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Tempered and Hadamard-Type Fractional Calculus with Respect to Functions

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Abstract

Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann–Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.

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References

  1. Abdalla, A.M., Salem, H.A.H., Cichoń, K.: On positive solutions of a system of equations generated by Hadamard fractional operators. Adv. Differ. Equ. 2020, 267 (2020)

    Article  MathSciNet  Google Scholar 

  2. Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    Article  MathSciNet  Google Scholar 

  3. Baleanu, D., Fernandez, A.: On fractional operators and their classifications. Mathematics 7(9), 830 (2019)

    Article  Google Scholar 

  4. Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)

    Article  MathSciNet  Google Scholar 

  5. Cao, J., Li, C., Chen, Y.: On tempered and substantial fractional calculus. In: 2014 IEEE/ASME 10th International Conference on Mechatronic and Embedded Systems and Applications (MESA) 2014 Sep 10. IEEE, pp. 1–6

  6. Caputo, M.: Linear model of dissipation whose q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  7. Cichoń, M., Salem, H.A.H.: On the solutions of Caputo–Hadamard Pettis-type fractional differential equations. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. (RACSAM) 113, 3031–3053 (2019)

  8. Cichoń, M., Salem, H.A.H.: On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems. J. Pseudo-Differ. Oper. Appl. 11(4), 1869–1895 (2020)

  9. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    Book  Google Scholar 

  10. Erdelyi, A.: An integral equation involving Legendre functions. J. Soc. Ind. Appl. Math. 12(1), 15–30 (1964)

    Article  MathSciNet  Google Scholar 

  11. Fahad, H.M., ur Rehman, M.: Generalized substantial fractional operators and well-posedness of Cauchy problem. Bull. Malays. Math. Sci. Soc. 44, 1501–1524 (2021)

  12. Fernandez, A., Ustaoglu, C.: On some analytic properties of tempered fractional calculus. J. Comput. Appl. Math. 366, 112400 (2020)

    Article  MathSciNet  Google Scholar 

  13. Fernandez, A., Özarslan, M.A., Baleanu, D.: On fractional calculus with general analytic kernels. Appl. Math. Comput. 354, 248–265 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Friedrich, R., Jenko, F., Baule, A., Eule, S.: Anomalous diffusion of inertial, weakly damped particles. Phys. Rev. Lett. 96, 230601 (2006)

    Article  Google Scholar 

  15. Hadamard, J.: Essai sur l’étude des fonctions, données par leur développement de Taylor. J. Math. Pures Appl. 8, 101–186 (1892)

    MATH  Google Scholar 

  16. Herzallah, M.A.E., El-Sayed, A.M.A., Baleanu, D.: On the Fractional-Order Diffusion-Wave Process. Rom. J. Phys. 55(3–4), 274–284 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2000)

    Book  Google Scholar 

  18. Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 1, 142 (2012)

    Article  MathSciNet  Google Scholar 

  19. Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 226, 3457–3471 (2018)

    Article  Google Scholar 

  20. Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations (North-Holland Mathematical Studies), vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  23. Li, C., Deng, W., Zhao, L.: Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. Disc. Cont. Dyn. Sys. B 24(4), 1989–2015 (2019)

    MATH  Google Scholar 

  24. Love, E.R., Young, L.C.: On fractional integration by parts. Proc. Lond. Math. Soc. 2(1), 1–35 (1938)

    Article  MathSciNet  Google Scholar 

  25. Ma, L., Li, C.: On Hadamard fractional calculus. World Sci Fract 25(3) (2017)

  26. Meerschaert, M.M., Sabzikar, F., Chen, J.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)

    Article  MathSciNet  Google Scholar 

  27. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, San Diego (1974)

    MATH  Google Scholar 

  28. Osler, T.J.: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 18(3), 658–674 (1970)

    Article  MathSciNet  Google Scholar 

  29. Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Int. J. Theor. Appl. 5(4) (2002)

  30. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  31. Salem, H.A.H.: Hadamard-type fractional calculus in Banach spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. (RACSAM) 113, 987–1006 (2019)

    Article  MathSciNet  Google Scholar 

  32. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  33. Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, HEP (2011)

    Google Scholar 

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Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for their helpful suggestions. In particular, we are grateful to one of the reviewers for suggesting to us the result and proof that became our Theorem 4.4.

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Authors and Affiliations

  1. Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan

    Hafiz Muhammad Fahad, Mujeeb ur Rehman & Maham Siddiqi

  2. Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, Northern Cyprus, Via Mersin 10, Turkey

    Hafiz Muhammad Fahad & Arran Fernandez

  3. Department of Space Science, Institute of Space Technology, Islamabad, Pakistan

    Maham Siddiqi

Authors
  1. Hafiz Muhammad Fahad
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  2. Arran Fernandez
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  3. Mujeeb ur Rehman
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  4. Maham Siddiqi
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Correspondence to Arran Fernandez or Maham Siddiqi.

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Maham Siddiqi was a visiting researcher at the School of Natural Sciences, NUST, Islamabad, Pakistan.

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Fahad, H.M., Fernandez, A., Rehman, M.u. et al. Tempered and Hadamard-Type Fractional Calculus with Respect to Functions. Mediterr. J. Math. 18, 143 (2021). https://doi.org/10.1007/s00009-021-01783-9

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  • DOI: https://doi.org/10.1007/s00009-021-01783-9

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