AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026030010
Submit to IJOCTA
Cite this article
8
Download
28
Views
Journal Browser
Volume | Year
Issue
Search
Forthcoming Issue
Current Issue
View All
News and Announcements
View All
RESEARCH ARTICLE

On the integral transform of generalized k-Hilfer–Prabhakar fractional derivative with applications to fractional type advection–dispersion equations

Ved Prakash Dubey1 ,  Jagdev Singh2∗ ,  Jitesh Pati Tripathi3 ,  Sarvesh Dubey4 ,  Dumitru Baleanu5 ,  Devendra Kumar6
Show Less
1 Department of Bachelor of Computer Application, Faculty of Science, Laxmi Narain Dubey College (Affiliated to Babasaheb Bhimrao Ambedkar Bihar University), Motihari, Bihar, India
2 Department of Mathematics, School of Sciences, JECRC University, Jaipur, Rajasthan, India
3 University Department of Mathematics, Faculty of Science, Babasaheb Bhimrao Ambedkar Bihar University, Muzaffarpur, Bihar, India
4 University Department of Physics, Faculty of Science, Babasaheb Bhimrao Ambedkar Bihar University, Muzaffarpur, Bihar, India
5 Department of Computer Science and Mathematics, School of Arts and Sciences, Lebanese American University, Beirut, Lebanon
6 Department of Mathematics, Faculty of Science, University of Rajasthan, Jaipur, Rajasthan, India
IJOCTA, 026030010 https://doi.org/10.36922/IJOCTA026030010
Received: 18 January 2026 | Revised: 1 March 2026 | Accepted: 4 March 2026 | Published online: 30 April 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Download PDF
XML
Cite
Abstract

The Hilfer-Prabhakar (HP) derivatives are advance class of fractional operators with nonlocal features, memory effects, and nonexponential decay. These derivatives are generally used in study of  heterogeneous systems, anomalous diffusion, dielectric spectroscopy, free electron lasers, fractional heat equations, and Cauchy problems. In this study, we derive the Kharrat–Toma (KT) transforms of the kernel function of the k-Prabhakar integral (k-PI), k-Prabhakar fractional derivative, its regularized form, and the k-Hilfer–Prabhakar fractional derivative (k-HPFD). We also establish the relationship between the k-Prabhakar fractional derivative and its regularized form for an absolutely continuous function using KT transform operations. Similarly, the KT transforms of k -HPFD and its regularized variant have been computed. Moreover, we establish the relationship between the k-HPFD and its regularized form using KT transform operations. Furthermore, we present solutions to Cauchy problems and generalized Cauchy problems for a fractional heat model involving the k -HPFD using the KT transform combined with the Fourier transform. Finally, we propose an integral technique combining the KT and Fourier transforms to construct solutions for the fractional advection–dispersion equation governed by the k -HPFD. The solutions of the Cauchy problems and advection–dispersion equations involving the k -HPFD and its Caputo form are expressed in terms of a generalized Mittag-Leffler function through sequential application of integral transform techniques. It is observed that the solutions for the generalized Cauchy problems and advection–dispersion equations involving the k -HPFD operator reduce to those involving the Hilfer fractional derivative, Riemann–Liouville fractional derivative, and Caputo derivative for specific parameter values.

Keywords
Generalized Mittag-Leffler function
k-Hilfer–Prabhakar fractional derivative
Cauchy equations
Kharrat–Toma transform
Funding
None.
Conflict of interest
The authors declare they have no competing interests.
