Analytic solutions of fractional kinetic equations involving incomplete $\aleph$-function

1. Introduction

Fractional calculus explores derivatives and integrals of any order for impactful mathematical analysis. Utilizing fractional calculus is crucial for assessing differ-integral equations of fractional order, a potent mathematical tool well established and advanced in engineering and scientific domains. Fractional differential equations have revolutionized biology, physics, chemistry, applied science, and engineering, imparting profound advancements to diverse fields, amplifying their impact on scientific and technological domains.^1,2,3

Special functions are advanced mathematical functions in physics, engineering, and applied sciences, extending beyond elementary functions like polynomials and exponentials. Examples include the Gamma function and Bessel functions. The H-function generalizes many special functions via Mellin–Barnes integrals, which are helpful in complex differential equations. The I-function and $\aleph$-function further extend the H-function’s versatility. These functions are vital for solving intricate mathematical problems and advancing various scientific fields.^4,5,6

Recently, researchers and mathematicians have been extensively applying special functions and differential calculus in their research. Generalized glucose supply models via incomplete I and $\aleph$-functions have recently been proposed.^7,8 Shekhawat et al.⁹ conducted a comprehensive study on generalized elliptic integrals involving incomplete Fox–Wright functions, demonstrating their potential to address advanced mathematical challenges. Similarly, Shukla et al.¹⁰ expanded the scope of fractional calculus by introducing generalized fractional calculus operators involving multivariable Aleph functions, opening new avenues for solving complex differential equations. Shukla et al.¹¹ investigated special functions of the Jacobi polynomial and incomplete H-function in modeling heat states within non-homogeneous dynamic rectangular parallelepipeds, showing practical applications in thermal analysis. Shukla et al.¹² also delved deeper into partial derivatives of multivariable incomplete H-functions, highlighting their mathematical implications.

Kumar et al.¹³ extended the investigation of finite integrals to include incomplete Aleph functions, which are essential in various fields, including signal processing and statistical mechanics. Bhatter et al.¹⁴ made significant contributions to the study of fractional differential equations involving the incomplete I-function as the kernel, providing new insights into integral equations. Bhatter et al.¹⁵ also explored the impact of environmental pollution on biological populations through mathematical modeling. Furthermore, Kumawat et al.¹⁶ explored the Khalouta transformation and its applications in Caputo-fractional differential equations, contributing to the theoretical development of fractional differential operators.

In the context of epidemiological modeling, Alqahtani and Mishra¹⁷ analyzed Streptococcus suis infection in a pig–human population using the Riemann–Liouville (R–L) fractional operator, demonstrating the applicability of fractional calculus in epidemiology. Similarly, Areshi et al.¹⁸ conducted a comparative study of blood glucose–insulin models using fractional derivatives, highlighting the role of fractional calculus in medical research. Meena et al.¹⁹ explored the fractionalized modeling of tuberculosis disease. Shyamsunder and Purohit²⁰ examined the effect of vaccination on pneumonia through a fractional approach, underscoring the importance of fractional calculus in public health.

Alazman et al.²¹ applied fractional operators to analyze rabies dynamics, demonstrating the versatility of these operators in solving epidemiological models. Furthermore, recent work has expanded into modeling infectious diseases using fractional operators. Meena et al.²² investigated the dynamics of the hepatitis B virus using a fractional operator with a nonlocal kernel, while Venkatesh et al.²³ investigated time-fractional Mpox models using Caputo fractional derivatives.

The exploration and expansion of kinetic fractional equations,²⁴ incorporating numerous fractional operators, have garnered increased attention in applied sciences. This growth extends beyond mathematics to encompass physics, dynamical systems, control systems, and engineering, where fractional differential equations play a pivotal role in modeling diverse physical phenomena. This enables the mathematical representation of diverse physical processes, such as porous media diffusion and viscoelastic media kinetics,²⁵ fostering impactful research. Fractional kinetic equations (KEs), exhibiting diverse formulations, have found widespread application in contemporary decades for delineating and tackling a myriad of consequential problems in astrophysics and physics.²⁶ Employing special functions and their applications is imperative when solving these differential equations. Owing to their efficacy, kinetic fractional equations are pertinent to a myriad of mathematical physics and diverse areas, significantly contributing to astrophysical computations.

