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Generalized Fractional Cauchy–Riemann Operator Associated with the Fractional Cauchy–Riemann Operator

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Abstract

In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy–Riemann operator in the Riemann–Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.

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Notes

  1. A function h is \({\mathbb {R}}\)-analytic in an open interval I if for every \(x\in I\), it admits a Taylor expansion with center x and converges in a neighborhood \(I_x\subset I\) of x.

References

  1. Alsaedi, A., Nieto, J., Venktesh, V.: Fractional electrical circuits. Advances in Mechanical Engineering (2015)

  2. Ariza, E., Di Teodoro, A., Vanegas, C.J.: \(\psi \)-weighted Cauchy-Riemann operators and some associated integral representation. Quaest. Math. 43(3), 1–26 (2019)

    MathSciNet  Google Scholar 

  3. Baleanu, D., Golmankhaneh, A.K., Golmankhaneh, A.K.: The dual action and fractional multi time Hamilton equations. Int. J. Theor. Phys. 48, 2558–2569 (2009)

    Article  Google Scholar 

  4. Baleanu, D., Golmankhaneh, A.K., Golmankhaneh, A.K., Nigmatullin, R.R.: Newtonian law with memory. Nonlinear Dyn. 60, 81–86 (2010)

    Article  Google Scholar 

  5. Baleanu, D., Guvenc, Z.B., Machado, J.A.T. (eds.): New trends in nanotechnology and fractional calculus applications, XI, 531, 1st edn. Springer, New York (2010)

  6. Caputo, M.: The role of memory in modeling social and economic cycles of extreme events. In: Forte, F., Navarra, P., Mudambi, R. (eds.). Edward Elgar Publishing, pp. 245–259 (2014)

  7. Caputo, M., Cametti, C.: Memory diffusion in two cases of biological interest. J. Theor. Biol. 254, 697–703 (2008)

    Article  Google Scholar 

  8. Caputo, M., Cametti, C., Ruggiero, V.: Time and spatial concentration profile inside a membrane by means of a memory formalism. Phys. A 387, 2010–2018 (2007)

    Article  Google Scholar 

  9. Caputo, M., Fabrizio, M.: Damage and fatigue described by a fractional derivative model, in press. J. Comput. Phys. 293(C), 400–408 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  10. Cesarone, F., Caputo, M., Cametti, C.: Memory formalism in the passive diffusion across a biological membrane. J. Membr. Sci. 250, 79–84 (2004)

    Article  Google Scholar 

  11. Diethelm, K.: The analysis of fractional differential equations, an application oriented, exposition using differential operators of caputo type. Lecture Notes in Mathematics nr, vol. 2004. Springer, Heidelbereg (2010)

  12. El Shaed, M.: A fractional calculus model of semilunar heart valve vibrations, Challenging the Boundaries of Symbolic Computation, pp. 57–64 (2003)

  13. Di Teodoro, A., Ferreira, M., Vieira, N.: Fundamental solution for natural powers of the fractional laplace and dirac operators in the Riemann–Liouville sense. Adv. Appl. Clifford Algebras 30, 3 (2020)

    Article  MathSciNet  Google Scholar 

  14. Ferreira, M., Vieira, N.: Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case. Complex Anal. Oper. Theory 10, 1081–1100 (2016)

    Article  MathSciNet  Google Scholar 

  15. Ferreira, M., Vieira, N.: Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives. Complex Variables Elliptic Eqn. 62(9), 1237–1253 (2017)

    Article  MathSciNet  Google Scholar 

  16. Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic functions in the plane and n-dimensional. Birkhäuser, Basel (2008)

  17. Hilfer, R.: Applications of fractional calculus in physics. World Scientific Publishing Co, Singapore (2003)

    MATH  Google Scholar 

  18. John, F.: Partial differential equations. Springer, New York (1982)

    Book  Google Scholar 

  19. Jumarie, G.: New stochastic fractional models of the Malthusian growth, the Poissonian birth process and optimal management of populations. Math. Comput. Modell. 44, 231–254 (2006)

    Article  MathSciNet  Google Scholar 

  20. Kähler, U., Vieira, N.: Fractional Clifford analysis. In: Bernstein, S., Kähler, U., Sabadini, I., Sommen, F. (eds.) Hypercomplex analysis: new perspectives and applications. Trends in mathematics, pp. 191–201. Birkähuser, Basel (2014)

  21. Kilbas, A.A., Marzan, S.A.: Non linear differential equations with caputo fractional derivative in the space of continuously differentiable functions. Differ. Equations 41, 84–95 (2005)

