**Title:** A general family of hybrid-type fractional-order kinetic equations involving the Liouville-Caputo fractional derivative

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-23-00033; **Volume 6 / Issue 3 / Year 2024**, Pages 48-61

**Document Type:** Research Paper

**Author(s):** Hari Mohan Srivastava ^{a, b, c, d, e, f}

^{a}Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

^{b}Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

^{c}Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea

^{d}Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City 320314, Taiwan, Republic of China

^{e}Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

^{f}Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy

Received: 17 October 2023, Accepted: 17 October 2023, Published: 28 October 2023

**Corresponding Author:** Hari Mohan Srivastava (Email address: harimsri@math.uvic.ca)

**Full Text:** PDF

**Abstract**

This article is motivated essentially by the fact that, in the current literature, many different kinds of operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) have been and continue to be successfully applied in the modeling and analysis of a considerably large number of applied scientific and real-world problems in the mathematical, physical, biological, engineering and statistical sciences, and indeed in other scientific disciplines as well. We aim here at investigating a general family of hybrid-type fractional-order kinetic equations, which is associated with the Liouville-Caputo (LC) fractional derivative. Our main results are sufficiently general in character and are capable of furnishing solutions of a remarkably large number of simpler fractional-order kinetic equations.

**Keywords:** Riemann-Liouville and related fractional integral and fractional derivative operators, Liouville-Caputo fractional derivative operator, Hypergeometric functions, special (or higher transcendental) functions, Fox-Wright hypergeometric function, Mittag-Leffler type functions, general Fox-Wright function, Zeta and related functions, Lerch transcendent (or Hurwitz-Lerch zeta function)

**References:**

- S. Bansal, A. Kumar and S. Aggarwal,
*Application of Anuj transform for the solution of Bacteria growth model*, GIS Sci. J.**9**, 1465–1472, 2022. - E. W. Barnes,
*The asymptotic expansion of integral functions defined by Taylor’s series*, Philos. Trans. Roy. Soc. A**206**, 249–297, 1906. - M. Caputo,
*Elasticità e dissipazionne*, Zanichelli, Bologna, 1969. - A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,
*Higher transcendental functions*(Volume I), McGraw-Hill Book Company, New York, Toronto and London, 1953. - A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,
*Tables of integral transforms*(Volume II), McGraw-Hill Book Company, New York, Toronto and London, 1954. - R. Gorenflo, F. Mainardi and H. M. Srivastava,
*Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena*, In: Proceedings of the Eighth International Colloquium on Differential Equations (Ed. by D. Bainov), (Plovdiv, Bulgaria; August 18–23, 1997), pp. 195–202, VSP Publishers, Utrecht and Tokyo, 1998. - H. J. Haubold and A. M. Mathai,
*The fractional kinetic equation and thermonuclear functions*, Astrophys. Space Sci.**273**, 53–63, 2000. - H. Jafari,
*A new general integral transform for solving integral equations*, J. Adv. Res.**32**, 133–138, 2021. - F. S. Khan and M. Khalid,
*Fareeha transform: A new generalized Laplace transform*, Math. Meth. Appl. Sci.**46**, 11043–11057, 2023. - A. A. Kilbas and M. Saigo,
, Analytical Methods and Special Functions: An International Series of Monographs in Mathematics (Volume 9), Chapman and Hall (A CRC Press Company), Boca Raton, London and New York, 2004.*H*-transforms: Theory and applications - A. A. Kilbas, H. M. Srivastava and J. J. Trujillo,
*Theory and applications of fractional differential equations*(Volume 204), North-Holland Mathematical Studies , Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. - A. Kumar, S. Bansal and S. Aggarwal,
*A new novel integral transform “Anuj transform" with application*, Design Engrg.**2021**, 12741–12751, 2021. - D. Kumar, J. Choi and H. M. Srivastava,
*Solution of a general family of kinetic equations associated with the Mittag-Leffler function*, Nonlinear Funct. Anal. Appl.**23**, 455–471, 2018. - R. Kumar, J. Chandel and S. Aggarwal,
*A new integral transform “Rishi transform" with application*, J. Sci. Res.**14**, 521–532, 2022. - J. Liouville,
*Mémoire sur quelques quéstions de géometrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces quéstions*, J. Éc. polytech. Math.**13 (21)**, 1–69, 1832. - M. Meddahi, H. Jafari and X.-J. Yang,
*Towards new general double integral transform and its applications to differential equations*, Math. Meth. Appl. Sci.**45**, 1915–1933, 2021. - N. W. McLachlan,
*Modern operational calculus with applications in technical mathematics*, Macmillan Book Company, London and New York, 1948. - I. Podlubny,
*Fractional differential equations*:*An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications*, Mathematics in Science and Engineering (Volume 198), Academic Press, New York, London, Sydney, Tokyo and Toronto, 1999. - S. G. Samko, A. A. Kilbas and O. I. Marichev,
*Fractional integrals and derivatives: Theory and applications*, Gordon and Breach Science Publishers, Yverdon (Switzerland), 1993. - H. M. Srivastava,
*Charles Fox*, Bull. London Math. Soc.**12**, 67–70, 1980. - H. M. Srivastava,
*A new family of the*, Appl. Math. Inform. Sci.*λ*-generalized Hurwitz-Lerch zeta functions with applications**8**, 1485–1500, 2014. - H. M. Srivastava,
*The Zeta and related functions: Recent developments*, J. Adv. Engrg. Comput.**3**, 329–354, 2019. - H. M. Srivastava,
*Some general families of the Hurwitz-Lerch Zeta functions and their applications: Recent developments and directions for further researches*, Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerbaijan**45**, 234–269, 2019. - H. M. Srivastava,
*Fractional-order derivatives and integrals: Introductory overview and recent developments*, Kyungpook Math. J.**60**, 73–116, 2019. - H. M. Srivastava,
*An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions*, J. Adv. Engrg. Comput.**5**, 135–166, 2021. - H. M. Srivastava,
*Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations*, J. Nonlinear Convex Anal.**22**, 1501–1520, 2021. - H. M. Srivastava,
*A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics*, Symmetry**13**, 2021; Article ID: 2294. - H. M. Srivastava,
*Some general families of integral transformations and related results*, Appl. Math. Comput. Sci.**6**, 27–41, 2022. - H. M. Srivastava and J. Choi,
*Series associated with the Zeta and related functions*, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001. - H. M. Srivastava and J. Choi,
*Zeta and*, Elsevier Science Publishers, Amsterdam, London and New York, 2012.*q*-Zeta functions and associated series and integrals - H. M. Srivastava and P. W. Karlsson,
*Multiple Gaussian hypergeometric series*, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. - H. M. Srivastava, R. K. Saxena, T. K. Pogány and R. Saxena,
*Integral and computational representations of the extended Hurwitz-Lerch Zeta function*, Integral Transforms Spec. Funct.**22**, 487–506, 2011. - E. M. Wright,
*The asymptotic expansion of integral functions defined by Taylor series. I*, Philos. Trans. Roy. Soc. London Ser. A Math. Phys. Sci.**238**, 423–451 1940. - E. M. Wright,
*The asymptotic expansion of integral functions defined by Taylor series. II*,*Philos. Trans. Roy. Soc. London Ser. A Math. Phys. Sci.***239**, 217–232, 1941. - E. M. Wright,
*The asymptotic expansion of integral functions and of the coefficients in their Taylor series*, Trans. Amer. Math. Soc.**64**, 409–438, 1948. - X.-J. Yang,
*Theory and applications of special functions for scientists and engineers*, Springer Nature Singapore Private Limited, Singapore, 2021.