The incorporation of incomplete special functions in the novel fractional extension of the KEs^27,28 introduces an extra dimension to the investigation. The Katugampola fractional KE is represented as a product of M-series and $\aleph$-function, accompanied by incomplete H-function, incomplete Meijer’s G-function, and incomplete Fox–Wright function. Employing the $\tau$-Laplace transformation method yields the solution for these fractional KEs. Additionally, specific instances are briefly examined.

The earlier study of KE provides the rate $d\mathcal{𝒩}/d𝔱$ as

where $\mathcal{𝒩}=\mathcal{𝒩}(𝔱)$ represents rate of change of reaction, $\mathrm{\Lambda }=\mathrm{\Lambda }({\mathcal{𝒩}}_{𝔱})$ represents the destruction rate and $\mathrm{{\rm Y}}=\mathrm{{\rm Y}}({\mathcal{𝒩}}_{𝔱})$ represents the production rate.

Or

moreover, ${\mathcal{𝒩}}_{𝔱}$ is stated as ${\mathcal{𝒩}}_{𝔱}({𝔱}^{\ast })=\mathcal{𝒩}(𝔱-{𝔱}^{\ast })$ for ${𝔱}^{\ast }>0$.

The fractional KE given by Haubold and Mathai²⁹ is as follows (see Refs. 30 and 31) :

we also obtained some appropriate case of (3), when the quantity of homogeneities or spatial variation of $\mathcal{𝒩}(𝔱)$ is ignored, is provided by the subsequent equation : where ${\mathcal{𝒩}}_k(𝔱=0)={\mathcal{𝒩}}_0$ express the variety of species density k at $t=0$, ${\epsilon }_k>0$ time. Ignoring the index k and integrating Eq. (4), we obtain where ${}_0{\mathcal{𝒟}}_{𝔱}^{-1}$ is the particular instance of the R–L fractional integral operator ${}_0{\mathcal{𝒟}}_{𝔱}^{-\mu }$ and it is described as

Moreover, Saxena and Kalla³² thought of the following fractional KE :

where $g(𝔱)\in 𝔏(0,\infty )$, $\mathcal{𝒩}(𝔱)$ represents the species’ population density at time $𝔱$ and ${\mathcal{𝒩}}_0=\mathcal{𝒩}(0)$ is the species’ population density at time $𝔱=0$.

In this paper, to solve the Katugampola fractional KE, authors will consider the $\tau$-Laplace transform³³ that involves incomplete $\aleph$-function, incomplete I-function, incomplete H-function, and incomplete Meijer G-functions described within the same group of circumstances defined in Refs. 31 and 34. This paper creatively examines the fractional KE using various mathematical functions, emphasizing its relevance across scientific fields. It explores outcomes from Katugampola’s equations, incorporating the generalized M-series and incomplete $\aleph$-function, and utilizes the $\tau$-Laplace transform to derive solutions. Additionally, the graphical representation of the behavior of the obtained solutions enhances the impact of the findings, providing a visual understanding of the system dynamics. Overall, integrating diverse mathematical techniques and synthesis of results contributes to the originality and novelty of the study.

This paper structure is as follows. Section 2 introduces some fundamental definitions and formulas. Section 3 presents the solution of the fractional kinetics equation. Section 4 provides application-based examples. Numerical simulations and graphical discussions are presented in Sec. 5. Section 6 concludes.

2. Preliminaries

Incomplete Gamma Function: The following is the definition of the incomplete Gamma function $\gamma (𝔠,s)$ and $\mathrm{\Gamma }(𝔠,s)$ represented by³⁵

and satisfy the subsequent rule of decomposition : where $\Re (𝔠)$ stands for real part of the parameter $𝔠$. Moreover, if authors set $s=0$, then authors have $\mathrm{\Gamma }(𝔠,s)=\mathrm{\Gamma }(𝔠)$.

Incomplete $\aleph$-Functions: The classical definition of the incomplete $\aleph$-functions ${}^{\gamma }{\aleph }_{r_l,s_l,f_l;\wp }^{u,v}(\varrho )$ and ${}^{\mathrm{\Gamma }}{\aleph }_{r_l,s_l,f_l;\wp }^{u,v}(\varrho )$ follows as defined³⁶:

and where for $\varrho \ne 0$, $z\geqq 0$, the incomplete $\aleph$-functions ${}^{\gamma }{\aleph }_{r_l,s_l,f_l;\wp }^{u,v}(\varrho )$ and ${}^{\mathrm{\Gamma }}{\aleph }_{r_l,s_l,f_l;\wp }^{u,v}(\varrho )$ in (8) and (9) exist under conditions.^36,37

Generalized M-Series: The generalized M-series is given by Sharma and Jain³⁸ is defined as follows:

For $𝔤$, $\alpha$, $\beta \in ℂ$, $\Re (\alpha )>0$,

Definition 2.1. The R–L integral operator is generalized into a distinct form by the Katugampola fractional operator, which was given by Katugampola³⁹ such as for $r\in ℂ$, then

The left-sided fractional integral is the name given to the above integral. The right-sided fractional integral is the name given to the above integral.