    Article  MathSciNet  Google Scholar 

  22. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

  23. Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calculus Appl. Anal. 20, 7–51 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Lewy, H.: An example of a smooth linear partial differential equation without solution. Ann. Math. 66, 155–158 (1957)

    Article  MathSciNet  Google Scholar 

  25. Magin, R.L.: Fractional calculus in bioengineering. Begell House Inc. Publishers, Danbury (2006)

    Google Scholar 

  26. Miller, K.S., Ross, B.: An introduction to fractional calculus and fractional differential equations. Wiley, New York (1993)

    MATH  Google Scholar 

  27. Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3339–3352 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  28. Ortigueira, M.D., Machado, J.T.: A critical analysis of the Caputo–Fabrizio operator. Commun. Nonlinear Sci. Numer. Simul. 59, 608–611 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  29. Osler, T.J.: A correction to Leibniz rule for fractional derivatives. SIAM J. Math. Anal. 4(3), 456–459 (1973)

    Article  MathSciNet  Google Scholar 

  30. 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 

  31. Osler, T.J.: The integral analog of the Leibniz rule. Math. Comput. 26(120), 903–915 (1972)

    MATH  Google Scholar 

  32. Podlubny, I.: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering. Academic Press, San Diego (1999)

  33. Pozrikidis, C.: The Fractional Laplacian. CRC Press Taylor & Francis Group, Boca Raton (2016)

    Book  Google Scholar 

  34. Sabatier, J., Agraval, O.P., Teneiro Machado, J.A.: Advances in Fractional Calculus. Springer, New York (2009)

    Google Scholar 

  35. 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 

  36. Son, L.H., Tutschke, W.: First order differential operator associated to the Cauchy-Riemann Operator in the Plane. Complex Var. 48, 797–801 (2003)

    MathSciNet  MATH  Google Scholar 

  37. Stein, E., Shakarchi, R.: Complex analysis. Princeton University Press, Princeton (2003)

  38. Tarasov, V.E.: No violation of the Liebniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18, 2945–2948 (2013)

  39. Torres, D.F.M., Malinowska, A.B.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012)

    MATH  Google Scholar 

  40. Tremblay, R., Gaboury, S., Fugere, B.-J.: A new Leibniz rule and its integral analogue for fractional derivatives. Integr. Transforms Spec.Funct. 24(2), 111–128 (2013)

    Article  MathSciNet  Google Scholar 

  41. Tutschke, W.: Associated spaces—a new tool of real and complex analysis, pp. 253–268. National University Publishers, Hanoi (2008)

  42. Tutschke, W.: Solution of initial value problems in classes of generalized analytic functions. Springer, Berlin, Heidelberg (1989)

    Book  Google Scholar 

  43. Vekua, I.N.: Generalized analytic functions. Pergamon Press, Oxford, New York (1962)

    MATH  Google Scholar 

  44. Vieira, N.: Fischer decomposition and Cauchy–Kovalevskaya extension in fractional Clifford analysis: the Riemann–Liouville case. Proc. Edinb. Math. Soc. II. 60(1), 251–272 (2017)

    Article  MathSciNet  Google Scholar 

  45. Wei, Y., Liu, D.-Y., Tse, P.W., Wang, Y.: Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation. Integr. Transforms Speci. Funct. 31(4), 304–322 (2020)

    Article  MathSciNet  Google Scholar 

  46. Zavada, P.: Relativistic wave equations with fractional derivatives and pseudodifferential operators. J. Appl. Math. 2(4), 163–197 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the reviewers for many useful comments which lead to substantial improvements of the paper.

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Authors and Affiliations

  1. Escuela de Ciencias Físicas y Matemáticas., Universidad de Las Américas. Quito, Quito, Ecuador

    Johan Ceballos

  2. Departamento de Matemáticas, Colegio de Ciencias e Ingenierías, Universidad San Francisco de Quito, Diego de Robles y vía Interoceánica, Quito, Ecuador

    Nicolás Coloma, Antonio Di Teodoro & Diego Ochoa–Tocachi

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  1. Johan Ceballos
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  2. Nicolás Coloma
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  3. Antonio Di Teodoro
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  4. Diego Ochoa–Tocachi
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Correspondence to Antonio Di Teodoro.

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This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29-August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen.

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Ceballos, J., Coloma, N., Di Teodoro, A. et al. Generalized Fractional Cauchy–Riemann Operator Associated with the Fractional Cauchy–Riemann Operator. Adv. Appl. Clifford Algebras 30, 70 (2020). https://doi.org/10.1007/s00006-020-01096-2

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