Solutions for Fractional Kinetic Equations withGeneralized Galue Type Struve Functions

Solutions for Fractional Kinetic Equations withGeneralized Galue Type Struve Functions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Solutions for Fractional Kinetic Equations withGeneralized Galue Type Struve Functions Sakshi Gupta, Hemlata Saxena, Pulkit Gahlot This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7430855/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this paper, we propose a novel and comprehensive method for solving generalized fractional kinetic equations that incorporate the generalized Galué-type Struve function. These equations are significant due to their wide-ranging applications in various fields of applied mathematical sciences, including physics, engineering, and biological modeling. By employing the Laplace transform technique, we successfully derive analytical solutions to these complex equations, offering a deeper understanding of the dynamic behaviors described by fractional models. The generalized Galué-type Struve function used in our formulation enables a more flexible representation of physical phenomena, extending the applicability of traditional Struve-based models. Our approach not only generalizes several known results but also introduces new solution forms for previously unexamined equations. Further, to illustrate the behavior of these solutions, we provide graphical visualizations generated using the latest version of MATLAB. These visual aids support the analytical findings and offer practical insights for researchers Applied Mathematics Fractional Kinetic equations Laplace transform Generalized Galué type Struve function Inverse Laplace transform Figures Figure 1 Figure 2 Figure 3 1. Introduction Recently, numerous studies have delved into resolving generalized fractional kinetic equations (GFKE) by employing diverse sets of special functions. Examples include tackling GFKE solutions entailing M-series [ 3 ], the Aleph function [ 5 ], the generalized Bessel function of the first kind [ 13 ], and the generalized Struve function of the first kind [ 17 ]. Kinetic equations play a crucial role in mathematical physics and natural sciences, governing the continuous motion of substances. Researchers have expanded and diversified these equations by incorporating numerous fractional operators [ 3 , 4 , 6 , 7 , 9 , 10 , 13 , 22 , 23 , 24 , 25 , 26 , 34 ]. Acknowledging the pivotal role of kinetic equations in addressing astrophysical phenomena, the authors have introduced a further generalized version, incorporating the generalized Struve function of the first kind. The standard kinetic equation is \(\:\frac{d}{dt}\:{N}_{i}\left(t\right)=-{C}_{i}{N}_{i}\left(t\right)\) , \(\:({C}_{i}>0)\) ...(1.1) is studied in the following form [ 10 ] $$\:N\left(t\right)-{N}_{0}=\:-{c}^{\upsilon\:}{{}_{0}D}_{t}^{-\upsilon\:}N\left(t\right)$$ 1.2 ,… where \(\:{{}_{0}D}_{t}^{-\upsilon\:}f\left(t\right)\) is the Riemann-Liouville operator defined as [ 14 ] \(\:{{}_{0}D}_{t}^{-\upsilon\:}f\left(t\right)=\frac{1}{Г\left(\upsilon\:\right)}\:\underset{0}{\overset{t}{\int\:}}{\left(t-u\right)}^{\upsilon\:-1}f\left(u\right)du\) , \(\:\upsilon\:>0\) …(1.3) with \(\:{{}_{0}D}_{t}^{0}f\left(t\right)=f\left(t\right)\) . Further, Saxena and Kalla [ 26 ] considered the subsequent fractional kinetic equation as \(\:N\left(t\right)-{N}_{0}f\left(t\right)=\:-{c}^{\upsilon\:}{{}_{0}D}_{t}^{-\upsilon\:}N\left(t\right)\) , \(\:(Re\left(v\right)>0\) …(1.4) where \(\:N\left(t\right)\) denotes the number density of a given species at time t, \(\:{N}_{0}=N\left(0\right)\) is the number density of that species at time \(\:t=0\:,\:c\) is constant and \(\:f\in\:L(0,\infty\:)\) . The Laplace transform of the operator defined in (1.3) is given by [ 8 ] $$\:L\left[{{}_{0}D}_{t}^{-\upsilon\:}f\left(t\right);p\right]={p}^{-\upsilon\:\:}F\left(p\right)$$ 1.5 .… This fractional kinetic Eq. ( 1.2 ) is generalized and studied in [ 23 , 24 ]. We arise the solution of the fractional kinetic equation including generalized Galue type Struve function. The Struve function of order p given by $$\:{H}_{p}\left(z\right)={\left(\frac{z}{2}\right)}^{p+1}\:\sum\:_{k=0}^{\infty\:}\frac{{(-1)}^{k}}{Г\left(k+\frac{3}{2}\right)Г(k+p+\frac{1}{2})}\:{\left(\frac{z}{2}\right)}^{2k}$$ 1.6 .… The Struve function and its more generalization are found in many papers [ 1 , 2 , 12 , 27 , 16 , 28 , 29 , 30 , 31 ]. The generalized Struve function given by [ 1 ] \(\:{H}_{l}^{\lambda\:}\left(z\right)=\sum\:_{k=0}^{\infty\:}\frac{{(-1)}^{k}}{Г\left(\lambda\:k+l+\frac{3}{2}\right)Г(k+\frac{3}{2})}\:\:{\left(\frac{z}{2}\right)}^{2k+l+1}\) , λ > 0 …(1.7) and by [ 12 ] \(\:{H}_{l}^{\lambda\:\:,\alpha\:}\left(z\right)=\sum\:_{k=0}^{\infty\:}\frac{{(-1)}^{k}}{Г\left(\lambda\:k+l+\frac{3}{2}\right)Г(\alpha\:k+\frac{3}{2})}\:\:{\left(\frac{z}{2}\right)}^{2k+l+1}\) . λ > 0, \(\:\alpha\:>0\) …(1.8) Alternative generalized form studied by [ 27 ] as follows \(\:{H}_{l\:,\:\:\xi\:}^{\lambda\:}\left(z\right)=\sum\:_{k=0}^{\infty\:}\frac{{(-1)}^{k}}{Г\left(\lambda\:k+\frac{l}{\xi\:}+\frac{3}{2}\right)Г(k+\frac{3}{2})}\:\:{\left(\frac{z}{2}\right)}^{2k+l+1}\) . \(\:\xi\:>0\:,\:{\lambda\:}\:>\:0\) …(1.9) The four-parameter generalized Struve function was given by [ 28 ] (also see in [ 16 ]) as $$\:{H}_{p,\mu\:}^{\lambda\:\:,\alpha\:}\left(x\right)=\sum\:_{k=0}^{\infty\:}\frac{{(-1)}^{k}}{Г\left(\lambda\:k+p+\frac{3}{2}\right)Г(\alpha\:k+\mu\:)}\:\:{\left(\frac{x}{2}\right)}^{2k+p+1},\:p,\:\lambda\:\in\:C$$ 1.10 … where \(\:\lambda\:>0,\alpha\:>0\) and \(\:\mu\:\:\) is an arbitrary parameter. Alternative generalization of Struve function studied by Orhan and Yagmur [ 19 , 20 ] $$\:{H}_{p,b,c}\left(z\right)=\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}}{Г(k+\frac{3}{2})Г\left(k+p+\frac{b}{2}+1\right)}\:{\left(\frac{z}{2}\right)}^{2k+p+1}.\:p,b,c\in\:C$$ 1.11 … More generalization form of Struve function called as generalized Galue type Struve function (GTSF) defined by Nisar et.al. [ 18 ] as follows $$\:{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left(z\right)=\:\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}}{Г(\alpha\:k+\mu\:)Г\left(ak+\frac{p}{\xi\:}+\frac{b+2}{2}\right)}\:{\left(\frac{z}{2}\right)}^{2k+p+1},\:a\in\:N,p,b,c\in\:C$$ 1.12 … where \(\:\alpha\:>0,\xi\:>0\) and \(\:\mu\:\) is an arbitrary parameter and studied fractional integral representations of generalized GTSF. In the solution of generalized fractional kinetic equation including generalized GTSF we need the following definitions such as In 1903, the Swedish mathematician Gosta Mittag –Leffler introduced the function $$\:{E}_{\propto\:}\left(z\right)=\sum\:_{n=0}^{\infty\:}\frac{{z}^{n}}{\varGamma\:(\propto\:n+1)}$$ 1.13 ,… where z is a complex variable and \(\:\propto\:\ge\:0\) . The Mittag – Leffler function is a direct generalization of exponential function to which it reduces for \(\:\propto\:=1\) see in [ 15 ]. The generalization of \(\:{E}_{\propto\:}\left(z\right)\) was studied by Wiman [ 33 ] and defined the function as \(\:{E}_{\propto\:,\:\beta\:}\left(z\right)=\sum\:_{n=0}^{\infty\:}\frac{{z}^{n}}{\varGamma\:(\propto\:n+\beta\:)}\) . \(\:(\propto\:,\beta\:\in\:C,Re(\propto\:)>0,Re(\beta\:)>0)\) …(1.14) In 1971, Prabhaker [ 21 ] introduced the function \(\:{E}_{\alpha\:,\beta\:}^{\gamma\:}\left[z\right]\) in the form of \(\:{E}_{\alpha\:,\beta\:}^{\gamma\:}\left[z\right]=\sum\:_{0}^{\infty\:}\frac{{\left(\gamma\:\right)}_{n}}{Г(\alpha\:n+\beta\:)}\:\frac{{z}^{n}}{n!