Definition 2.2. Let $𝔤:[0,\infty )\to ℝ$ be a real-valued function which is piecewise continuous and is of $\tau$-exponential order $exp(-w\frac{t^{\tau }}{\tau })$, where w is a non-negative constant, then its $\tau$-Laplace transform exists for w and is defined as

There are numerous physical implementations that depend on the convolution of the functions of $𝔤(t)$ and $𝔥(t)$, that are expressed for $t>0$. The following integral gives the $\tau$-Laplace convoluation of functions $𝔤(t)$ and $𝔥(t)$ :

which exists if the function $𝔤(t)$ and $𝔥(t)$ are at least piecewise continuous. The fact that the $\tau$-Laplace transform of a convoluation of two functions is the product of their transforms is one of the most crucial characteristics procured by the convoluation in connection with the $\tau$-Laplace transform (see, e.g., Refs. 33 and 40).

$\tau$-Laplace Convoluation Theorem: If $𝔤(t)$ and $𝔥(t)$ are two piecewise continuous functions on $[0,\infty )$ and have exponential order w, when $t\to \infty$, then

The $\tau$-Laplace transform for Katugampola fractional integral²⁷ : by using the identity where $L_{\tau }^{-1}$ is considered as the $\tau$-inverse Laplace transform.

3. Solution of Fractionalized Kinetic Equations

Theorem 3.1. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(20)

$$\times {}^{\mathrm{\Gamma }}{\aleph }_{r_l+1,s_l+1,f_l;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\hfill \\ \hfill \left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},\hfill \\ \hfill {[f_m({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})]}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\hfill \\ \hfill \left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {[f_m({\mathrm{\Psi }}_{ml},B_{ml})]}_{u+1,s_l}\hfill \end{array}\right.\right].$$(21)

Proof. By applying $\tau$-Laplace transform both side of Eq. (19), one obtains

By using Eq. (17) in Eq. (22) and replacing M-series and incomplete $\aleph$-function by Eqs. (10) and (9), respectively, one obtains After some adjustment of terms and use of ${(1+x)}^{-1}={\sum }_{r=0}^{\infty }{(-1)}^r{x}^r$ in Eq. (23), we may write By applying $\tau$-inverse Laplace transform both sides of Eq. (24) and use Eq. (18), one obtains After some adjustment of terms, one obtains the intended outcomes. □

Theorem 3.2. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(27)

$$\times {}^{\gamma }{\aleph }_{r_l+1,s_l+1,f_l;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\left(1-(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},{[f_m({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})]}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {[f_m({\mathrm{\Psi }}_{ml},B_{ml})]}_{u+1,s_l}\hfill \end{array}\right.\right].$$(28)

Proof. The proof is the immediate consequence of the definitions (8), (10), and parallel to Theorem 3.1. Consequently, we exclude the proof. □

(i) $\aleph$-Function: On setting $z=0$, then Eq. (9) reduces to the $\aleph$-function proposed by Südland^41,42:

Corollary 3.3. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(31)

$$\times {\aleph }_{r_l+1,s_l+1,f_l;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill \left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),{({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{1,v},\hfill \\ \hfill {[f_m({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})]}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {[f_m({\mathrm{\Psi }}_{ml},B_{ml})]}_{u+1,s_l}\hfill \end{array}\right.\right].$$(32)

(ii) Incomplete I-Function: Again, setting $f_l=1$ in (8) and (9), then it becomes to the incomplete I-function suggested by Bansal and Kumar⁴³:

and

Corollary 3.4. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(36)

$$\times {}^{\mathrm{\Gamma }}{I}_{r_l+1,s_l+1;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},{({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Psi }}_{ml},B_{ml})}_{u+1,s_l}\hfill \end{array}\right.\right].$$(37)

(iii) I-Function: Next, setting $z=0$ and $f_l=1$ in (9), then it becomes to the I-function suggested by Saxena⁴⁴:

