UDC 517.9:519.6

ON SOME GENERALIZATIONS OF THE BI-ORDINAL
HILFER’S FRACTIONAL DERIVATIVE

REFERENCES

1. Podlubny I. Fractional differential equations. New York: Academic Press, 1999. 341 p.


2. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations. Amsterdam: Elsevier, 2006. 523 p.


3. Sandev T., Tomovsky Z. Fractional equations and models. Theory and applications. Cham, Switzerland: Springer Nature Switzerland AG, 2019. 344 p.


4. Uchaikin V.V. Fractional derivatives for phusicsts and engineers. Berlin, Heidelberg; Springer, 2013. 512 p.


5. Mainardi F. Fractional calculus and waves in linear viscoelasticity. London: Imperial College Press, 2010. 368 p.


6. Povstenko Yu. Linear fractional diffusion-wave equation for scientists and engineers. Switzerland: Springer Int. Publ., 2015. 460 p.


7. Magin R.L. Fractional calculus in bioengineering. Connecticut, USA: Begel House Publishers Inc., 2006. 355 p.


8. Ninghu Su. Fractional calculus for hydrology, soil science and geomechanics. CRC Press, Taylor & Francis Group, 2021. 336 p.


9. Deseri L., Zingales M. A mechanical picture of fractional-order Darcy equation. Commun. Nonlin. Sci. Numer. Simulat. 2015. Vol. 20. P. 940–949.


10. Tomovski Z., Sandev T., Metzler R., Dubbeldam J.L. Generalized space-time fractional diffusion equation with composite fractional time derivative. Physica A. 2012. Vol. 391. P. 2527– 2542.


11. Al-Homidan S., Ghanam R.A., Tatar N. On a generalized diffusion equation arising in petroleum engineering. Advances in Differential Equations. 2013. Vol. 349. P. 1–14.


12. Bogayenko V.O., Bulavatsky V.M., Khimich O.M. Mathematical and computer modeling in problems of hydrogeomigratory dynamics. Kyiv: Nauk. dumka, 2022. 249 p.


13. Bulavatsky V.M., Kryvonos I.G. Mathematical models with a control function for investigation of fractional-differential dynamics of geomigration processes. Journal of Automation and Information Science. 2014. Vol. 46, N 6. P. 1–11.


14. Bulavatsky V.M. Mathematical models and problems of fractional differential dynamics of some relaxation filtration processes. Cybernetics and Systems Analysis. 2018. Vol. 54, N 5. P. 727–736.


15. Almeida R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlin. Sci. Numer. Simul. 2017. Vol. 44. P. 160–181.


16. J. Vanterler da C. Sousa, E. Capelas de Oliveira. On the -Hilfer fractional derivative function. Commun. Nonlin. Sci. Numer. Simul. 2018. Vol. 60. P. 72–91.


17. J. Vanterler da C. Sousa, E. Capelas de Oliveira. A Gronwall inequality and the Cauchy-type problem by means of -Hilfer operator. Differ. Equat. Appl. 2019. Vol. 11(1). P. 87–106.


18. Kuccehe K.D., Mali A.D., J. Vanterler da C. Sousa. On the nonlinear -Hilfer fractional differential equations. Comput. Appl. Math. 2019. Vol. 38 . P. 73–98. doi.org/10.1007/s40314-019-0833-5.


19. Hilfer R. Fractional time evolution. Applications of Fractional Calculus in Physics. R. Hilfer (ed.). Singapore: World scientific, 2000. P. 87–130.


20. Oliveira D.S., E. Capelas de Oliveira. Hilfer-Katugampola fractional derivative. arXiv: 1705.07733v1 [math. CA], 15 May 2017.


21. Katugampola U.N. A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 2014. N 6. P. 1–15.


22. Ishfaq Mallah, Idris Ahmed, Ali Akgul, Fahd Jarad, Subhash Alha. On -Hilfer generalized proportional fractional operators. AIMS Mathematics. 2021. Vol. 7(1). P. 82–103.


23. Fand Jarad, Thabet Abdeljawad, Jehad Alzabut. Generalized fractional derivatives generated by a class of local proportional derivatives. Europ. Phys. J. Spetial Topics. 2017. Vol. 226. P. 3457–3471.


24. Fand Jarad, Thabet Abdeljawad, Saima Rashid, Zakia Hammouch. More properties of the proportional fractional integral and derivatives of a function with respect to another function. Adv. Differ. Equat. 2020. Vol. 303. P. 1–16. doi.org.10.1186/s13662-020-02767-x .


25. Weerawat Sudsutad, Chatthai Thaiprayoon, Bounmy Khaminsou, Jehad Alzabut, Jutarat Kongson. A Gronwall inequality and its applications to the Cauchy-type problem under -Hilfer proportional fractional operators. Journal of Inequalities and Applications. 2023. Vol. 2023:20. P. 1–35. doi.org.10.1186/s13660-023-02929-x .


26. Bulavatsky V.M. Closed form of the solutions of some boundary-value problems for anomalous diffusion equation with Hilfer’s generalized derivative. Cybernetics and Systems Analysis. 2014. Vol. 50, N 4. P. 570–577.


27. Karimov E.T., Toshtemirov B.H. Non-local boundary value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional differential operators. Uzbek Mathematical Journal. 2021. Vol. 65(2). P. 61–77.


28. Karimov E.T., Ruzhansky M., Toshtemirov B.H. A boundary-value problem for a mixed type equation involving hyper-Bessel fractional differential operator and Hilfer’s bi-ordinal fractional derivative. arXiv:2103.08989v2 [math. AP], 15 Jul 2021.


29. Karimov E.T. Boundary value problems with integral transmitting conditions and inverse problems for integer and fractional order differential equations (DSc Thesis). Vol. 1. Tashkent: Romanovskiy Institute of Mathematics. Uzbekistan Academy of Sciences, 2020. 207 p.


30. Abramovitz M., Stegun I.A. Handbook of mathematical functions. New York: Dover, 1965. 831 p.


31. Fand Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems. Ser. S. 2020. Vol. 13, N 3. P. 709–722.


32. Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag-Leffler functions, related topics and applications. Berlin: Springer-Verlag, 2014. 454 p.


33. Chitalkar-Dhaigude C.P., Sandeep P. Bhairat, Dhaigude D.B. Solution of fractional differential equations involving Hilfer fractional derivative. Bulletin of the Marathwada Mathematical Society. 2017. Vol. 18, N 2. P. 1–13.


34. Sneddon Ian N. The use of integral transform. New York: Mc. Graw-Hill Book Comp., 1972. 539 p.