References
  1. Carr, E. J. New Semi-analytical Solutions for Advection–dispersion Equations in Multilayer Porous Media. Transp Porous Med. 2020;135:39- 58. https://www.doi.org/10.1007/s11242-020-01468-z
  2. Suzuki, J. L., Gulian, M., Zayernouri, M, & D’Elia, M. Fractional Modeling in Action: a Sur- vey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials. J Peridyn Nonlocal Model. 2023;5:392-459. https://www.doi.org/10.1007/s42102-022-00085- 2
  3. Zheng, X., Ervin, V. J, & Wang, H. Analysis and Petrov–Galerkin Numerical Approximation for Variable Coefficient Two-sided Fractional Diffu- sion, Advection, Reaction Equations. J Comput Appl Math. 2023;425:115033. https://www.doi.org/10.1016/j.cam.2022.115033
  4. Garra, R., Garrappa, R. The Prabhakar or Three- parameter Mittag–Leffler Function: Theory and Application. Commun Nonlinear Sci Numer Simul. 2018;56:314-329. https://www.doi.org/10.1016/j.cnsns.2017.08.018
  5. Giusti, A., Colombaro, I., Garra, R, Garrappa, R., Polito F., Popolizio, M., et al. A Practical Guide to Prabhakar Fractional Calculus. Fract Calc Appl Anal. 2020;23:9-54. https://www.doi.org/10.1515/fca-2020-0002
  6. Garrappa, R., Kaslik, E. Stability of Fractional- order Systems with Prabhakar Derivatives. Non- linear Dyn. 2020;102:567-578. https://www.doi.org/10.1007/s11071-020-05897- 9
  7. Kilbas, A. A., Saigo, M., & Saxena, R. K. Gen- eralized Mittag-Leffler Function and Generalized Fractional Calculus Operators. Integral Trans- forms Spec Funct. 2004;15(1):31-49. https://www.doi.org/10.1080/10652460310001600717
  8. Garra, R., Gorenflo, R., Polito, F., & Tomovski, Z. Hilfer–Prabhakar Derivatives and Some Applications. Appl Math Comput. 2014;242:576-589. https://www.doi.org/10.1016/j.amc.2014.05.129
  9. Tomovski, Z., Hilfer, R., & Srivastava, H. M. Frac- tional and Operational Calculus with Generalized Fractional Derivative Operators and Mittag- Leffler Type Functions. Integral Transforms Spec Funct. 2010;21(11):797-814. https://www.doi.org/10.1080/10652461003675737
  10. Polito, F., Tomovski, Z. Some Properties of Prabhakar-type Fractional Calculus. Fract Dif- fer Calc. 2016;6(1):73-94. https://www.doi.org/10.7153/fdc-06-05
  11. Sandev, T. Generalized Langevin Equation and the Prabhakar Derivative. Mathematics. 2017;5(4):66. https://www.doi.org/10.3390/math5040066
  12. Tomovski, Z., Dubbeldam, J. L. A., & Korbel, J. Applications of Hilfer–Prabhakar Operator to Option Pricing Financial Model. Fract Calc Appl Anal. 2020;23:996-1012. https://www.doi.org/10.1515/fca-2020-0052
  13. Samraiz, M., Perveen, Z., Rahman, G., Nisar, K. S., & Kumar, D. On the (k,s)-Hilfer–Prabhakar Fractional Derivative with Applications to Math- ematical Physics. Front Phys. 2020;8:309. https://www.doi.org/10.3389/fphy.2020.00309
  14. Sousa J. D. V., de Oliveira, E. C. On the Ψ- Hilfer Fractional Derivative, Commun Nonlinear Sci Numer Simul. 2018;60:72–91. https://www.doi.org/10.1016/j.cnsns.2018.01.005
  15. Nisar, K. S., Rahman, G., Baleanu, D., Mubeen, S., & Arshad, M. The (k, s)-Fractional Calculus of k -Mittag-Leffler Function. Adv Differ Equat. 2017;2017:118. doi: 10.1186/s13662-017-1176-4
  16. Magar, S. K., Dole, P. V., Ghadle, & K. P. Prab- hakar and Hilfer–Prabhakar Fractional Deriva- tives in the Setting of Ψ-Fractional Calcu- lus and its Applications. Kragujevac J Math. 2024;48(4):515-533. https://www.doi.org/10.46793/KgJMat2404.515M
  17. Samraiz, M., Umer, M., Naheed, S., & Baleanu, D. On Weighted Fractional Operators with Applica- tions to Mathematical Models Arising in Physics. In: Singh, J., Anastassiou, G. A., Baleanu, D., Kumar, D., eds. Advances in Mathematical Mod elling, Applied Analysis and Computation. Lecture Notes in Networks and Systems. 2023;666, Springer. https://www.doi.org/10.1007/978-3-031-29959-93
