# Application of the Laplace Transform to a New Form of Fractional Kinetic Equation Involving the Composition of the Galúe Struve Function and the Mittag–Leffler Function

Authors: Komal Prasad Sharma; Alok Bhargava; D. L. Suthar
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The great importance of the fractional kinetic equations (FKEs) can be seen in the very recent research work of proposing such new equations and finding their solutions by different analytical and numerical methods. The paramount purpose of the present work is to proffer and study new FKEs involving the composition of the generalized Galúe Struve function of first kind and k-Mittag–Leffler function and apply a very influential effective analytical approach based on Laplace transform to find their analytical solutions. The obtained solutions have been presented with some real values and the simulation done via MATLAB. Furthermore, the numerical and graphical interpretations are also mentioned to illustrate the main results. Some special cases are to be taken under consideration to get the results in simpler form.


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## Body

## 1. Introduction

In ordinary calculus, Riemann–Liouville, Euler, and Fourier have done a lot of contribution for discovering beneficial results in the eighteenth century. Several researchers have made a great contribution to the area of fractional calculus (FC) at that time. Due to their applications, scholars and scientists have shown more interest on analysed noninteger order of differential and integral calculus.The differential equations of fractional order are proved to be of more importance in several areas of physical science and engineering in the past few decades. These equations have several significant applications in various areas of sciences and engineering. Kinetic equations of fractional order attracted the researchers to work on it due to their wide applications in science and applied mathematics. The aim of this work is to study the generalized form of the FKE involving generalized form Struve function of the first kind and k-Mittag–Leffler function. Many authors work on fractional differential equations and on these FKEs to extend their utilisation in various areas of applied mathematical sciences. The work of Haubold and Mathai [1] and Saxena and Kalla [2] was the milestone in this direction. Thereafter, a frequent study of these equations was carried out by several researchers to explore the new applications. Saxena et al. [3], Gupta and Parihar [4], Gupta et al. [5], Habenom et al. [6, 7], Suthar et al. [8–11], Nisar et al. [12], Saichev and Zaslavsky [13], Zaslavsky [14], Agarwal and Bhargava [15], and Agrawal et al. [16], Prasad Sharma and Bhargava [17], and Jain and Bhargava [18], Din [19], Din et. al. [20–22], etc., are few of them.The significance of differential equations of fractional order is not only in mathematics but also in physics, dynamical system, and control system in engineering to create the mathematical model of many physical phenomena; especially, the kinetic equation describes the continuity of motion of substance and is the basic equations of mathematical physics and natural science. The extension and generalization of fractional kinetic equations involving many fractional operators were found.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 Struve function of the first kind. The fractional differential equation between rate of change of reaction was established by Haubold and Mathai [1], and the destruction rate and the production rate are calculated as follows:(1)dNdt=−dNt+pNt,where N=Nt the rate of reaction, d=dN the rate of destruction, p=pN the rate of production, and Nt denotes the function defined by Ntt∗=Nt−t∗,t∗>0.The sentence may be written as: for spatial fluctuations or inhomogeneities inNt the quantity is neglected. So, as the special case of (1), the equation (1) is reducing as(2)dNdt=−ciNit,with the initial condition that Nit=0=N0 is the number of density of species iattimet=0andci>0. If we remove the index i and integrate the standard kinetic equation, we have(3)Nt−N0=−c0Dt−1Nt,where Dt−1 is the special case of Riemann–Liouville integral operator Dt−1 defined as follows:(4)0Dt−1ft=1Γv∫0tt−sv−1fsds.The fractional generalization of the standard kinetic equation is given by Habould and Mathai [1] which is as follows:(5)Nt−N0=−cvDt−1Nt.Furthermore, Saxena and Kalla [2] gave the generalized equation as follows:(6)Νt−Ν0ft=−cv0Dt−vΝt,Rev>0,where Νt is the number density of species at time t, N0:Nt=0, c is the constant, ft∈L0,∞, and Dt−v is the fractional integral operator [23], which is defined as follows:(7)Dt−v0fx=1Γv∫0tt−uv−1fudu,t>0,Rev>0,where, in this manuscript, we consider the function ft as a composition of the generalized Galúe Struve function and k-Mittag–Leffler function and find a solution by the approach of Laplace transform [24].Habenom et al. [7] studied and presented the following form of Galúe Struve function [25].(8)wp,b,β,γα,μat=∑n=0∞−βnΓαn+μΓan+p/γ+b+2/2t22n+p+1,where α>0,γ>0,a∈ℕ,p,b,β∈ℂ and μ is an arbitrary parameter.The well-known classical Mittag–Leffler function plays a significant role to obtain the analytical solution of differential and integral equations of arbitrary order. The k-Mittag–Leffler function has significant role in fractional calculus, and their interesting and very vast useful physical applications are in several areas of engineering and applied science. The Mittag–Leffler function is a special function and not a part of fractional calculus but closely attached with the fractional calculus, as most of the theoretical developments in fractional calculus were defined with the help of this function. Furthermore, its significant applications are to obtain the analytical solution in the applied science, engineering, biological, and physical problems. The function appears obviously in the solution of integral and differential equations of fractional order and especially in the analysis of the generalization kinetic equation of fractional order. Thus, it plays a vital role in the area of fractional calculus.Thek-Mittag–Leffler function [26]is as follows:(9)Ek,ζ,τξ,ηt=∑m=0∞ξmη,kk!Γkmζ+τtm.For convergence conditions and more results on Mittag–Leffler function, we refer [27, 28].By using the definition of fractional calculus [23] and Laplace transform [24], we can have the following results:(10)Dtλ0wp,b,β,γα,μatEk,ζ,τξ,ηt=∑n=0∞∑m=0∞ξmη,kk!Γkmζ+τ−βnΓ2n+p+m+2t2n+p+m−λ+1Γαn+μΓan+p/γ+b+2/2Γ2n+p+m−λ+2122n+p+1,LDtλ0wp,b,β,γα,μatEk,ζ,τξ,ηt=∑n=0∞∑m=0∞ξmη,kk!Γkmζ+τ−βnΓ2n+p+m+2s−2n+p+m−λ+2Γαn+μΓan+p/γ+b+2/2122n+p+1.The object of this paper is to derive the analytical solution of the fractional kinetic equation involving generalized Struve function of the first kind and k-Mittag–Leffler function. The results obtained are rather general in nature and can simply construct several known results and new FKEs.


## 2. Results and Discussion

In this section, we proffer some new FKEs and find their analytical solutions involving generalized Struve function of the first kind and k-Mittag–Leffler function by Laplace transform technique and interpret graphical presentation of the results by 2-dimensional and 3-dimensional graphs.Theorem 1.
The proposed fractional kinetic equation is given by(11)Νt−Ν0wp,b,β,γα,μatEk,ζ,τξ,ηt=−cvDt−v0Νt,provided c>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0andμ is an arbitrary parameter, and then, its solution is(12)Νt=N0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2t22n+p+1∑m=0∞ξmη,ktmk!Γkmζ+τEv,2n+p+m+2−cvtv,where Ev,2n+p+m+2. is the Mittag–Leffler function.Proof 1.
Using Laplace transform on equation (11), we have(13)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+2+mΓαn+μΓan+p/γ+b+2/2122n+p+1∑m=0∞ξmη,kk!Γkmζ+τ1s2n+p+m+2×1+cvs−v−1,where N¯s is the Laplace transform of Nt.(14)=Ν0∑n=0∞−βnΓ2n+p+2+mΓαn+μΓan+p/γ+b+2/2122n+p+1∑m=0∞ξmη,kk!Γkmζ+τ1s2n+p+m+2∑k=0∞−cvs−vk=Ν0∑n=0∞−βnΓ2n+p+2+mΓαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,kk!Γkmζ+τ∑k=0∞−cvks−2n+p+m+vk+2.
Now, taking the inverse Laplace transform of equation (14), we get(15)Nt=N0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2t22n+p+1×∑m=0∞ξmη,ktmk!Γkmζ+τ∑k=0∞−cvtvkΓvk+2n+p+m+2.
Now, using the definition equation (4), we acquire equation (12).


### 2.1. Graphical Interpretation 1

A quantitative analysis with different and suitable assignments of parametric values has been done (presented in Table1) on proposed solution equation (12), and based on that, Figures 1(a) and 1(b) are the 3-dimensional and 2-dimensional graphs, respectively.Theorem 2.
The proposed FKE is(16)Νt−Ν0wp,b,β,γα,μadvtvEk,ζ,τξ,ηcvtv=−cvDt−v0Νt,provided that c>0,d>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,d≠c,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0 and μ is an arbitrary parameter, and then, the solution of equation (16) is(17)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2.dvtv22n+p+1×∑m=0∞ξmη,ktmk!Γkmζ+τEv,2n+p+1v+m+1−cvtv.Proof.
Using Laplace transform on equation (16), we have(18)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τs−2n+p+1v+m+11+cvs−v−1=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1∑m=0∞ξmη,kk!Γkmζ+τ×∑k=0∞−cvs−vks−2n+p+1v+m+1=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1∑m=0∞ξmη,kk!Γkmζ+τ×∑k=0∞−cvs−vks−2n+p+1v+m+1.
The inverse Laplace transform gives(19)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dvtv22n+p+1∑m=0∞ξmη,ktmk!Γkmζ+τ×∑k=0∞−cvtvkΓvk+2n+p+1v+m+1.
Now, using the definition equation (4), we acquire equation (17).Table 1
Analysis for (12) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.27.8933161650.7322089570.1590277730.0379775630.4319.903793936.336229099.8616338843.0327458520.62830.110823364.7054134111.918238739.634515060.813388.290671900.306313633.8038571247.09396751.044892.667246910.9486482452.8657181027.4620791.2121009.458320018.765577462.8655983308.3050491.4280467.599949563.1850819236.237778932.0561651.6581895.3517109399.193643924.5027721204.87051.81109174.866221221.154491448.9159245642.066582.01977296.432417521.8058177045.109490955.047742.23338697.809745318.3781323239.7018170334.25622.45390076.6551270803.459562354.1765303094.58822.68379665.9712085107.707939652.4694516764.2552.812614963.913311395.1491517274.488849715.88683.018470910.465113549.9272379129.2361354460.463.226398504.357706756.4853636959.5452101751.253.436933854.4111370323.525437837.1193185677.5483.650707659.9616463156.147973403.4364729967.613.868455115.6323442340.7511491241.136895768.8834.091026236.2132885374.0116308844.359891232.828Figure 1
3D (a) and 2D (b) analysis for (12).
(a)(b)


### 2.2. Graphical Interpretation 2

Figures2(a) and 2(b) are the 3-dimensional and 2-dimensional graphs for equation (17) with different and suitable assignments of parametric values (presented in Table 2).Theorem 3.
The proposed FKE is(20)Nt−N0Dtλ0wp,b,β,γα,μatEk,ζ,τξ,ηt=−cvDt−v0Nt,provided that c>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,λ≠v,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0andμ is an arbitrary parameter, then its solution is(21)Nt=N0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2.t22n+p+1×∑m=0∞ξmη,ktm−λk!Γkmζ+τEv,2n+p+m−λ+2−cvtv.Proof.
Using Laplace Transform on equation (20) and using equation (6), we have(22)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,ks−2n+p+m−λ+2k!Γkmζ+τ1+cvs−v−1=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,ks−2n+p+m−λ+2k!Γkmζ+τ∑k=0∞−cvs−vk=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,kk!Γkmζ+τ∑k=0∞−cvks−2n+p−λ+1+vk.
The inverse Laplace Transform gives(23)Νt=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,kt2n+p+m−λ+1k!Γkmζ+τ∑k=0∞−cvtvkΓvk+2n+p+m−λ+2.
Now, using the definition equation (4), we acquire equation (21).Figure 2
3D (a) and 2D (b) analysis for (17).
(a)(b)Table 2
Analysis for (17) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.278.30793701119.1439083180.242716175.07735090.46940.49453212360.331222757.4389128299.968720.699427.76661194321.6835394969.4137561997.69960.8673555.98541409995.9483044757.4074737330.8581.03024963.486698524.39315057309.9524996691.141.210474154.8724352278.8556268677.2298151361.781.430314231.2773632285.1173398704.8314502693.71.676965070.36194612401.4464220962.2868874579.31.8176861119.1464288265.4111661096521443296872.03757950141021972344246910837348431715332.2749703626.721081997965102005563101848155312.4142024019641229908629971101948201965412712.62576957001771148154718600354201381313803492.845085687161388780802033354978311690578981303.0764663153524211917732578234634141.2068E + 113.21262612911041038943303973458735762.04461E + 113.420368997431678672438101.59737E + 113.37151E + 113.6321986704921.0982E + 112.5627E + 115.42832E + 113.8499964478911.74303E + 114.02997E + 118.55651E + 114.0764140840912.71906E + 116.22521E + 111.32342E + 12


### 2.3. Graphical Interpretation 3

Figures3(a) and 3(b) are the 3-dimensional and 2-dimensional graphs for equation (21) with different and suitable assignments of parametric values (presented in Table 3).Theorem 4.
The proposed FKE is(24)Νt−Ν0Dtλ0wp,b,β,γα,μadvtvEk,ζ,τξ,ηcvtv=−cvDt−v0Νt,provided that c>0,d>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,d≠c,λ≠v,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0 and μ is an arbitrary parameter, then its solution is(25)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2.dvtv22n+p+1×∑m=0∞ξmη,ktm−λk!Γkmζ+τEv,2n+p+1v+m−λ+1−cvtv.Proof.
Using Laplace Transform on equation (24), we get(26)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τs−2n+p+1v+m−λ+11+cvuv−1=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τs−2n+p+1v+m−λ+1∑k=0∞−cvuvk=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τ∑k=0∞−cvks−2n+p+k+1v+m−λ+1.
Furthermore, using the inverse Laplace transform and the definition of Mittag–Leffler function (as in Theorems1–3), we acquire (25).Figure 3
3D (a) and 2D (b) analysis for (21).
(a)(b)Table 3
Analysis for (21) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.2141060232.921314100.277352443.5492838644.4240.4224834421.642535832.917160108.368071442.9470.6318863933.470659361.4430752547.5215930827.870.8429562655.9108399409.349415615.8927145202.431.0562164847.715888604974765816.6742729038.51.2722294959.3226008500.9108869284.964017683.211.4916501664.9314659847.6154371396.792741469.751.61152622248431000394.2214655668.71311273601.81440137765582768886.7294042536.7182030134.62.01790571457779658964.4398038997.6249099058.52.222179587061033777534533652194.1336988415.32.427394114941360203966709782977.8451622739.72.633758006581777672564937719229.3600530398.12.8415258188823094053031231753327793262586.53.051007953682984127473160995380110419190163.262582738443837305580209512810613618067373.476710996304912655018271602178917722649493.693953576706263974674350880953022976965063.8114992392197957377212451894577729688565404.0140655591441007399745258034576163824460562


### 2.4. Graphical Interpretation 4

Figures4(a) and 4(b) are the 3-dimensional and 2-dimensional graphs for equation (25) with different and suitable assignments of parametric values (presented in Table 4).Figure 4
3D (a) and 2D (b) analysis for (25).
(a)(b)Table 4
Analysis for (25) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.2−3370.76401117430.747282839136.00426728741.540.4−2132.225529132042.90966957034.16972196732.210.63132.285129410419.743213021712.6135073068.30.818464.98123966312.569521766496.21217972548.41.055356.629471970528.2234145776.84324881128.81.2133893.03073672657.03151409826.94460995758.91.4287491.64476430289.02875193828.84632842848.51.6569824.206710747713.83107630280.2848509527.91.81064770.73417326896.1215149206911179585032.01900533.99927134349.23210374670.614534270212.23269393.56641488552.8428892788618699212912.45455008.92262173842.8393150100.823858240402.68869704.7191588307.35530761237.530236384292.814104800.59132935228.2711674522.838108973123.021997797.33190470096.9948592045.247812735073.233721113.75269818324.6125775503159759346823.450898101.12378382567165988704874451959233.675753258.27525863268.9218137722692505345423.8111304948.3724921757.22855761420114670454984.0161610493.4992022197.7372557238714186432513


## 2.1. Graphical Interpretation 1

A quantitative analysis with different and suitable assignments of parametric values has been done (presented in Table1) on proposed solution equation (12), and based on that, Figures 1(a) and 1(b) are the 3-dimensional and 2-dimensional graphs, respectively.Theorem 2.
The proposed FKE is(16)Νt−Ν0wp,b,β,γα,μadvtvEk,ζ,τξ,ηcvtv=−cvDt−v0Νt,provided that c>0,d>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,d≠c,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0 and μ is an arbitrary parameter, and then, the solution of equation (16) is(17)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2.dvtv22n+p+1×∑m=0∞ξmη,ktmk!Γkmζ+τEv,2n+p+1v+m+1−cvtv.Proof.
Using Laplace transform on equation (16), we have(18)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τs−2n+p+1v+m+11+cvs−v−1=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1∑m=0∞ξmη,kk!Γkmζ+τ×∑k=0∞−cvs−vks−2n+p+1v+m+1=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1∑m=0∞ξmη,kk!Γkmζ+τ×∑k=0∞−cvs−vks−2n+p+1v+m+1.
The inverse Laplace transform gives(19)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dvtv22n+p+1∑m=0∞ξmη,ktmk!Γkmζ+τ×∑k=0∞−cvtvkΓvk+2n+p+1v+m+1.
Now, using the definition equation (4), we acquire equation (17).Table 1
Analysis for (12) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.27.8933161650.7322089570.1590277730.0379775630.4319.903793936.336229099.8616338843.0327458520.62830.110823364.7054134111.918238739.634515060.813388.290671900.306313633.8038571247.09396751.044892.667246910.9486482452.8657181027.4620791.2121009.458320018.765577462.8655983308.3050491.4280467.599949563.1850819236.237778932.0561651.6581895.3517109399.193643924.5027721204.87051.81109174.866221221.154491448.9159245642.066582.01977296.432417521.8058177045.109490955.047742.23338697.809745318.3781323239.7018170334.25622.45390076.6551270803.459562354.1765303094.58822.68379665.9712085107.707939652.4694516764.2552.812614963.913311395.1491517274.488849715.88683.018470910.465113549.9272379129.2361354460.463.226398504.357706756.4853636959.5452101751.253.436933854.4111370323.525437837.1193185677.5483.650707659.9616463156.147973403.4364729967.613.868455115.6323442340.7511491241.136895768.8834.091026236.2132885374.0116308844.359891232.828Figure 1
3D (a) and 2D (b) analysis for (12).
(a)(b)


## 2.2. Graphical Interpretation 2

Figures2(a) and 2(b) are the 3-dimensional and 2-dimensional graphs for equation (17) with different and suitable assignments of parametric values (presented in Table 2).Theorem 3.
The proposed FKE is(20)Nt−N0Dtλ0wp,b,β,γα,μatEk,ζ,τξ,ηt=−cvDt−v0Nt,provided that c>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,λ≠v,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0andμ is an arbitrary parameter, then its solution is(21)Nt=N0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2.t22n+p+1×∑m=0∞ξmη,ktm−λk!Γkmζ+τEv,2n+p+m−λ+2−cvtv.Proof.
Using Laplace Transform on equation (20) and using equation (6), we have(22)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,ks−2n+p+m−λ+2k!Γkmζ+τ1+cvs−v−1=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,ks−2n+p+m−λ+2k!Γkmζ+τ∑k=0∞−cvs−vk=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,kk!Γkmζ+τ∑k=0∞−cvks−2n+p−λ+1+vk.
The inverse Laplace Transform gives(23)Νt=Ν0∑n=0∞−βnΓ2n+p+m+2Γαn+μΓan+p/γ+b+2/2122n+p+1×∑m=0∞ξmη,kt2n+p+m−λ+1k!Γkmζ+τ∑k=0∞−cvtvkΓvk+2n+p+m−λ+2.
Now, using the definition equation (4), we acquire equation (21).Figure 2
3D (a) and 2D (b) analysis for (17).
(a)(b)Table 2
Analysis for (17) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.278.30793701119.1439083180.242716175.07735090.46940.49453212360.331222757.4389128299.968720.699427.76661194321.6835394969.4137561997.69960.8673555.98541409995.9483044757.4074737330.8581.03024963.486698524.39315057309.9524996691.141.210474154.8724352278.8556268677.2298151361.781.430314231.2773632285.1173398704.8314502693.71.676965070.36194612401.4464220962.2868874579.31.8176861119.1464288265.4111661096521443296872.03757950141021972344246910837348431715332.2749703626.721081997965102005563101848155312.4142024019641229908629971101948201965412712.62576957001771148154718600354201381313803492.845085687161388780802033354978311690578981303.0764663153524211917732578234634141.2068E + 113.21262612911041038943303973458735762.04461E + 113.420368997431678672438101.59737E + 113.37151E + 113.6321986704921.0982E + 112.5627E + 115.42832E + 113.8499964478911.74303E + 114.02997E + 118.55651E + 114.0764140840912.71906E + 116.22521E + 111.32342E + 12


## 2.3. Graphical Interpretation 3

Figures3(a) and 3(b) are the 3-dimensional and 2-dimensional graphs for equation (21) with different and suitable assignments of parametric values (presented in Table 3).Theorem 4.
The proposed FKE is(24)Νt−Ν0Dtλ0wp,b,β,γα,μadvtvEk,ζ,τξ,ηcvtv=−cvDt−v0Νt,provided that c>0,d>0,v>0,a∈ℕ,p,b,β,ζ,η,ξ∈ℂ,p>0,α>0,γ>0,d≠c,λ≠v,k∈ℝ,η∈0,1∪ℕ,ℜξ>0,ℜζ>0,ℜτ>0 and μ is an arbitrary parameter, then its solution is(25)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2.dvtv22n+p+1×∑m=0∞ξmη,ktm−λk!Γkmζ+τEv,2n+p+1v+m−λ+1−cvtv.Proof.
Using Laplace Transform on equation (24), we get(26)Ν¯s=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τs−2n+p+1v+m−λ+11+cvuv−1=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τs−2n+p+1v+m−λ+1∑k=0∞−cvuvk=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2dv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τ∑k=0∞−cvks−2n+p+k+1v+m−λ+1.
Furthermore, using the inverse Laplace transform and the definition of Mittag–Leffler function (as in Theorems1–3), we acquire (25).Figure 3
3D (a) and 2D (b) analysis for (21).
(a)(b)Table 3
Analysis for (21) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.2141060232.921314100.277352443.5492838644.4240.4224834421.642535832.917160108.368071442.9470.6318863933.470659361.4430752547.5215930827.870.8429562655.9108399409.349415615.8927145202.431.0562164847.715888604974765816.6742729038.51.2722294959.3226008500.9108869284.964017683.211.4916501664.9314659847.6154371396.792741469.751.61152622248431000394.2214655668.71311273601.81440137765582768886.7294042536.7182030134.62.01790571457779658964.4398038997.6249099058.52.222179587061033777534533652194.1336988415.32.427394114941360203966709782977.8451622739.72.633758006581777672564937719229.3600530398.12.8415258188823094053031231753327793262586.53.051007953682984127473160995380110419190163.262582738443837305580209512810613618067373.476710996304912655018271602178917722649493.693953576706263974674350880953022976965063.8114992392197957377212451894577729688565404.0140655591441007399745258034576163824460562


## 2.4. Graphical Interpretation 4

Figures4(a) and 4(b) are the 3-dimensional and 2-dimensional graphs for equation (25) with different and suitable assignments of parametric values (presented in Table 4).Figure 4
3D (a) and 2D (b) analysis for (25).
(a)(b)Table 4
Analysis for (25) with fix v.
tNtNtNtNtFixv=0.01Fixv=0.05Fixv=0.09Fixv=0.13000000.2−3370.76401117430.747282839136.00426728741.540.4−2132.225529132042.90966957034.16972196732.210.63132.285129410419.743213021712.6135073068.30.818464.98123966312.569521766496.21217972548.41.055356.629471970528.2234145776.84324881128.81.2133893.03073672657.03151409826.94460995758.91.4287491.64476430289.02875193828.84632842848.51.6569824.206710747713.83107630280.2848509527.91.81064770.73417326896.1215149206911179585032.01900533.99927134349.23210374670.614534270212.23269393.56641488552.8428892788618699212912.45455008.92262173842.8393150100.823858240402.68869704.7191588307.35530761237.530236384292.814104800.59132935228.2711674522.838108973123.021997797.33190470096.9948592045.247812735073.233721113.75269818324.6125775503159759346823.450898101.12378382567165988704874451959233.675753258.27525863268.9218137722692505345423.8111304948.3724921757.22855761420114670454984.0161610493.4992022197.7372557238714186432513


## 3. Special Cases

(i)
Puttingd=c in equation (16), the equation quashes in a new simpler form as(27)Νt−Ν0wp,b,β,γα,μacvtvEk,ζ,τξ,ηcvtv=−cvDt−v0Νt,and its solution is given by(28)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1Γαn+μΓan+p/γ+b+2/2.cvtv22n+p+1×∑m=0∞ξmη,ktmk!Γkmζ+τEv,2n+p+1v+1−cvtv.(ii)
Puttingd=c in equation (24), the equation quashes in a new simpler form as(29)Νt−Ν0Dtλ0wp,b,β,γα,μacvtvEk,ζ,τξ,ηcvtv=−cvDt−v0Νt,and its solution is given by(30)Νt=Ν0∑n=0∞−βnΓ2n+p+1v+m+1t2n+p+1v+m−λΓαn+μΓan+p/γ+b+2/2.cv22n+p+1×∑m=0∞ξmη,kk!Γkmζ+τEv,2n+p+1v+m−λ+1−cvtv.(iii)
If we convertEk,ζ,τξ,ηtv=1 by suitable parametric changes in equation (11), then we arrive at the known result acquired by Nisar et al. [12].


## 4. Conclusion

Here, in this work, we propose the solution of a new form of generalized fractional kinetic equation by using the approach of Laplace transform in the form of Mittag–Leffler function.The main observations of the results are as follows:(i)
A quantitative analysis with different and suitable assignments of parametric values has been done (presented in Tables1–4) on the proposed solutions equations (12), (17), (21), (25), and based on that, Figures 1(a), 2(a), 3(a), 4(a), 1(b), 2(b), 3(b), and 4(b) are the 3-dimensional and 2-dimensional graphs, respectively(ii)
The quantitative and graphical analysis shows the monotonic increasing nature of the solutions in the range of prescribed values of the parametersThe importance of the results has been recognized in the field of pure mathematics and applied frameworks and touches several branches of research.



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*Source: 5668579-2022-08-25.xml*