Advertisement

Some properties of pseudo-fractional operators

  • Azizollah Babakhani
  • Milad Yadollahzadeh
  • Abdolali Neamaty
Article
  • 130 Downloads

Abstract

In the present paper, we discuss the pseudo-fractional calculus, including two fields of fractional calculus and pseudo-analysis. We also provide pseudo-fractional integral/derivative operators on a semiring \(([a,b],\oplus ,\odot )\). Then, some basic properties of these operators are proposed such as the Leibniz rule, chain rule and g-Laplace transform formulas.

Keywords

Pseudo-multiplication Pseudo-addition Semiring Pseudo-fractional integrals Pseudo-fractional derivatives The Leibniz rule Pseudo-Laplace transform 

Mathematics Subject Classification

47GXX 26A33 47AXX 

Notes

Acknowledgements

The authors would like to thank to the anonymous referees for their valuable comments and suggestions.

References

  1. 1.
    Agahi, H., Babakhani, A., Mesiar, R.: Pseudo-fractional integral inequality of Chebyshev type. Inf. Sci. 301, 161–168 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agahi, H., Alipour, M.: On pseudo-Mittag-Leffler functions and applications. Fuzzy Sets Syst. (2017). doi: 10.1016/j.fss.2016.11.011
  3. 3.
    Agahi, H., Mesiar, R., Ouyang, Y.: Chebyshev type inequalities for pseudo-integrals. Nonlinear Anal. 72, 2737–2743 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y., Gao, H., Sun, C.: The stochastic fractional power dissipative equations in any dimension and applications. J. Math. Anal. Appl. 425, 1240–1256 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gaul, L., Klein, P., Kempfle, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991)CrossRefGoogle Scholar
  6. 6.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hosseini, M., Babakhani, A., Agahi, H., Rasouli, S.H.: On pseudo-fractional integrals inequalities related to Hermite–Hadamard type. Soft Comput. 20(7), 2521–2529 (2017)Google Scholar
  8. 8.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Netherlands (2006)zbMATHGoogle Scholar
  9. 9.
    Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kuich, W.: Semirings, Automata, Languages. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Mesiar, R., Pap, E.: Idempotent integral as limit of \(g\) -integrals. Fuzzy Sets Syst. 102, 385–392 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mesiar, R., Rybárik, J.: Pseudo-arithmetical operations. Tatra Mt. Math. Publ. 2, 185–192 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Metzler, F., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)CrossRefGoogle Scholar
  14. 14.
    Pap, E.: An integral generated by decomposable measure. Univ. Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 20(1), 135–144 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pap, E.: \(g\)-calculus. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23(1), 145–156 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pap, E., Ralević, N.: Pseudo-Laplace transform. Nonlinear Anal. 33, 553–560 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pap, E.: Applications of the generated pseudo-analysis on nonlinear partial differential equations. In: Litvinov, G.L., Maslov, V.P. (eds.) Proceedings of the conference on idempotent mathematics and mathematical physics, contemporary mathematics 377, American Mathematical Society, 239–259 (2005)Google Scholar
  18. 18.
    Pap, E., Štrboja, M.: Generalization of the Jensen inequality for pseudo-integral. Inf. Sci. 180, 543–548 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pap, E., Štrboja, M., Rudas, I.: Pseudo-L\(^{p}\) space and convergence. Fuzzy Sets Syst. 238, 113–128 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pap, E.: Pseudo-additive measures and their applications. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 1403–1468. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  21. 21.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  22. 22.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivatives (Theory and Application). Gordon and Breach, Switzerland (1993)zbMATHGoogle Scholar
  23. 23.
    Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Azizollah Babakhani
    • 1
  • Milad Yadollahzadeh
    • 2
  • Abdolali Neamaty
    • 2
  1. 1.Department of MathematicsBabol Noshirvani University of TechnologyBabolIran
  2. 2.Department of Mathematics, Faculty of mathematical SciencesUniversity of MazandaranBabolsarIran

Personalised recommendations