The Second-Level General Fractional Derivatives and Applications

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DOI: https://doi.org/10.28924/2291-8639-24-2026-273
Published: Aug 24, 2026
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H. P. Masiha, M. Ghasemi, Manuel De La Sen, The Second-Level General Fractional Derivatives and Applications, Int. J. Anal. Appl., 24 (2026), 273.

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H. P. Masiha, M. Ghasemi, Manuel De La Sen

Abstract

In the theory of fractional calculus, we focus on the general fractional calculus because it provides a more precise and flexible modeling tool for a wide range of complex dynamical systems. We initially consider the Sonin and Kochubei kernels along with their key properties. The primary purpose of this article is to develop general fractional derivatives (GFDs) of the first level up to the second level. Compared with the 1st level GFDs, the second-level operators are constructed from the convolution of three kernels. We also present specific features of the second-level GFDs, including the first and second fundamental theorems of fractional calculus with an additional condition and appropriately defined general fractional integrals. Afterward, we establish the Laplace transform of the second-level GFDs, and we find a solution for the fractional relaxation equations with these second-level operators.

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References

  1. N.H. Abel, Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler, Mag. Naturvidenskaberne 1 (1823), 55–68.
  2. N.H. Abel, Auflösung einer mechanischen Aufgabe, J. Reine Angew. Math. 1 (1826), 153–157. https://doi.org/10.1515/crll.1826.1.153.
  3. N. Sonine, Sur la généralisation d'une formule d'Abel, Acta Math. 4 (1884), 171–176. https://doi.org/10.1007/BF02418416.
  4. Y. Luchko, R. Gorenflo, An Operational Method for Solving Fractional Differential Equations with the Caputo Derivatives, Acta Math. Vietnam. 24 (1999), 207–233.
  5. S.G. Samko, R.P. Cardoso, Integral Equations of the First Kind of Sonine Type, Int. J. Math. Math. Sci. 2003 (2003), 3609–3632. https://doi.org/10.1155/S0161171203211455.
  6. A.N. Kochubei, General Fractional Calculus, Evolution Equations, and Renewal Processes, Integral Equ. Oper. Theory 71 (2011), 583–600. https://doi.org/10.1007/s00020-011-1918-8.
  7. Y. Luchko, Fractional Derivatives and the Fundamental Theorem of Fractional Calculus, Fract. Calc. Appl. Anal. 23 (2020), 939–966. https://doi.org/10.1515/fca-2020-0049.
  8. Y. Luchko, General Fractional Integrals and Derivatives with the Sonine Kernels, Mathematics 9 (2021), 594. https://doi.org/10.3390/math9060594.
  9. Y. Luchko, The 1st Level General Fractional Derivatives and Some of Their Properties, J. Math. Sci. 266 (2022), 709–722. https://doi.org/10.1007/s10958-022-06055-9.
  10. I.H. Dimovski, Operational Calculus for a Class of Differential Operators, C. R. Acad. Bulg. Sci. 19 (1966), 1111–1114.
  11. V.E. Tarasov, General Fractional Dynamics, Mathematics 9 (2021), 1464. https://doi.org/10.3390/math9131464.
  12. V.E. Tarasov, General Fractional Vector Calculus, Mathematics 9 (2021), 2816. https://doi.org/10.3390/math9212816.
  13. V.E. Tarasov, General Non-Local Continuum Mechanics: Derivation of Balance Equations, Mathematics 10 (2022), 1427. https://doi.org/10.3390/math10091427.
  14. V.E. Tarasov, Nonlocal Probability Theory: General Fractional Calculus Approach, Mathematics 10 (2022), 3848. https://doi.org/10.3390/math10203848.
  15. V.E. Tarasov, Nonlocal Statistical Mechanics: General Fractional Liouville Equations and Their Solutions, Phys. A: Stat. Mech. Appl. 609 (2023), 128366. https://doi.org/10.1016/j.physa.2022.128366.
  16. V.E. Tarasov, General Non-Markovian Quantum Dynamics, Entropy 23 (2021), 1006. https://doi.org/10.3390/e23081006.
  17. V.E. Tarasov, Nonlocal Classical Theory of Gravity: Massiveness of Nonlocality and Mass Shielding by Nonlocality, Eur. Phys. J. Plus 137 (2022), 1336. https://doi.org/10.1140/epjp/s13360-022-03512-x.
  18. V. Miskovic-Stankovic, T.M. Atanackovic, On a System of Equations with General Fractional Derivatives Arising in Diffusion Theory, Fractal Fract. 7 (2023), 518. https://doi.org/10.3390/fractalfract7070518.
  19. M. Alkandari, Y. Luchko, Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications, Mathematics 12 (2024), 2626. https://doi.org/10.3390/math12172626.
  20. R. Zacher, Boundedness of Weak Solutions to Evolutionary Partial Integro-Differential Equations with Discontinuous Coefficients, J. Math. Anal. Appl. 348 (2008), 137–149. https://doi.org/10.1016/j.jmaa.2008.06.054.
  21. Y. Luchko, On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative, Mathematics 8 (2020), 1561. https://doi.org/10.3390/math8091561.
  22. E. Bazhlekova, I. Bazhlekov, Identification of a Space-Dependent Source Term in a Nonlocal Problem for the General Time-Fractional Diffusion Equation, J. Comput. Appl. Math. 386 (2021), 113213. https://doi.org/10.1016/j.cam.2020.113213.
  23. K. Górska, A. Horzela, Subordination and Memory Dependent Kinetics in Diffusion and Relaxation Phenomena, Fract. Calc. Appl. Anal. 26 (2023), 480–512. https://doi.org/10.1007/s13540-023-00141-8.
  24. Y. Luchko, General Fractional Integrals and Derivatives of Arbitrary Order, Symmetry 13 (2021), 755. https://doi.org/10.3390/sym13050755.
  25. S.G. Samko, B.G. Vakulov, On Equivalent Norms in Fractional Order Function Spaces of Continuous Functions on the Unit Sphere, Fract. Calc. Appl. Anal. 3 (2000), 401–433.
  26. M.V. Kukushkin, Abstract Fractional Calculus for m-Accretive Operators, Int. J. Appl. Math. 34 (2021), 1–42. https://doi.org/10.12732/ijam.v34i1.1.
  27. J. Mendiola-Fuentes, E. Guerrero-Ruiz, J. Rosales-García, Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator, Mathematics 12 (2024), 1502. https://doi.org/10.3390/math12101502.
  28. X.-J. Yang, F. Gao, Y. Ju, General Fractional Derivatives with Applications in Viscoelasticity, Academic Press, Elsevier, 2020.
  29. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, 1993.
  30. R.L. Schilling, R. Song, Z. Vondraček, Bernstein Functions: Theory and Applications, 2nd ed., De Gruyter, 2012. https://doi.org/10.1515/9783110269338.