$\alpha$-Differentiable functions in complex plane

R. Pashaei ; A. Pishkoo ; M. S. Asgari ; D. Ebrahimi Bagha

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 379-389.

Résumé

In this paper, the conformable fractional derivative of order $\alpha$ is defined in complex plane. Regarding to multi-valued function $z^{1-\alpha}$, we obtain fractional Cauchy–Riemann equations which in case of $\alpha=1$ give classical Cauchy–Riemann equations. The properties relating to complex conformable fractional derivative of certain functions in complex plane have been considered. Then, we discuss about two complex conformable differential equations and solutions with their Riemann surfaces. For some values of order of derivative, $\alpha$, we compare their plots.

Détail

Keywords: conformable fractional derivative, limit based fractional derivative.
Mots-clés : Cauchy–Riemann equations

Bibliographie

[1] Tenreiro Machado J., Kiryakova V., Mainardi F., “A poster about the recent history of fractional calculus”, Fract. Calc. Appl. Anal., 13:3 (2010), 329–334 http://eudml.org/doc/219594 | MR | Zbl

[2] Samko S. G., Kilbas A. A., Marichev O. I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publ., Neq York, 1993, xxxvi+976 pp. | MR | Zbl

[3] Oldham K., Spanier J., The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, 111, Academic Press, New York, London, 1974, xiii+234 pp. | MR | Zbl

[4] de Oliveira E. C., Solved Exercises in Fractional Calculus, Studies in Systems, Decision and Control, 240, Springer, Cham, 2019, xviii+321 pp. | DOI | MR | Zbl

[5] Tenreiro Machado J., Kiryakova V., Mainardi F., “Recent history of fractional calculus”, Comm. Nonlinear Science Numerical Simulation, 16:3 (2011), 1140–1153 | DOI | MR | Zbl

[6] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006, xv+523 pp. | MR | Zbl

[7] Podlubny J., 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, 198, Academic Press, San Diego, CA, 1999, xxiv+340 pp. | MR | Zbl

[8] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993, xiii+366 pp. | MR | Zbl

[9] Pashaei R., Asgari M., Pishkoo A., “Conformable derivatives in Laplace equation and fractional Fourier series solution”, Int. Ann. Sci., 9:1 (2019), 1–7 | DOI | MR

[10] Ortigueira M. D., Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011, xiv+152 pp. | DOI | MR | Zbl

[11] Khalil R., Al Horani M., Yousef A., Sababheh M., “A new definition of fractional derivative”, J. Comp. Appl. Math., 264 (2014), 65–70 | DOI | MR | Zbl

[12] Katugampola U. N., A new fractional derivative with classical properties, 2014, arXiv: [math.CA] 1410.6535 | MR | Zbl

[13] Ortigueira M. D., Tenreiro Machado J. A., “What is a fractional derivative?”, J. Comp. Phys., 293 (2015), 4–13 | DOI | MR | Zbl

[14] Tarasov V. E., “No nonlocality. No fractional derivative”, Commun. Nonlinear Sci. Numer. Simulat., 62 (2018), 157–163, arXiv: [math.CA] 1803.00750 | DOI | MR

[15] Tarasov V. E., “Electromagnetic fields on fractals”, Modern Phys. Lett. A., 21:20 (2006), 1587–1600, arXiv: 0711.1783 | DOI | Zbl

[16] Li J., Ostoja-Starzewski M., “Fractal solids, product measures and fractional wave equations”, Proc. R. Soc. A., 465:2108, 2521–2536 | DOI | MR | Zbl

[17] Ostoja-Starzewski M., Li J., “Fractal materials, beams, and fracture mechanics”, Z. angew. Math. Phys., 60:6 (2009), 1194–1205 | DOI | MR | Zbl

[18] Ostoja-Starzewski M., “Electromagnetism on anisotropic fractal media”, Z. angew. Math. Phys., 64:2 (2013), 381–390 | DOI | MR | Zbl

[19] Ostoja-Starzewski M., Li J., Joumaa H., Demmie P. N., “From fractal media to continuum mechanics”, Z. angew. Math. Mech., 94:5, 373–401 | DOI | MR | Zbl