Corollary 3.5. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(40)

$$\times I_{r_l+1,s_l+1;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill \left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},{({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Psi }}_{ml},B_{ml})}_{u+1,s_l}\hfill \end{array}\right.\right].$$(41)

(iv) Incomplete H-Function: Further setting $f_l=1$ and $\wp =1$ in (8) and (9), then it becomes to the incomplete H-function suggested by Srivastava et al.³⁴:

and

Corollary 3.6. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(45)

$$\times {}^{\mathrm{\Gamma }}{H}_{r+1,s+1}^{u,v+1}\left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\hfill \\ \hfill \left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\hfill \\ \hfill \left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right)\hfill \end{array}\right.\right].$$(46)

(v) H-Function: Next, we taking $z=0$, $f_l=1,$ and $\wp =1$ in (9), then it becomes to the H-function suggested by Srivastava $($see, Ref. 45, p. $10)$:

Corollary 3.7. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta }>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(49)

$$\times H_{r+1,s+1}^{u,v+1}\left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill \left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{1,v}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\hfill \\ \hfill \left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right)\hfill \end{array}\right.\right].$$(50)

(vi) Incomplete Meijer G-Function: Next, we taking $f_l=1,{\mathrm{{\rm Y}}}_m=1,B_m=1,$ and $\wp =1$ in (9), then it becomes to the G-function suggested by Meijer⁴⁶:

Corollary 3.8. For all $\varrho ,\varpi ,𝔤,\mathrm{\text{x}},\epsilon ,\mathrm{\Theta },>0$, $\alpha ,\beta ,\zeta \in ℂ$, $r,l,p,q\in \mathbb{N}$, and $\Re (\alpha )>0$, then the equation

has solution

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(53)

$$\times {}^{\mathrm{\Gamma }}{G}_{r+1,s+1}^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1:z),\left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota })}_{2,v}\hfill \\ \hfill {({\mathrm{\Psi }}_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right)\hfill \end{array}\right.\right].$$(54)

4. Applications

A few implications and uses of the aforementioned findings are addressed in this section. By properly specializing the generalized M-series to produce a wide number of existing series, certain unique instances of the resultant discoveries can be developed. We look at the following instances to illustrate this.

Example 4.1. The solution of the problem

Solution: If we set $\beta =1$ in Eq. (19), the generalized M-series is the M-series from Sharma,⁴⁷ then solution of Eq. (55) is

Similar to the above example, we can obtain assertions following Theorem 3.2, Corollaries 3.3–3.8, respectively.

Example 4.2. The solution of the problem

Solution: If we set $\alpha =\beta =1$, (i.e. ${}_p\stackrel{1,1}{M}{}_q\to {}_p{F}_q$, where ${}_p{F}_q$ is generalized hypergeometric function) and making use of the connection, that is,

in Eq. (19), then solution of Eq. (58) is

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p{(a_i)}_k}{{\prod }_{i=1}^q{(b_i)}_k}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k+1)}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(60)

$$\times {}^{\mathrm{\Gamma }}{\aleph }_{r_l+1,s_l+1,f_l;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},{[f_m({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})]}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {[f_m({\mathrm{\Psi }}_{ml},B_{ml})]}_{u+1,s_l}\hfill \end{array}\right.\right].$$(61)

Similar to the above example, we can obtain assertions following Theorem 3.2, Corollaries 3.3–3.8, respectively.

Example 4.3. The solution of the problem

Solution: If we set $p=q=1$, $a=\rho \in ℂ$, and $b=1$ (i.e. ${}_1\stackrel{\alpha ,\beta }{M}{}_1\to E_{\alpha ,\beta }^{\rho }$, where $E_{\alpha ,\beta }^{\rho }$ is generalized Mittag-Leffler function introduced by Prabhakar and studied by Kilbas et al.⁴⁸) then

in Eq. (19), then solution of Eq. (62) is

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{(\rho )}_k}{k!}\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\times \sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(64)

$$\times {}^{\mathrm{\Gamma }}{\aleph }_{r_l+1,s_l+1,f_l;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},{[f_m({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})]}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {[f_m({\mathrm{\Psi }}_{ml},B_{ml})]}_{u+1,s_l}\hfill \end{array}\right.\right].$$(65)

Similar to the above example, we can obtain assertions following Theorem 3.2, Corollaries 3.3–3.8, respectively.