}\) , \(\:\left(\propto\:,\beta\:,\gamma\:\in\:C,Re\left(\propto\:\right)>0,Re\left(\beta\:\right),Re\left(\gamma\:\right)>0\right)\) …(1.15) where \(\:{\left(\gamma\:\right)}_{n}\) is the Pochhammer symbol (Rainville (1960)) $$\:\:\:\:\:\:\:\:\:\:\:\:{\left(\gamma\:\right)}_{n}=\frac{Г\left(\gamma\:+n\right)}{Г\left(\gamma\:\right)},\:\:{\left(\gamma\:\right)}_{0}=1,\:$$ $$\:{\left(\gamma\:\right)}_{n}=\gamma\:\left(\gamma\:+1\right)\left(\gamma\:+2\right)\dots\:\left(\gamma\:+n-1\right).\:\:\:n\ge\:1$$ The Laplace transform of Riemann-Liouville fractional integral operator defined by [ 8 ] and [ 32 ] as follows $$\:L\left[{{}_{0}D}_{t}^{-\upsilon\:}f\left(t\right);p\right]={p}^{-\upsilon\:\:}F\left(p\right),$$ 1.16 … where \(\:F\left(p\right)\) is the Laplace transform of \(\:f\left(t\right)\) is given by $$\:F\left(p\right)=L\left\{f\left(t\right);p\right\}={\int\:}_{0}^{\infty\:}{e}^{-pt}f\left(t\right)dt,$$ 1.17 … $$\:=\begin{array}{c}lim\\\:\tau\:\to\:\infty\:\end{array}{\int\:}_{0}^{\tau\:}{e}^{-pt}f\left(t\right)dt$$ , whenever the limit exists. The following Laplace transforms are also required in the sequel see in [ 11 ] $$\:L\left[{\left(1+{c}^{\upsilon\:}{{}_{0}D}_{t}^{-\upsilon\:}\right)}^{n}N\left(t\right);p\right]=\sum\:_{r=0}^{n}\left(\begin{array}{c}n\\\:r\end{array}\right){c}^{\upsilon\:r}L\{\left[{{}_{0}D}_{t}^{-\upsilon\:}N\left(t\right)\right];p\},$$ 1.18 … where \(\:\left(\begin{array}{c}n\\\:r\end{array}\right)=\frac{\left(n\right)!}{\left(n-r\right)!\left(r\right)!}\) , …(1.19) which in view of (1.16) gives $$\:L\left[{\left(1+{c}^{\upsilon\:}{{}_{0}D}_{t}^{-\upsilon\:}\right)}^{n}N\left(t\right);p\right]={\left(1+{c}^{\upsilon\:}{p}^{-\upsilon\:}\right)}^{n}N\left(p\right).$$ 1.20 … The Laplace transform in Eq. ( 1.20 ) is obtained with the help of binomial expansion and term by term integration in view of Eq. ( 1.16 ). The Laplace transform of power function is defined as $$\:L\left\{{t}^{\mu\:-1}\right\}=\frac{Г\left(\mu\:\right)}{{p}^{\mu\:}}$$ 1.21 .… Then the following inverse Laplace transform in view of Eq. ( 1.21 ) are required as $$\:{L}^{-1}\left\{{p}^{-\mu\:}\right\}=\frac{{t}^{\mu\:-1}}{Г\left(\mu\:\right)}.$$ 1.22 … The results can be applied to many areas in applied mathematics and science, such as materials with memory viscoelasticity, processes where particles move irregularly (anomalous diffusion), reaction rates in nuclear physics, and various problems in engineering and physical sciences. The method we present is flexible and can be used for many known types of fractional kinetic equations. It also opens new possibilities for solving problems that have not been studied before. This makes our work a valuable resource for both researchers and professionals working with fractional calculus and its practical applications. 2. Results Theorem 1 If \(\:e,\:t,v\:\in\:{R}^{+}\:,\:a,b,c,l\:\in\:C\:and\:R\left(l\right)>-1\:\) then the solution of the equation $$\:N\left(t\right)-{N}_{o}{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left(t\right)=-\left\{\sum\:_{r=1}^{n}\left(\begin{array}{c}n\\\:r\end{array}\right){\left({e}^{v}\right)}^{r}{{}_{0}D}_{t}^{-vr}\right\}\:N\left(t\right),$$ 2.1 … i.e. \(\:{\left(1+{e}^{v}{{}_{0}D}_{t}^{-v}\right)}^{n}N\left(t\right)={N}_{0}\) \(\:{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left(t\right)\) , …(2.2) is given by $$\:N\left(t\right)=\:{N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2k+l+2\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right)}\:{\left(\frac{t}{2}\right)}^{2k+l+1}\:.{E}_{v,2k+l+2}^{n}\left({-e}^{v}{t}^{v}\right)$$ 2.3 .… Proof Taking Laplace transform of Eq. (2.2) on both the side in view of Eq. ( 1.12 ) and Eq. ( 1.20 ) $$\:(1+{e}^{v}{p}^{-v}{)}^{n}\:\:N\left(p\right)={N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2k+l+2\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right){\left(2\right)}^{2k+l+1}}.\frac{1}{{p}^{2k+l+2}}$$ , $$\:N\left(p\right)=\:{N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2k+l+2\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right){\left(2\right)}^{2k+l+1}}.\frac{1}{{p}^{2k+l+2}}\sum\:_{r=0}^{\infty\:}\frac{{\left(n\right)}_{r}}{\left(r\right)!}\:\:{(-{e}^{v})}^{r}{p}^{-vr}$$ . Taking inverse Laplace transform we have, $$\:N\left(t\right)={N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2k+l+2\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right){\left(2\right)}^{2k+l+1}}.\sum\:_{r=0}^{\infty\:}\frac{{\left(n\right)}_{r}}{\left(r\right)!}\:\:{(-{e}^{v})}^{r}\:{L}^{-1}\left\{{p}^{-\left(vr+2k+l+2\right)}\right\}$$ , which in view of Eq. ( 1.22 ) gives $$\:N\left(t\right)={N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2k+l+2\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right){\left(2\right)}^{2k+l+1}}.\sum\:_{r=0}^{\infty\:}\frac{{\left(n\right)}_{r}}{\left(r\right)!}\:\:{(-{e}^{v})}^{r}\frac{{t}^{vr+2k+l+1}}{Г(vr+2k+l+2)}$$ . On interpreting the resulting series with the help of Eq. (1.15) we at once arrive at the solution in Eq. ( 2.3 ). Theorem 2 If \(\:e,\:t,v\:\in\:{R}^{+}\:,\:a,b,c,l\:\in\:C\:and\:R\left(l\right)>-1,\:\) then the solution of the equation $$\:N\left(t\right)-{N}_{o}{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left({e}^{v}{t}^{v}\right)=-\left\{\sum\:_{r=1}^{n}\left(\begin{array}{c}n\\\:r\end{array}\right){\left({e}^{v}\right)}^{r}{{}_{0}D}_{t}^{-vr}\right\}\:N\left(t\right)$$ 2.4 ,… i.e. \(\:{\left(1+{e}^{v}{{}_{0}D}_{t}^{-v}\right)}^{n}N\left(t\right)={N}_{0}\) \(\:{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left({e}^{v}{t}^{v}\right)\) , …(2.5) is given by $$\:N\left(t\right)={N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2kv+lv+v+1\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right)}\:{\left(\frac{{e}^{v}{t}^{v}}{2}\right)}^{2k+l+1}.{E}_{v,\left(2k+l+1\right)v+1}^{n}\left({-e}^{v}{t}^{v}\right)$$ 2.6 .… Proof We can derive parallel as theorem 1 . Theorem 3 If \(\:e,\:t,v\:\in\:{R}^{+}\:,\:a,b,c,l\:\in\:C\:and\:R\left(l\right)>-1\:,\:a\ne\:e,\) then the solution of the equation $$\:N\left(t\right)-{N}_{o}{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left({e}^{v}{t}^{v}\right)=-\left\{\sum\:_{r=1}^{n}\left(\begin{array}{c}n\\\:r\end{array}\right){\left({a}^{v}\right)}^{r}{{}_{0}D}_{t}^{-vr}\right\}\:N\left(t\right)$$ 2.7 ,… i.e. \(\:{\left(1+{a}^{v}{{}_{0}D}_{t}^{-v}\right)}^{n}N\left(t\right)={N}_{0}\) \(\:{{}_{a}W}_{p,b,c,\:\:\xi\:}^{\alpha\:,\mu\:}\left({e}^{v}{t}^{v}\right)\) , …(2.8) is given by $$\:N\left(t\right)={N}_{0}\sum\:_{k=0}^{\infty\:}\frac{{(-c)}^{k}Г\left(2kv+lv+v+1\right)}{Г\left(\propto\:k+\mu\:\right)Г\left(ak+\frac{l}{\xi\:}+\frac{b+2}{2}\right)}\:{\left(\frac{{e}^{v}{t}^{v}}{2}\right)}^{2k+l+1}.{E}_{v,\left(2k+l+1\right)v+1}^{n}\left({-a}^{v}{t}^{v}\right)$$ 2.9 .… Proof We can derive parallel as theorem 1 . Special Cases : If we take \(\:r=1\) in theorem 1 we at once arrive the known result of [18, pp. 9, Eq. 34, 38]. If we take \(\:r=1\) in theorem 2 we at once arrive the known result of [18, pp. 9, Eq. 39, 40] If we take \(\:r=1\) in theorem 3 we at once arrive the known result of [18, pp. 9, Eq. 41, 42]. 3. Conclusion The present study investigates fractional kinetic equations involving the generalized Galué-type Struve function. The solutions to these generalized fractional kinetic equations are obtained using the Laplace transform. It is observed that the Laplace transform provides an effective method for solving such equations, with several applications and explicit solutions derived. Notably, the results presented in Theorem 1 , Theorem 2 , and Theorem 3 are expressed in terms of the Mittag-Leffler function. Furthermore, to illustrate the behavior and properties of the solutions, graphical representations have been provided using MATLAB (Fig. (1–3)). These visualizations support the analytical findings and offer insight into the dynamic nature of the solutions. This work opens the door for further research, where other researchers may explore alternative integral transforms and special functions to derive new or existing forms of fractional kinetic equations. Declarations Conflict of Interest: The authors declare that they have no conflict of interest Funding: No funds, grants, or other support was received from funding agencies except host University References Bhowmick KN (1962) Some relations between a generalized Struve’s function and hypergeometric functions. Vijnana Parishad Anusandhan Patrika 5:93–99 Bhowmick KN (1993) A generalized Struve’s function and its recurrence formula. Vijnana Parishad Anusandhan Patrika 6:1–11 Chaurasia VBL, Kumar D (2010) On the solutions of generalized fractional kinetic equations. 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Math Ed (Siwan) 23:30–36 Srivastava HM, Saxena RK (2001) Operators of fractional integration and their applications. Appl Math Comput 118:1–52 Wiman A (1905) Uber den fundamental satz in der theorie der funktionen \:{E}_{\alpha\:}\left(z\right) . Acta Math 29:191–201 Zaslavsky GM (1994) Fractional kinetic equation for Hamiltonian chaos. Phys D. 76, 110 – 122 Additional Declarations The authors declare no competing interests. Supplementary Files PaperHemlataSakshiPulkit.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7430855","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":503989019,"identity":"349e7221-453e-46dd-8d1f-18d8017b234e","order_by":0,"name":"Sakshi Gupta","email":"","orcid":"","institution":"Career Point University, Kota","correspondingAuthor":false,"prefix":"","firstName":"Sakshi","middleName":"","lastName":"Gupta","suffix":""},{"id":503989020,"identity":"d1588559-c848-4e38-8799-be7109874fd3","order_by":1,"name":"Hemlata Saxena","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABC0lEQVRIiWNgGAWjYPACZgYDBgbGA0CWHIh74AGRWhgOMCQwGIO1JJCiJbEBxMenRb79jOHjghrrfHP27oQDP3/Ypc8PO/wQaIudnG4Ddi0GZ9KSjWccS7fc2XN2w8GehOTcjbfTDIBako3NDuDQwpB8TJqH7bCBwY3cDQd4EphzN85OAGk5kLgNhxb5/oftv3n+AbXcf7vh4J+E+nTD2ekf8GphuJF8jJm3DWQL74bDPAmHE+Slc/DbYnDjWbI0b1+6gcGZ3A2HZdKOG26Qzik4kGCA2y/y/TmGn3m+WRsYHD+78eEbm2p5+dnpmz98qLCTw6UFi70HIMFCApBvIEX1KBgFo2AUjAQAADQeaFa8aI3AAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0003-1037-4746","institution":"Career Point Uversity,Kota","correspondingAuthor":true,"prefix":"","firstName":"Hemlata","middleName":"","lastName":"Saxena","suffix":""},{"id":503989214,"identity":"dec7aeb5-61d3-4ff9-9661-6c0098d86213","order_by":2,"name":"Pulkit Gahlot","email":"","orcid":"","institution":"Career Point University, Kota","correspondingAuthor":false,"prefix":"","firstName":"Pulkit","middleName":"","lastName":"Gahlot","suffix":""}],"badges":[],"createdAt":"2025-08-22 05:15:25","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-7430855/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7430855/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89790024,"identity":"6f7e612f-b37f-4fd5-9b53-fef5d5eb0b35","added_by":"auto","created_at":"2025-08-25 05:31:23","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":43324,"visible":true,"origin":"","legend":"\u003cp\u003eSolution \u003cem\u003eN\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) of the Theorem 1 for different combinations of the values of \u003cem\u003ev\u003c/em\u003e and \u003cem\u003eα\u003c/em\u003e.\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7430855/v1/6a54f3c179cef0bba7793247.jpeg"},{"id":89790549,"identity":"1109dcf3-b0ac-4d5d-951f-a9872d565199","added_by":"auto","created_at":"2025-08-25 05:39:23","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":46823,"visible":true,"origin":"","legend":"\u003cp\u003eSolution \u003cem\u003eN\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) of the Theorem 2 for different combinations of the values of \u003cem\u003ev\u003c/em\u003e and \u003cem\u003eα\u003c/em\u003e.\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7430855/v1/53bf17f37745793b84cd07c3.jpeg"},{"id":89790028,"identity":"156efca3-185e-42e8-b6e7-64ab9eb98e6d","added_by":"auto","created_at":"2025-08-25 05:31:23","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":49698,"visible":true,"origin":"","legend":"\u003cp\u003eSolution \u003cem\u003eN\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) of the Theorem 3 for different combinations of the values of \u003cem\u003ev\u003c/em\u003e and \u003cem\u003eα\u003c/em\u003e.