[20] Ortigueira M. D., Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011, xiv+154 pp. | DOI | MR | Zbl

[21] Olver P. J., Complex Analysis and Conformal Mapping, Lecture Notes, 2018, 84 pp. http://www-users.math.umn.edu/~olver/ln_/cml.pdf

[22] Abdeljawada T., Al-Mdallal Q. M., Jarad F., “Fractional logistic models in the frame of fractional operators generated by conformable derivatives”, Chaos, Solitons and Fractals, 119 (2019), 94–101 | DOI | MR

[23] Ibrahim R. W., Meshram C., Hadid S. B., Momani S., “Analytic solutions of the generalized water wave dynamical equations based on time-space symmetric differential operator”, J. Ocean Eng. Sci., 5:2 (2020), 186–195 | DOI

[24] Ebaid A., Masaedeh B., El-Zahar E., “A new fractional model for the falling body problem”, Chinese Phys. Lett., 34:2, 020201 | DOI

[25] Alharbi F.M., Baleanu D., Ebaid A., “Physical properties of the projectile motion using the conformable derivative”, Chinese J. Phys., 58 (2019), 18–28 | DOI

[26] Kaabar M.K.A., Novel methods for solving the conformable wave equation, 2019 https://hal.archives-ouvertes.fr/hal-02267015

[27] Chung W. S., “Fractional Newton mechanics with conformable fractional derivative”, J. Comp. Appl. Math., 290 (2015), 150–158 | DOI | MR | Zbl

[28] Morales-Delgado V. F., Gómez-Aguilar J. F., Taneco-Hernández M. A., “Mathematical modeling approach to the fractional Bergman's model”, Discrete Contin. Dyn. Syst. Ser. S, 13:3 (2020), 805–821 | DOI | MR | Zbl

[29] C̨erdik Yaslan H., Mutlu F., “Numerical solution of the conformable differential equations via shifted Legendre polynomials”, Int. J. Comp. Math., 97:5 (2020), 1016–1028 | DOI | MR

[30] Meng S., Cui Y., “The extremal solution to conformable fractional differential equations involving integral boundary condition”, Mathematics, 7:2 (2019), 186 | DOI

[31] Sibatov R. T., “Anomalous grain boundary diffusion: Fractional calculus approach”, Adv. Math. Phys., 2019, 8017363, 9 pp. | DOI | MR | Zbl

[32] Alikhanov A. A., “A new difference scheme for the time fractional diffusion equation”, J. Comp. Phys., 280 (2015), 424–438, arXiv: [math.NA] 1404.5221 | DOI | MR | Zbl

[33] Amanov D., Ashyralyev A., “Initial-boundary value problem for fractional partial differential equations of higher order”, Abstract and Applied Analysis, 2012, 973102, 16 pp. | DOI | MR | Zbl

[34] Application of Fractional Calculus in Physics, ed. R. Hilfer, World Scientific, Singapore, 2000 | DOI | MR

[35] Muslih S. I., Baleanu D., “Fractional multipoles in fractional space”, Nonlinear Anal. RWA, 8:1 (2007), 198–203 | DOI | MR | Zbl

[36] Tavazoei M. S., Haeri M., “Stabilization of unstable fixed points of chaotic fractional order systems by a state fractional PI controller”, Eur. J. Control, 14:3 (2008), 247–257 | DOI | MR | Zbl

[37] Ortigueira M. D., “A coherent approach to non-integer order derivatives”, Signal Processing, 86:10 (2006), 2505–2515 | DOI | Zbl

[38] Li C., Dao X, Guoa P., “Fractional derivatives in complex planes”, Nonlinear Analysis: Theory, Methods, Applications, 71:5–6 (2009), 1857–1869 | DOI | MR | Zbl

[39] Owa S., “Some properties of fractional calculus operators for certain analytic functions”, RIMS Kôkyûroku, 1626 (2009), 86–92 http://hdl.handle.net/2433/140314 | MR

[40] Abdeljawad T., “On conformable fractional calculus”, J. Comp. Appl. Math., 279 (2015), 57–66 | DOI | MR | Zbl