  18. Rahman, G., Mubeen, S., & Nisar, K. S. On Generalized k -Fractional Derivative Opera- tor. AIMS Math. 2020;5(3):1936–1945. Doi: 10.3934/math.2020129
  19. Marynets, K., Tomovski, Z. Fractional Periodic Boundary Value and Cauchy Problems with Hilfer–Prabhakar Operator. Comput Appl Math. 2024;43:130. https://www.doi.org/10.1007/s40314-024-02644-3
  20. Feng, Y.-Y., Yang, X.-J., Liu, J.-G., & Chen, Z.-Q. Study on the Generalized k -Hilfer–Prabhakar Fractional Viscoelastic–plastic Model. Math Mech Solids. 2021;27(3). https://www.doi.org/10.1177/10812865211025813
  21. Sucu, D. Y., Karako¸c, S. B. G., & Gu¨ng¨or M. Investigation on the New Numerical Soli- ton Solutions of FitzHugh–Nagumo Equation with Collocation Method. Fundam J Math Appl. 2025;8(3):180-186. https://www.doi.org/10.33401/fujma.1617563
  22. Kumar, V. On a Generalized Mittag-Leffler Function and Fractional Integrals. Fundam J Math Appl. 2024;7(1):12-25. https://www.doi.org/10.33401/fujma.1378534
  23. Natalini, P., Caratelli, D., & Ricci, P. E. A Note on Fractional Parametric-type Laplace Trans- forms. Adv Anal Appl Math. 2025;2(2):89-106. https://www.doi.org/10.62298/advmath.33
  24. Sawar, S., Hussain, S., Ayaz, M. Secure Im- age Transmission Using Fractional Variable Or- der Memristive Hyperchaotic System with Non- linear Synchronization. Adv Anal Appl Math. 2025;2(1):32-43. https://www.doi.org/10.62298/advmath.30
  25. Reynolds, R. Double and Single Integrals of the Mittag-Leffler Function: Derivation and Evalua- tion. Adv Anal Appl Math. 2025;2(1):58-72. https://www.doi.org/10.62298/advmath.24
  26. Ma, Y. K., Raja, M. M., Shukla, A., Vijayakumar, V., Nisar, K. S., & Thilagavathi, K. New Results on Approximate Controllability of Fractional De- lay Integrodifferential Systems of Order 1<r<2 with Sobolev-type. Alex Eng J. 2023;81:501-518. https://www.doi.org/10.1016/j.aej.2023.09.043
  27. Ma, Y. K., Raja, M. M., Nisar, K. S., Shukla, A., & Vijayakumar, V. Results on Controllability for Sobolev Type Fractional Differential Equations of Order 1<r<2 with Finite Delay. AIMS Math. 2022;7(6):10215-10233. https://www.doi.org/10.3934/math.2022568
  28. Panchal SK, Khandagale AD, Dole PV. Sumudu Transform of Hilfer-Prabhakar Fractional Derivatives and Applications. Preprint posted online 2016. https://www.doi.org/10.48550/arXiv.1608.08017
  29. Dhillon, M., Vishwakarma, P., & Kataria, K. K. Fractional Linear Birth-death Process Involving Hilfer-Prabhakar Derivative, Stochastics. 2026:1- 28. https://www.doi.org/10.1080/17442508.2025.2609661
  30. Mashoof, M., Sheikhani, A. H. R., & Najafi, H. S. Stability Analysis of Distributed-Order Hilfer–Prabhakar Systems Based on Inertia The- ory. Math Notes. 2018;104:74–85. https://www.doi.org/10.1134/S000143461807009X
  31. Dubey, V. P., Singh, J., Dubey, S., & Kumar, D. Analysis of Cauchy Problems and Diffusion Equations Associated with the Hilfer–Prabhakar Fractional Derivative via Kharrat–Toma Trans- form. Fractal Fract. 2023;7(5):413. https://www.doi.org/10.3390/fractalfract7050413
  32. Dorrego, G. A. Generalized Riemann-Liouville Fractional Operators Associated with a General- ization of the Prabhakar Integral Operator. Progr Fract Differ Appl. 2016;2(2):131-140. https://www.doi.org/10.18576/pfda/020206
  33. Panchal SK, Khandagale AD, Dole PV. K-Hilfer- Prabhakar Fractional Derivatives and Applica- tions. Preprint posted online 2016. https://www.doi.org/10.48550/arXiv.1609.05696
  34. Kharrat, B. N., Toma, G. A. A New Integral Transform: Kharrat-Toma Transform and its Properties. World Appl Sci J. 2020;38(5):436-443.