\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7430855/v1/331e723042b7637804b048fa.jpeg"},{"id":89791763,"identity":"576a0952-03cf-43ba-8590-9ca1ae6908b8","added_by":"auto","created_at":"2025-08-25 05:55:25","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1345165,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7430855/v1/879ab0d1-e390-454b-a4c5-1972de2900e6.pdf"},{"id":89790928,"identity":"9119d5e5-b032-456d-8a27-8d6ba47ce282","added_by":"auto","created_at":"2025-08-25 05:47:23","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":184786,"visible":true,"origin":"","legend":"","description":"","filename":"PaperHemlataSakshiPulkit.docx","url":"https://assets-eu.researchsquare.com/files/rs-7430855/v1/8810fc1a71d40b92ed28e8ff.docx"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eSolutions for Fractional Kinetic Equations with\u003c/p\u003e\u003cp\u003eGeneralized Galue Type Struve Functions\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eRecently, numerous studies have delved into resolving generalized fractional kinetic equations (GFKE) by employing diverse sets of special functions. Examples include tackling GFKE solutions entailing M-series [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], the Aleph function [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], the generalized Bessel function of the first kind [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], and the generalized Struve function of the first kind [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eKinetic equations play a crucial role in mathematical physics and natural sciences, governing the continuous motion of substances. Researchers have expanded and diversified these equations by incorporating numerous fractional operators [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e25\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e26\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e34\u003c/span\u003e]. Acknowledging the pivotal role of kinetic equations in addressing astrophysical phenomena, the authors have introduced a further generalized version, incorporating the generalized Struve function of the first kind.\u003c/p\u003e\u003cp\u003eThe standard kinetic equation is\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{d}{dt}\\:{N}_{i}\\left(t\\right)=-{C}_{i}{N}_{i}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({C}_{i}\u0026gt;0)\\)\u003c/span\u003e\u003c/span\u003e ...(1.1)\u003c/p\u003e\u003cp\u003eis studied in the following form [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:N\\left(t\\right)-{N}_{0}=\\:-{c}^{\\upsilon\\:}{{}_{0}D}_{t}^{-\\upsilon\\:}N\\left(t\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.2\u003c/div\u003e\u003c/div\u003e,\u0026hellip;\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{}_{0}D}_{t}^{-\\upsilon\\:}f\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e is the Riemann-Liouville operator defined as [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{}_{0}D}_{t}^{-\\upsilon\\:}f\\left(t\\right)=\\frac{1}{Г\\left(\\upsilon\\:\\right)}\\:\\underset{0}{\\overset{t}{\\int\\:}}{\\left(t-u\\right)}^{\\upsilon\\:-1}f\\left(u\\right)du\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\upsilon\\:\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e \u0026hellip;(1.3)\u003c/p\u003e\u003cp\u003ewith \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{}_{0}D}_{t}^{0}f\\left(t\\right)=f\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eFurther, Saxena and Kalla [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e26\u003c/span\u003e] considered the subsequent fractional kinetic equation as\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:N\\left(t\\right)-{N}_{0}f\\left(t\\right)=\\:-{c}^{\\upsilon\\:}{{}_{0}D}_{t}^{-\\upsilon\\:}N\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(Re\\left(v\\right)\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e \u0026hellip;(1.4)\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:N\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e denotes the number density of a given species at time t, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{0}=N\\left(0\\right)\\)\u003c/span\u003e\u003c/span\u003e is the number density of that species at time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:t=0\\:,\\:c\\)\u003c/span\u003e\u003c/span\u003e is constant and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:f\\in\\:L(0,\\infty\\:)\\)\u003c/span\u003e\u003c/span\u003e. The Laplace transform of the operator defined in (1.3) is given by [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:L\\left[{{}_{0}D}_{t}^{-\\upsilon\\:}f\\left(t\\right);p\\right]={p}^{-\\upsilon\\:\\:}F\\left(p\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.5\u003c/div\u003e\u003c/div\u003e.\u0026hellip;\u003c/p\u003e\u003cp\u003eThis fractional kinetic Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1.2\u003c/span\u003e) is generalized and studied in [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e24\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eWe arise the solution of the fractional kinetic equation including generalized Galue type Struve function.\u003c/p\u003e\u003cp\u003eThe Struve function of order p given by\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{H}_{p}\\left(z\\right)={\\left(\\frac{z}{2}\\right)}^{p+1}\\:\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-1)}^{k}}{Г\\left(k+\\frac{3}{2}\\right)Г(k+p+\\frac{1}{2})}\\:{\\left(\\frac{z}{2}\\right)}^{2k}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.6\u003c/div\u003e\u003c/div\u003e.\u0026hellip;\u003c/p\u003e\u003cp\u003eThe Struve function and its more generalization are found in many papers [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e28\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e29\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. The generalized Struve function given by [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{H}_{l}^{\\lambda\\:}\\left(z\\right)=\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-1)}^{k}}{Г\\left(\\lambda\\:k+l+\\frac{3}{2}\\right)Г(k+\\frac{3}{2})}\\:\\:{\\left(\\frac{z}{2}\\right)}^{2k+l+1}\\)\u003c/span\u003e\u003c/span\u003e, λ\u0026thinsp;\u0026gt;\u0026thinsp;0 \u0026hellip;(1.7)\u003c/p\u003e\u003cp\u003eand by [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{H}_{l}^{\\lambda\\:\\:,\\alpha\\:}\\left(z\\right)=\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-1)}^{k}}{Г\\left(\\lambda\\:k+l+\\frac{3}{2}\\right)Г(\\alpha\\:k+\\frac{3}{2})}\\:\\:{\\left(\\frac{z}{2}\\right)}^{2k+l+1}\\)\u003c/span\u003e\u003c/span\u003e. λ\u0026thinsp;\u0026gt;\u0026thinsp;0, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e\u0026hellip;(1.8)\u003c/p\u003e\u003cp\u003eAlternative generalized form studied by [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e27\u003c/span\u003e] as follows\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{H}_{l\\:,\\:\\:\\xi\\:}^{\\lambda\\:}\\left(z\\right)=\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-1)}^{k}}{Г\\left(\\lambda\\:k+\\frac{l}{\\xi\\:}+\\frac{3}{2}\\right)Г(k+\\frac{3}{2})}\\:\\:{\\left(\\frac{z}{2}\\right)}^{2k+l+1}\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\xi\\:\u0026gt;0\\:,\\:{\\lambda\\:}\\:\u0026gt;\\:0\\)\u003c/span\u003e\u003c/span\u003e \u0026hellip;(1.9)\u003c/p\u003e\u003cp\u003eThe four-parameter generalized Struve function was given by [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e28\u003c/span\u003e] (also see in [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]) as\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{H}_{p,\\mu\\:}^{\\lambda\\:\\:,\\alpha\\:}\\left(x\\right)=\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-1)}^{k}}{Г\\left(\\lambda\\:k+p+\\frac{3}{2}\\right)Г(\\alpha\\:k+\\mu\\:)}\\:\\:{\\left(\\frac{x}{2}\\right)}^{2k+p+1},\\:p,\\:\\lambda\\:\\in\\:C$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.10\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\lambda\\:\u0026gt;0,\\alpha\\:\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mu\\:\\:\\)\u003c/span\u003e\u003c/span\u003e is an arbitrary parameter. Alternative generalization of Struve function studied by Orhan and Yagmur [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{H}_{p,b,c}\\left(z\\right)=\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}}{Г(k+\\frac{3}{2})Г\\left(k+p+\\frac{b}{2}+1\\right)}\\:{\\left(\\frac{z}{2}\\right)}^{2k+p+1}.\\:p,b,c\\in\\:C$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.11\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003eMore generalization form of Struve function called as generalized Galue type Struve function (GTSF) defined by Nisar et.al. [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] as follows\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left(z\\right)=\\:\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}}{Г(\\alpha\\:k+\\mu\\:)Г\\left(ak+\\frac{p}{\\xi\\:}+\\frac{b+2}{2}\\right)}\\:{\\left(\\frac{z}{2}\\right)}^{2k+p+1},\\:a\\in\\:N,p,b,c\\in\\:C$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.12\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:\u0026gt;0,\\xi\\:\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mu\\:\\)\u003c/span\u003e\u003c/span\u003e is an arbitrary parameter and studied fractional integral representations of generalized GTSF. In the solution of generalized fractional kinetic equation including generalized GTSF we need the following definitions such as\u003c/p\u003e\u003cp\u003eIn 1903, the Swedish mathematician Gosta Mittag \u0026ndash;Leffler introduced the function\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:{E}_{\\propto\\:}\\left(z\\right)=\\sum\\:_{n=0}^{\\infty\\:}\\frac{{z}^{n}}{\\varGamma\\:(\\propto\\:n+1)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.13\u003c/div\u003e\u003c/div\u003e,\u0026hellip;\u003c/p\u003e\u003cp\u003ewhere z is a complex variable and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\propto\\:\\ge\\:0\\)\u003c/span\u003e\u003c/span\u003e. The Mittag \u0026ndash; Leffler function is a direct generalization of exponential function to which it reduces for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\propto\\:=1\\)\u003c/span\u003e\u003c/span\u003e see in [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe generalization of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{\\propto\\:}\\left(z\\right)\\)\u003c/span\u003e\u003c/span\u003e was studied by Wiman [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e33\u003c/span\u003e] and defined the function as\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{\\propto\\:,\\:\\beta\\:}\\left(z\\right)=\\sum\\:_{n=0}^{\\infty\\:}\\frac{{z}^{n}}{\\varGamma\\:(\\propto\\:n+\\beta\\:)}\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(\\propto\\:,\\beta\\:\\in\\:C,Re(\\propto\\:)\u0026gt;0,Re(\\beta\\:)\u0026gt;0)\\)\u003c/span\u003e\u003c/span\u003e \u0026hellip;(1.14)\u003c/p\u003e\u003cp\u003eIn 1971, Prabhaker [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] introduced the function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{\\alpha\\:,\\beta\\:}^{\\gamma\\:}\\left[z\\right]\\)\u003c/span\u003e\u003c/span\u003e in the form of\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{\\alpha\\:,\\beta\\:}^{\\gamma\\:}\\left[z\\right]=\\sum\\:_{0}^{\\infty\\:}\\frac{{\\left(\\gamma\\:\\right)}_{n}}{Г(\\alpha\\:n+\\beta\\:)}\\:\\frac{{z}^{n}}{n!}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(\\propto\\:,\\beta\\:,\\gamma\\:\\in\\:C,Re\\left(\\propto\\:\\right)\u0026gt;0,Re\\left(\\beta\\:\\right),Re\\left(\\gamma\\:\\right)\u0026gt;0\\right)\\)\u003c/span\u003e\u003c/span\u003e \u0026hellip;(1.15)\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\left(\\gamma\\:\\right)}_{n}\\)\u003c/span\u003e\u003c/span\u003e is the Pochhammer symbol (Rainville (1960))\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:{\\left(\\gamma\\:\\right)}_{n}=\\frac{Г\\left(\\gamma\\:+n\\right)}{Г\\left(\\gamma\\:\\right)},\\:\\:{\\left(\\gamma\\:\\right)}_{0}=1,\\:$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:{\\left(\\gamma\\:\\right)}_{n}=\\gamma\\:\\left(\\gamma\\:+1\\right)\\left(\\gamma\\:+2\\right)\\dots\\:\\left(\\gamma\\:+n-1\\right).\\:\\:\\:n\\ge\\:1$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe Laplace transform of Riemann-Liouville fractional integral operator defined by [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e32\u003c/span\u003e] as follows\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:L\\left[{{}_{0}D}_{t}^{-\\upsilon\\:}f\\left(t\\right);p\\right]={p}^{-\\upsilon\\:\\:}F\\left(p\\right),$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.16\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:F\\left(p\\right)\\)\u003c/span\u003e\u003c/span\u003e is the Laplace transform of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:f\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e is given by\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:F\\left(p\\right)=L\\left\\{f\\left(t\\right);p\\right\\}={\\int\\:}_{0}^{\\infty\\:}{e}^{-pt}f\\left(t\\right)dt,$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.17\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:=\\begin{array}{c}lim\\\\\\:\\tau\\:\\to\\:\\infty\\:\\end{array}{\\int\\:}_{0}^{\\tau\\:}{e}^{-pt}f\\left(t\\right)dt$$\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e\u003cp\u003ewhenever the limit exists.\u003c/p\u003e\u003cp\u003eThe following Laplace transforms are also required in the sequel see in [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\:L\\left[{\\left(1+{c}^{\\upsilon\\:}{{}_{0}D}_{t}^{-\\upsilon\\:}\\right)}^{n}N\\left(t\\right);p\\right]=\\sum\\:_{r=0}^{n}\\left(\\begin{array}{c}n\\\\\\:r\\end{array}\\right){c}^{\\upsilon\\:r}L\\{\\left[{{}_{0}D}_{t}^{-\\upsilon\\:}N\\left(t\\right)\\right];p\\},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.18\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(\\begin{array}{c}n\\\\\\:r\\end{array}\\right)=\\frac{\\left(n\\right)!}{\\left(n-r\\right)!\\left(r\\right)!}\\)\u003c/span\u003e\u003c/span\u003e, \u0026hellip;(1.19)\u003c/p\u003e\u003cp\u003ewhich in view of (1.16) gives\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\:L\\left[{\\left(1+{c}^{\\upsilon\\:}{{}_{0}D}_{t}^{-\\upsilon\\:}\\right)}^{n}N\\left(t\\right);p\\right]={\\left(1+{c}^{\\upsilon\\:}{p}^{-\\upsilon\\:}\\right)}^{n}N\\left(p\\right).$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.20\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003eThe Laplace transform in Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e1.20\u003c/span\u003e) is obtained with the help of binomial expansion and term by term integration in view of Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e1.16\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eThe Laplace transform of power function is defined as\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\:L\\left\\{{t}^{\\mu\\:-1}\\right\\}=\\frac{Г\\left(\\mu\\:\\right)}{{p}^{\\mu\\:}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.21\u003c/div\u003e\u003c/div\u003e.\u0026hellip;\u003c/p\u003e\u003cp\u003eThen the following inverse Laplace transform in view of Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e1.21\u003c/span\u003e) are required as\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\:{L}^{-1}\\left\\{{p}^{-\\mu\\:}\\right\\}=\\frac{{t}^{\\mu\\:-1}}{Г\\left(\\mu\\:\\right)}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1.22\u003c/div\u003e\u003c/div\u003e\u0026hellip;\u003c/p\u003e\u003cp\u003eThe results can be applied to many areas in applied mathematics and science, such as materials with memory viscoelasticity, processes where particles move irregularly (anomalous diffusion), reaction rates in nuclear physics, and various problems in engineering and physical sciences.\u003c/p\u003e\u003cp\u003eThe method we present is flexible and can be used for many known types of fractional kinetic equations. It also opens new possibilities for solving problems that have not been studied before. This makes our work a valuable resource for both researchers and professionals working with fractional calculus and its practical applications.\u003c/p\u003e"},{"header":"2. Results","content":"\u003cp\u003e\u003cstrong\u003eTheorem 1\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:e,\\:t,v\\:\\in\\:{R}^{+}\\:,\\:a,b,c,l\\:\\in\\:C\\:and\\:R\\left(l\\right)\u0026gt;-1\\:\\)\u003c/span\u003e\u003c/span\u003e then the solution of the equation\u003c/p\u003e\n\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)-{N}_{o}{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left(t\\right)=-\\left\\{\\sum\\:_{r=1}^{n}\\left(\\begin{array}{c}n\\\\\\:r\\end{array}\\right){\\left({e}^{v}\\right)}^{r}{{}_{0}D}_{t}^{-vr}\\right\\}\\:N\\left(t\\right),$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2.1\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u0026hellip;\u003c/p\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003ei.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\left(1+{e}^{v}{{}_{0}D}_{t}^{-v}\\right)}^{n}N\\left(t\\right)={N}_{0}\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u0026hellip;(2.2)\u003c/p\u003e\u003c/div\u003e\u003cp\u003eis given by\u003c/p\u003e\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)=\\:{N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2k+l+2\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right)}\\:{\\left(\\frac{t}{2}\\right)}^{2k+l+1}\\:.{E}_{v,2k+l+2}^{n}\\left({-e}^{v}{t}^{v}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2.3\u003c/div\u003e\u003c/div\u003e\u003cp\u003e.\u0026hellip;\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eProof\u003c/strong\u003e\u003c/p\u003e\u003cp\u003eTaking Laplace transform of Eq.\u0026nbsp;(2.2) on both the side in view of Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1.12\u003c/span\u003e) and Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1.20\u003c/span\u003e)\u003c/p\u003e\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e$$\\:(1+{e}^{v}{p}^{-v}{)}^{n}\\:\\:N\\left(p\\right)={N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2k+l+2\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right){\\left(2\\right)}^{2k+l+1}}.\\frac{1}{{p}^{2k+l+2}}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e,\u003c/p\u003e\n\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e$$\\:N\\left(p\\right)=\\:{N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2k+l+2\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right){\\left(2\\right)}^{2k+l+1}}.\\frac{1}{{p}^{2k+l+2}}\\sum\\:_{r=0}^{\\infty\\:}\\frac{{\\left(n\\right)}_{r}}{\\left(r\\right)!}\\:\\:{(-{e}^{v})}^{r}{p}^{-vr}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e.\u003c/p\u003e\n\u003cp\u003eTaking inverse Laplace transform we have,\u003c/p\u003e\n\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)={N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2k+l+2\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right){\\left(2\\right)}^{2k+l+1}}.\\sum\\:_{r=0}^{\\infty\\:}\\frac{{\\left(n\\right)}_{r}}{\\left(r\\right)!}\\:\\:{(-{e}^{v})}^{r}\\:{L}^{-1}\\left\\{{p}^{-\\left(vr+2k+l+2\\right)}\\right\\}$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003e,\u003c/p\u003e\u003cp\u003ewhich in view of Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1.22\u003c/span\u003e) gives\u003c/p\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)={N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2k+l+2\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right){\\left(2\\right)}^{2k+l+1}}.\\sum\\:_{r=0}^{\\infty\\:}\\frac{{\\left(n\\right)}_{r}}{\\left(r\\right)!}\\:\\:{(-{e}^{v})}^{r}\\frac{{t}^{vr+2k+l+1}}{Г(vr+2k+l+2)}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e.\u003c/p\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eOn interpreting the resulting series with the help of Eq.\u0026nbsp;(1.15) we at once arrive at the solution in Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e2.3\u003c/span\u003e).\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 2\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:e,\\:t,v\\:\\in\\:{R}^{+}\\:,\\:a,b,c,l\\:\\in\\:C\\:and\\:R\\left(l\\right)\u0026gt;-1,\\:\\)\u003c/span\u003e\u003c/span\u003e then the solution of the equation\u003c/p\u003e\n\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)-{N}_{o}{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left({e}^{v}{t}^{v}\\right)=-\\left\\{\\sum\\:_{r=1}^{n}\\left(\\begin{array}{c}n\\\\\\:r\\end{array}\\right){\\left({e}^{v}\\right)}^{r}{{}_{0}D}_{t}^{-vr}\\right\\}\\:N\\left(t\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2.4\u003c/div\u003e\u003c/div\u003e\u003cp\u003e,\u0026hellip;\u003c/p\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003ei.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\left(1+{e}^{v}{{}_{0}D}_{t}^{-v}\\right)}^{n}N\\left(t\\right)={N}_{0}\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left({e}^{v}{t}^{v}\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u0026hellip;(2.5)\u003c/p\u003e\u003c/div\u003e\u003cp\u003eis given by\u003c/p\u003e\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)={N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2kv+lv+v+1\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right)}\\:{\\left(\\frac{{e}^{v}{t}^{v}}{2}\\right)}^{2k+l+1}.{E}_{v,\\left(2k+l+1\\right)v+1}^{n}\\left({-e}^{v}{t}^{v}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2.6\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e.\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProof\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe can derive parallel as theorem \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 3\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:e,\\:t,v\\:\\in\\:{R}^{+}\\:,\\:a,b,c,l\\:\\in\\:C\\:and\\:R\\left(l\\right)\u0026gt;-1\\:,\\:a\\ne\\:e,\\)\u003c/span\u003e\u003c/span\u003e then the solution of the equation\u003c/p\u003e\n\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)-{N}_{o}{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left({e}^{v}{t}^{v}\\right)=-\\left\\{\\sum\\:_{r=1}^{n}\\left(\\begin{array}{c}n\\\\\\:r\\end{array}\\right){\\left({a}^{v}\\right)}^{r}{{}_{0}D}_{t}^{-vr}\\right\\}\\:N\\left(t\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2.7\u003c/div\u003e\u003c/div\u003e\u003cp\u003e,\u0026hellip;\u003c/p\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003ei.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\left(1+{a}^{v}{{}_{0}D}_{t}^{-v}\\right)}^{n}N\\left(t\\right)={N}_{0}\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{}_{a}W}_{p,b,c,\\:\\:\\xi\\:}^{\\alpha\\:,\\mu\\:}\\left({e}^{v}{t}^{v}\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u0026hellip;(2.8)\u003c/p\u003e\u003c/div\u003e\u003cp\u003eis given by\u003c/p\u003e\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e$$\\:N\\left(t\\right)={N}_{0}\\sum\\:_{k=0}^{\\infty\\:}\\frac{{(-c)}^{k}Г\\left(2kv+lv+v+1\\right)}{Г\\left(\\propto\\:k+\\mu\\:\\right)Г\\left(ak+\\frac{l}{\\xi\\:}+\\frac{b+2}{2}\\right)}\\:{\\left(\\frac{{e}^{v}{t}^{v}}{2}\\right)}^{2k+l+1}.{E}_{v,\\left(2k+l+1\\right)v+1}^{n}\\left({-a}^{v}{t}^{v}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2.9\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e.\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProof\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe can derive parallel as theorem \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSpecial Cases\u003c/strong\u003e:\u003c/p\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003col style=\"list-style-type: lower-roman;\"\u003e\n \u003cli\u003eIf we take \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r=1\\)\u003c/span\u003e\u003c/span\u003e in theorem \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e we at once arrive the known result of [18, pp. 9, Eq. 34, 38].\u003c/li\u003e\n \u003cli\u003eIf we take \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r=1\\)\u003c/span\u003e\u003c/span\u003e in theorem \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e we at once arrive the known result of [18, pp. 9, Eq. 39, 40]\u003c/li\u003e\n \u003cli\u003eIf we take \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r=1\\)\u003c/span\u003e\u003c/span\u003e in theorem \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e we at once arrive the known result of [18, pp. 9, Eq. 41, 42].\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003c/p\u003e"},{"header":"3. Conclusion","content":"\u003cp\u003eThe present study investigates fractional kinetic equations involving the generalized Galu\u0026eacute;-type Struve function. The solutions to these generalized fractional kinetic equations are obtained using the Laplace transform. It is observed that the Laplace transform provides an effective method for solving such equations, with several applications and explicit solutions derived. Notably, the results presented in Theorem \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, Theorem \u003cspan refid=\"FPar3\" class=\"InternalRef\"\u003e2\u003c/span\u003e, and Theorem \u003cspan refid=\"FPar5\" class=\"InternalRef\"\u003e3\u003c/span\u003e are expressed in terms of the Mittag-Leffler function. Furthermore, to illustrate the behavior and properties of the solutions, graphical representations have been provided using MATLAB (Fig.\u0026nbsp;(1\u0026ndash;3)). These visualizations support the analytical findings and offer insight into the dynamic nature of the solutions. This work opens the door for further research, where other researchers may explore alternative integral transforms and special functions to derive new or existing forms of fractional kinetic equations.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003ch2\u003eConflict of Interest:\u003c/h2\u003e\u003cp\u003eThe authors declare that they have no conflict of interest\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eFunding:\u003c/h2\u003e\u003cp\u003eNo funds, grants, or other support was received from funding agencies except host\u003c/p\u003e\u003cp\u003eUniversity\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBhowmick KN (1962) Some relations between a generalized Struve\u0026rsquo;s function and hypergeometric functions. Vijnana Parishad Anusandhan Patrika 5:93\u0026ndash;99\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBhowmick KN (1993) A generalized Struve\u0026rsquo;s function and its recurrence formula. Vijnana Parishad Anusandhan Patrika 6:1\u0026ndash;11\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChaurasia VBL, Kumar D (2010) On the solutions of generalized fractional kinetic equations. Adv stud Theor Phy 4:773\u0026ndash;780\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChaurasia VBL, Pandey SC (2008) 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 317:213\u0026ndash;219\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChoi j, Kumar D (2015) Solutions of generalized fractional kinetic equations involving Aleph functions. Math Commun 20:113\u0026ndash;123\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChouhan A, Sarswat S (2012) On solution of generalized kinetic equation of fractional order. Int J Math Sci Appl 2(2):813\u0026ndash;818\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChouhan A, Purohit SD, Saraswat S (2013) An alternative method for solving generalized differential equations of fractional order kragujevac. J Math 37(2):299\u0026ndash;306\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eErdelyi A, Magnus W, Oberhettinger F, Tricomi FG (1954) Tables of Integral Transforms, Vol.-I. McGraw-Hill Book, New York, Toronto and London\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eGupta A, Parihar CL (2014) On solution of generalized kinetic equations of fractional order. Bol Soc Paran Mat 32(1):181\u0026ndash;189\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHaubold HJ, Mathai AM, Haubold HJ (2002) On fractional kinetic equations Astrophys. Space Sci 282:281\u0026ndash;287\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eJaimini BB, Saxena H (2007) Solutions of certain fractional differential equations. J Indian Acad Math 29(1):223\u0026ndash;236\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKanth BN (1981) Integrals involving generalized Struve\u0026rsquo;s function. Nepali Math Sci Rep 6:61\u0026ndash;64\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKumar D, Purohit SD, Secer A, Atangana A (2015) On generalized fractional kinetic equations involving generalized Bessel function of the first kind. Math Probl Eng 289387\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMiller KS, Ross B (1993) An Introduction to the Fractional Calculus and Fractional Differential Equation. A Willey Inter Science Publication, John Wiley \u0026amp; Sons, New York\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMittag-Leffler GM (1905) Sur la representation analytiqie d\u0026rsquo;une function monogene cinquieme note. Acta Math 29:101\u0026ndash;181\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNisar KS, Atangana A (2016) Exact solution of fractional kinetic equation in terms of Struve functions. (submitted)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNisar KS, Purohit SD, Mondal SR (2016b) Generalized fractional kinetic equations involving generalized Struve function of the first kind. J King Saud Univ Sci 28(2):167\u0026ndash;171\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNisar KS, Baleanu D, Qurashi MA (2016) Fractional calculus and application of generalized Struve function. SpringerPlus 5:910\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eOrhan H, Yagmur N (2013) Starlikeness and convexity of generalized Struve functions. Abstr Appl Anal Art ID 954513:6\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eOrhan H, Yagmur N (2014) Geometric properties of generalized Struve functions. Ann Alexandru loan Cuza Univ \u0026ndash;Math. Doi: 10. 2478/aicu-2014-0007\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePrabhakar TR (1971) A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J 19:7\u0026ndash;15\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eRainville ED (1960) Special Functions. Macmillan, New York\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSaichev A, Zaslavsky M (1997) Fractional kinetic equations: solutions and applications. Chaos 7:753\u0026ndash;764\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSaxena RK, Mathai AM, Haubold H (2002) J.,On fractional Kinetic equations. Astrophys Space Sci 282:281\u0026ndash;287\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSaxena RK, Mathai AM, Haubold H (2004) J.,On generalized fractional kinetic equations. Phys A 344:657\u0026ndash;664\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSaxena RK, Mathai AM, Haubold HJ (2006) Solution of generalized fractional rectional diffusion equations. Astrophys Space Sci 305:305\u0026ndash;313\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSaxena RK, Kalla SL (2008) On the solutions of certain fractional kinetic equations. Appl Math Comput 199:504\u0026ndash;511\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh RP (1974) Generalized Struve\u0026rsquo;s function and its recurrence relations. Ranchi Univ Math J 5:67\u0026ndash;75\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh RP (1985) Generalized Struve\u0026rsquo;s function and its recurrence equation. Vijnana Parishad Anusandhan Patrika 28:287\u0026ndash;292\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh RP (1988) Some integral representation of generalized Struve function. Math Ed (Siwan) 22:91\u0026ndash;94\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh RP (1988) On definite integrals involving generalized Struve\u0026rsquo;s function. Math Ed (Siwan) 22:62\u0026ndash;66\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh RP (1989) Infinite integrals involving generalized Struve function. Math Ed (Siwan) 23:30\u0026ndash;36\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSrivastava HM, Saxena RK (2001) Operators of fractional integration and their applications. Appl Math Comput 118:1\u0026ndash;52\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWiman A (1905) Uber den fundamental satz in der theorie der funktionen \u003cdiv id=\"IEq55\" class=\"InlineEquation\"\u003e\u003cdiv format=\"TEX\" class=\"mathinline\" id=\"FileID_IEq55\" name=\"EquationSource\"\u003e\u003cscript type=\"math/tex; mode=inline\"\u003e\\:{E}_{\\alpha\\:}\\left(z\\right)\u003c/script\u003e\u003c/div\u003e\u003c/div\u003e. Acta Math 29:191\u0026ndash;201\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZaslavsky GM (1994) Fractional kinetic equation for Hamiltonian chaos. Phys D. 76, 110 \u0026ndash;\u0026thinsp;122\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Fractional Kinetic equations, Laplace transform, Generalized Galué type Struve function, Inverse Laplace transform","lastPublishedDoi":"10.21203/rs.3.rs-7430855/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7430855/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we propose a novel and comprehensive method for solving generalized fractional kinetic equations that incorporate the generalized Galu\u0026eacute;-type Struve function. These equations are significant due to their wide-ranging applications in various fields of applied mathematical sciences, including physics, engineering, and biological modeling. By employing the Laplace transform technique, we successfully derive analytical solutions to these complex equations, offering a deeper understanding of the dynamic behaviors described by fractional models. The generalized Galu\u0026eacute;-type Struve function used in our formulation enables a more flexible representation of physical phenomena, extending the applicability of traditional Struve-based models. Our approach not only generalizes several known results but also introduces new solution forms for previously unexamined equations. Further, to illustrate the behavior of these solutions, we provide graphical visualizations generated using the latest version of MATLAB. These visual aids support the analytical findings and offer practical insights for researchers\u003c/p\u003e","manuscriptTitle":"Solutions for Fractional Kinetic Equations withGeneralized Galue Type Struve Functions","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-25 05:31:01","doi":"10.21203/rs.3.rs-7430855/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"cf55ff8f-c778-4e39-a30a-9066041a8b35","owner":[],"postedDate":"August 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":53545614,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2025-08-25T05:31:18+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-25 05:31:01","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7430855","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7430855","identity":"rs-7430855","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}