M.The fractional kinetic equation and thermonuclear functions20002731/4536310.1023/a:1002695807970SrivastavaH. M.SaxenaR. K.Operators of fractional integration and their applications2001118115210.1016/s0096-3003(99)00208-82-s2.0-0012072325SaxenaR. K.MathaiA. M.HauboldH. J.On fractional kinetic equations2002282128128710.1023/a:10211751089642-s2.0-0141570688SaxenaR. K.MathaiA. M.HauboldH. J.On generalized fractional kinetic equations20043443-465766410.1016/j.physa.2004.06.0482-s2.0-6344261966MehrezK.SitnikS. M.Functional inequalities for the mittag-leffler functions2017721-270371410.1007/s00025-017-0664-x2-s2.0-85014169246MehrezK.SitnikS. M.Turán type inequalities for classical and generalized mittag-leffler functions201844452154110.1007/s10476-018-0404-92-s2.0-85052084796WimanA.Uber de fundamental theorie der funktionen Eα (x)190529119120110.1007/bf024032022-s2.0-0001799540GarraR.GarrappaR.The Prabhakar or three parameter mittag-leffler function: theory and application20185631432910.1016/j.cnsns.2017.08.0182-s2.0-85028298587SneddonI. N.1979New Delhi, IndiaTata Mc-Graw HillTomovskiŽ.PogányT. K.Integral expressions for Mathieu-type power series and for the Butzer-Flocke-Hauss Ω function201114462363410.2478/s13540-011-0036-22-s2.0-84856320966WatugalaG. K.Sumudu transform: a new integral transform to solve differential equations and control engineering problems1993241354310.1080/00207399302401052-s2.0-0006219608SpiegelM. R.1986New York, NY, USAMcGraw HillSchaum’s Outline SeriesAsiruM. A.Sumudu transform and the solution of integral equations of convolution type200132690691010.1080/0020739013171478702-s2.0-77950958585BelgacemF. B. M.KaraballiA. A.Sumudu transform fundamental properties investigations and applications20062006239108310.1155/JAMSA/2006/910832-s2.0-33749525486BelgacemF. B. M.KaraballiA. A.KallaS. L.Analytical investigations of the Sumudu transform and applications to integral production equations20032003310311810.1155/s1024123x032070182-s2.0-2342468673BertrandJ.BertrandP.OvarlezJ. P.PoularkasA. D.1996Boca Raton, FL, USACRC PressKilicmanA.EltayebH.Some remarks on the Sumudu and Laplace transforms and applications to differential equations201220121359151710.5402/2012/591517KilicmanA.EltayebH.AtanK. A. M.A note on the comparison between Laplace and Sumudu transforms2011371131141KilicmanA.GadainH. E.On the applications of Laplace and Sumudu transforms2010347584886210.1016/j.jfranklin.2010.03.0082-s2.0-77955228730KimH.The intrinsic structure and properties of Laplace-typed integral transforms201720178176272910.1155/2017/17627292-s2.0-85021654183