  35. Schumer, R., Benson, D. A., Meerschaert, M. M., & Wheatcraft, S. W. Eulerian Derivation of the Fractional Advection-dispersion Equation. J Con- tam Hydrol. 2001;48(1-2):69-88. https://www.doi.org/10.1016/S0169-7722(00)00170-4
  36. Debnath, L., Bhatta, D. Integral Transforms and Their Applications. 2nd ed. Chapman and Hall/CRC; 2007. https://www.doi.org/10.1201/9781420010916
  37. Hilfer, R. Fractional Calculus and Regular Varia- tion in Thermodynamics. In: Hilfer, R., ed. Appli- cations of Fractional Calculus in Physics. World Scientific; 2000:429-463. https://www.doi.org/10.1142/97898128177470009
  38. Prabhakar, T. R. A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel. Yokohama Math J. 1971;19:7-15.
  39. Chu, Y.-M., Inc, M., Hashemi, M. S, Eshaghi, S., Analytical Treatment of Regularized Prab- hakar Fractional Differential Equations by Invari- ant Subspaces. Comput Appl Math. 2022;41:271. https://www.doi.org/10.1007/s40314-022-01977-1
  40. Diaz, R., Pariguan, E. On Hypergeometric Func- tions and k -Pochhammer Symbol. Divulg Math. 2007;15(2):179-192.
  41. Dorrego, G. A., Cerutti, R. A. The k -Mittag- Leffler Function. Int J Contemp Math Sci. 2012;7(15):705-716.
  42. Dorrego, A.,  Cerutti,  R.  A.  The  k-Fractional Hilfer Derivative. Int J Math Anal. 2013;7(11):543-550. https://www.doi.org/10.12988/ijma.2013.13051
  43. Sarikaya, M. Z., Karaca, A. On the k -Riemann- Liouville Fractional Integral and Applications. Int J Stat Math. 2014;1(3):33-43. https://www.doi.org/10.15672/HJ20164512484
  44. Dattoli, G., Giannessi, L., Mezi, L., Torre, A., & Caloi, R. FEL Time-evolution Operator. Nucl Instrum Methods Phys Res A. 1991;304(1-3):541-544. https://www.doi.org/10.1016/0168-9002(91)90926-H
  45. Miller, K. S. The Weyl Fractional Calculus. In: Fractional Calculus and Its Applications. Springer;1975:80-89. https://www.doi.org/10.1007/BFb0067098
  46. Saxena, R. K., Mathai, A. M., Haubold, H. J. Reaction-diffusion Systems and Nonlinear Waves. Astrophys Space Sci. 2006;305:297-303. https://www.doi.org/10.1007/s10509-006-9190- 0
  47. Agarwal, R., Jain, S., Agarwal, R. P. Analytic Solution of Generalized Space Time Advection- dispersion Equation with Fractional Laplace Op- erator. J Nonlinear Sci Appl. 2016;9:3545-3554. https://www.doi.org/10.22436/jnsa.009.06.09
Next article in this issue
Share
Back to top
An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept