Article ID: MTJPAM-D-23-00033

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

aDepartment of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

bDepartment of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

cCenter for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea

dDepartment of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City 320314, Taiwan, Republic of China

eDepartment of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

fSection 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:

Full Text: PDF


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)

  1. 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.
  2. E. W. Barnes, The asymptotic expansion of integral functions defined by Taylor’s series, Philos. Trans. Roy. Soc. A 206, 249–297, 1906.
  3. M. Caputo, Elasticità e dissipazionne, Zanichelli, Bologna, 1969.
  4. 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.
  5. 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.
  6. 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.
  7. H. J. Haubold and A. M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci. 273, 53–63, 2000.
  8. H. Jafari, A new general integral transform for solving integral equations, J. Adv. Res. 32, 133–138, 2021.
  9. F. S. Khan and M. Khalid, Fareeha transform: A new generalized Laplace transform, Math. Meth. Appl. Sci. 46, 11043–11057, 2023.
  10. A. A. Kilbas and M. Saigo, H-transforms: Theory and applications, 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.
  11. 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.
  12. A. Kumar, S. Bansal and S. Aggarwal, A new novel integral transform “Anuj transform" with application, Design Engrg. 2021, 12741–12751, 2021.
  13. 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.
  14. R. Kumar, J. Chandel and S. Aggarwal, A new integral transform “Rishi transform" with application, J. Sci. Res. 14, 521–532, 2022.
  15. 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.
  16. 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.
  17. N. W. McLachlan, Modern operational calculus with applications in technical mathematics, Macmillan Book Company, London and New York, 1948.
  18. 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.
  19. 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.
  20. H. M. Srivastava, Charles Fox, Bull. London Math. Soc. 12, 67–70, 1980.
  21. H. M. Srivastava, A new family of the λ-generalized Hurwitz-Lerch zeta functions with applications, Appl. Math. Inform. Sci. 8, 1485–1500, 2014.
  22. H. M. Srivastava, The Zeta and related functions: Recent developments, J. Adv. Engrg. Comput. 3, 329–354, 2019.
  23. 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.
  24. H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J. 60, 73–116, 2019.
  25. 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.
  26. 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.
  27. 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.
  28. H. M. Srivastava, Some general families of integral transformations and related results, Appl. Math. Comput. Sci. 6, 27–41, 2022.
  29. H. M. Srivastava and J. Choi, Series associated with the Zeta and related functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  30. H. M. Srivastava and J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  31. 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.
  32. 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.
  33. 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.
  34. 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.
  35. 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.
  36. X.-J. Yang, Theory and applications of special functions for scientists and engineers, Springer Nature Singapore Private Limited, Singapore, 2021.