Skip to main content
SpringerLink
Log in

Conformable fractional derivative in commutative algebras

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called φ-monogenic. A relation between the concepts of φ-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

  2. I. Podlubny, Fractional differential equations. Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

  3. R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” J. Comp. Appl. Math., 264, 65–70 (2014).

  4. T. Abdeljawad, “On conformable fractional calculus,” J. Comp. Appl. Math., 279, 57–66 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Atangana, D. Baleanu, and A. Alsaedi, “New properties of conformable derivative,” Open Math., 13, 57–63 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Martinez, I. Martinez, and S. Paredes, “Conformable Euler’s Theorem on homogeneous functions,” Comp. and Math. Methods, 1(5), 1–11 (2018).

    MathSciNet  Google Scholar 

  7. R. Almeida, M. Guzowska, and T. Odzijewicz, “A remark on local fractional calculus and ordinary derivatives,” Open Math., 14, 1122–1124 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “Fractional analytic functions,” Far East Journal of Mathematical Sciences, 103(1), 113–123 (2018).

  9. S. Ucar and N. Y. Ozgur, “Complex Conformable derivative,” Arabian Journal of Geosciences, 12(6), 1–6 (2019).

    Article  Google Scholar 

  10. M. D. Ortigueira, L. Rodríguez-Germá, and J. J. Trujillo, “Complex Grunwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions,” Commun Nonlinear Sci Numer Simulat, 16, 4174–4182 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Martínez, I. Martínez, M. K. A. Kaabar, R. Ortiz-Munuera, and S. Paredes, “Note on the Conformable Fractional Derivatives and Integrals of Complex-valued Functions of a Real Variable,” IAENG Int. J. Appl. Math., 50, 609–615 (2020).

    Google Scholar 

  12. F. Martínez, I. Martínez, M. K. A. Kaabar, and S. Paredes, “New results on complex conformable integral,” AIMS Mathematics, 5(6), 7695–7710 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. Y. Çenesiz and A. Kurt, “New Fractional Complex Transform for Conformable Fractional Partial Differential Equations,” JAMSI, 12(2), 41–47 (2016).

    MathSciNet  MATH  Google Scholar 

  14. M. Kaabar, “Novel methods for solving the conformable wave equation,” J. New Theory, 31, 56–85 (2020).

    Google Scholar 

  15. F. Martínez, I. Martínez, M. K. A. Kaabar, and S. Paredes, “On Conformable Laplace’s Equation,” Mathematical Problems in Engineering, 2021, Article ID 5514535, 10 p. (2021).

  16. M. A. Hammad and R. Khalil, “Abel’s formula and wronskian for conformable fractional differential equations,” International Journal of Differential Equations and Applications, 13(2), 177–183 (2014).

    MATH  Google Scholar 

  17. F. S. Silva, M. D. Moreira, and M. A. Moret, “Conformable Laplace transform of fractional differential equations,” Axioms, 7(55) (2018).

  18. Z. Korpinar, A. S. Alshomrani, M. Inc, and D. Baleanu, “The deterministic and stochastic solutions of the Schrodinger equation with time conformable derivative in birefrigent fibers,” AIMS Mathematics, 5(3), 2326–2345 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. M. A. Khater, Y.-M. Chu, R. A. M. Attia, M. Inc, and D. Lu, “On the analytical and numerical solutions in the quantum magnetoplasmas: the atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity,” Advances in Mathematical Physics, 2020, 1–10 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Unal, A. Gokogak, “Solution of conformable ordinary differential equations via differential transform method,” Optik, 128, 264–273 (2017).

    Article  Google Scholar 

  21. E. Cartan, “Les groupes bilinéares et les systèmes de nombres complexes,” Annales de la faculté des sciences de Toulouse, 12(1), 1–64 (1898).

    MathSciNet  Google Scholar 

  22. I. P. Mel’nichenko, “The representation of harmonic mappings by monogenic functions,” Ukr. Math. J., 27(5), 599–505 (1975).

  23. V. S. Shpakivskyi, “Monogenic functions in finite-dimensional commutative associative algebras,” Zb. Pr. Inst. Mat. NAN Ukr., 12(3), 251–268 (2015).

    MathSciNet  MATH  Google Scholar 

  24. V. S. Shpakivskyi, “Constructive description of monogenic functions in a finite-dimensional commutative associative algebra,” Adv. Pure Appl. Math., 7(1), 63–75 (2016).

    MathSciNet  MATH  Google Scholar 

  25. E. R. Lorch, “The theory of analytic function in normed abelin vector rings,” Trans. Amer. Math. Soc., 54, 414–425 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. A. Plaksa and R. P. Pukhtaievych, “Constructive description of monogenic functions in n-dimensional semi-simple algebra,” An. Şt. Univ. Ovidius Constanţa, 22(1), 221–235 (2014).

    MATH  Google Scholar 

  27. M. E. Luna-Elizarraras, M. Shapiro, D. C. Struppa, and A. Vajiac, Bicomplex holomorphic functions: the algebra, geometry and analysis of bicomplex numbers. Springer, 2015.

  28. S. V. Grishchuk and S. A. Plaksa, “Monogenic functions in a biharmonic algebra,” Ukr. Math. J., 61(12), 1865–1876 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  29. S. A. Plaksa and V. S. Shpakovskii, “Constructive description of monogenic functions in a harmonic algebra of the third rank,” Ukr. Math. J., 62(8), 1251–1266 (2011).

    Article  MathSciNet  Google Scholar 

  30. V. S. Shpakivskyi, “Integral theorems for monogenic functions in commutative algebras,” Zb. Pr. Inst. Mat. NAN Ukr., 12(4), 313–328 (2015).

    MATH  Google Scholar 

  31. V. S. Shpakivskyi, “Curvilinear integral theorems for monogenic functions in commutative associative algebras,” Advances in Applied Clifford Algebras, 26(1), 417–434 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  32. S. A. Plaksa and V. S. Shpakivskyi, “Cauchy theorem for a surface integral in commutative algebras,” Complex Variables and Elliptic Equations, 59(1), 110–119 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. V. S. Shpakivskyi, “On monogenic functions defined in different commutative algebras,” J. Math. Sci., 239(1), 92–109 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  34. V. S. Shpakivskyi, “On monogenic functions on extensions of commutative algebra,” Proceedings of the International Geometry Center, 11(3), 1–18 (2018).

    MathSciNet  Google Scholar 

  35. V. S. Shpakivskyi, “Hypercomplex method for solving linear partial differential equations,” Proc. of the Inst. Appl. Math. Mech. NAS Ukraine, 32, 147–168 (2018).

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

    Vitalii S. Shpakivskyi

Authors
  1. Vitalii S. Shpakivskyi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vitalii S. Shpakivskyi.

Additional information

Presented by V. Gutlyanskyĭ

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 20, No. 2, pp. 269–282, April–June, 2023.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shpakivskyi, V.S. Conformable fractional derivative in commutative algebras. J Math Sci 274, 392–402 (2023). https://doi.org/10.1007/s10958-023-06608-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06608-6

Keywords

Search

Navigation