Example 4.4. The solution of the problem

Solution: When there is no upper and lower parameters $p=q=0$ (i.e. ${}_0\stackrel{\alpha ,\beta }{M}{}_0\to E_{\alpha ,\beta }$, where $E_{\alpha ,\beta }$ is the Mittag-Leffler function) and making use of the connection, that is,

in Eq. (19), then solution of Eq. (66) is

$$\mathcal{𝒩}(t)={\mathcal{𝒩}}_0{t}^{\mathrm{\Theta }-1}\sum _{k=0}^{\infty }\frac{{(𝔤t^{\varrho })}^k}{\mathrm{\Gamma }(k\alpha +\beta )}\sum _{𝔥=0}^{\infty }{(-1)}^{𝔥}{(\epsilon t^{\tau })}^{\zeta 𝔥}{\tau }^{-\zeta 𝔥}$$(68)

$$\times {}^{\mathrm{\Gamma }}{\aleph }_{r_l+1,s_l+1,f_l;\wp }^{u,v+1}\times \left[\mathrm{\text{x}}{t}^{\varpi }\left|\begin{array}{c}\hfill ({\mathrm{\Theta }}_1,{\mathrm{{\rm Y}}}_1:z),\left(1-\left(\frac{\tau +\mathrm{\Theta }-1+\varrho k}{\tau }\right),\frac{\varpi }{\tau }\right),\hfill \\ \hfill {({\mathrm{\Theta }}_{\iota },{\mathrm{{\rm Y}}}_{\iota })}_{2,v},{[f_m({\mathrm{\Theta }}_{ml},{\mathrm{{\rm Y}}}_{ml})]}_{v+1,r_l}\hfill \\ \hfill {({\mathrm{\Psi }}_m,B_m)}_{1,u},\left(1-\frac{\mathrm{\Theta }+\varrho k-1+\zeta 𝔥\tau }{\tau },\frac{\varpi }{\tau }\right),\hfill \\ \hfill {[f_m({\mathrm{\Psi }}_{ml},B_{ml})]}_{u+1,s_l}\hfill \end{array}\right.\right].$$(69)

Similar to the above example, we can obtain assertions following Theorem 3.2, Corollaries 3.3–3.8, respectively.

5. Numerical Results and Discussion

In this section, authors use MATLAB to simulate the numerical results for fractional KE (19) at different values of various parameters presented in Figs. 1–3. The effect of time and order of the Katugampola integral operator of arbitrary order on the reaction rate are demonstrated in Figs. 1–3. It is observed from Figs. 1–3 that the reaction rate decreases with the enhancement of time and continuously depends on the value of the fractional parameter. The graphical results show that the solutions’ convergence region depends on the fractional parameter $\xi$. As a result, similar observations can be made for the behavior of solutions (26), (30)–(52), and (55)–(62) for different parameters and time intervals.

Figures 1–3 provide insight into the temporal dynamics of the system by showing how the reaction rates change over time. Additionally, the graphs highlight the effect of the order of the Katugampola integral operator, providing valuable information about its impact on the system’s behavior. Furthermore, the observed trends in reaction rates related to the fractional parameter $\xi$ highlight the system’s sensitivity to this parameter, emphasizing its importance in controlling the system’s dynamics. Furthermore, by identifying convergence regions, graphs contribute to stability analysis, helping to assess the model’s reliability. These findings provide predictive power regarding the behavior of systems under different conditions and suggest broad generalizations across various scenarios, increasing their applicability in scientific and engineering contexts. Overall, graphs serve as valuable tools for understanding the complexities of fractional KEs, guiding further research, and informing engineering applications that rely on reaction kinetics.

6. Concluding Remarks

This study aims to establish a novel fractional generalization of the classical KE and analyze its solution using the integral transformation method. The investigation covers a family of functions, including the incomplete $\aleph$-function, incomplete I-function, incomplete H-function, Meijer’s G-function, and generalized M-series, along with various other fractional KE and their corresponding solutions. The main conclusions presented in Theorems 3.1, 3.2, and their corollaries hold in a general context.

Similarly, numerous fractional KEs and their solutions documented in previous research can be viewed as special instances of the fundamental findings. Moreover, the generalized M-series extends to various other functions such as M-series, hypergeometric function, Mittag-Leffler, and several others. Consequently, the primary results can be employed to formulate a variety of KEs and their potential solutions by appropriately assigning arbitrary sequences and satisfying the necessary constraints. Future research will further explore the more generalized KEs and the proposed solutions.

ORCID

Manisha Meena  https://orcid.org/0009-0007-6887-714X

Mridula Purohit  https://orcid.org/0000-0001-9860-6593