# American Institute of Mathematical Sciences

July  2021, 14(7): 2357-2371. doi: 10.3934/dcdss.2020172

## Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel

 1 Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology (CHARUSAT), Changa, Anand-388421, Gujarat, India 2 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

* Corresponding author: Krunal B. Kachhia

Received  March 2019 Published  July 2021 Early access  September 2020

The concept of differential operator with variable order has attracted attention of many scholars due to their abilities to capture more complexities like anomalous diffusion. While these differential operators are useful in real life, they can only be handled numerically. In this work, we used a newly introduced variable order differential operators that can be used analytically and numerically, has connection with all integral transform to model some interesting mathematical models arising in electromagnetic wave in plasma and dielectric. The differential operators used are non-singular and have the crossover properties therefore the models studied can explain the propagation of the wave in two different layers which cannot be achieved with those differential variable order operators with singular kernels. Using the Laplace transform and its connection with the new differential operator, we derive the exact solution of the models under investigation.

Citation: Krunal B. Kachhia, Abdon Atangana. Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2357-2371. doi: 10.3934/dcdss.2020172
##### References:
 [1] B. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 547-551. [2] B. S. T. Alkahtani, I. Koca and A. Atangana, A novel approach of variable order derivative: Theory and methods, J. Nonlinear Sci. Appl., 9 (2016), 4867-4876.  doi: 10.22436/jnsa.009.06.122. [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [4] A. Atangana and J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 166. [5] A. Atangana and I. Koca, New direction in fractional differentiation, Math. Nat. Sci., 1 (2017), 18-25. [6] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. [7] M. Caputo and M. Fabrizio, On the notion of fractional derivative and applications to the hysteresis phenomena, Meccanica, 52 (2017), 3043-3052.  doi: 10.1007/s11012-017-0652-y. [8] W. C. Chew, Waves and Flelds in Inhomogenous Medias, IEEE Press, New York, 1995. [9] T.-C. Chiu and F. Erdogan, One-dimensional wave propagation in a functionally graded elastic medium, J. Sound vib., 222 (1999), 453-487.  doi: 10.1006/jsvi.1998.2065. [10] M. J. Gander, L. Halpern and F. Nata, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., 41 (2003), 1643-1681.  doi: 10.1137/S003614290139559X. [11] J. F. Gómez-Aguilar and D. Baleanu, Fractional transmission line with losses, Zeitschrift für Naturforschung A, 69 (2015), 539-546. [12] J. F. Gómez-Aguilar, J. J. Rosales-García, J. J. Bernal-alvarado, T. Córdova-fraga and R. Gujmán-cabrera, Fractional mechanical oscillators, Rev. Mex. Fis., 58 (2012), 348-352. [13] J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza, L. J. Morales-Mendoza and M. González-Lee, Universal character of the fractional space-time electromagnetic waves in dielectric media, J. Electromagnet. Wave, 29 (2015), 727-740. [14] J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, Curr. Dev. Math. Sci., Bentham Sci. Publ., Sharjah, 1 (2018), 269-341. [15] D. C. Labora, J. J. Nieto and R. Rodriguez-Lopez, Is it possible to construct a fractional derivative such that the index law holds?, Progr. Fract. Differ. Appl., 4 (2018), 1-3.  doi: 10.18576/pfda/040101. [16] T. R. Prabhakar, A singular integral equation with a genearlized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15. [17] M. B. Riaz, N. A. Asif, A. Atangana and M. I. Asjad, Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 645-664. [18] T. H. Stix, Waves in Plasmas, American Institute of Physics, 1992. [19] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Front. Phys., 5 (2017), 52. doi: 10.3389/fphy.2017.00052.

show all references

##### References:
 [1] B. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 547-551. [2] B. S. T. Alkahtani, I. Koca and A. Atangana, A novel approach of variable order derivative: Theory and methods, J. Nonlinear Sci. Appl., 9 (2016), 4867-4876.  doi: 10.22436/jnsa.009.06.122. [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [4] A. Atangana and J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 166. [5] A. Atangana and I. Koca, New direction in fractional differentiation, Math. Nat. Sci., 1 (2017), 18-25. [6] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. [7] M. Caputo and M. Fabrizio, On the notion of fractional derivative and applications to the hysteresis phenomena, Meccanica, 52 (2017), 3043-3052.  doi: 10.1007/s11012-017-0652-y. [8] W. C. Chew, Waves and Flelds in Inhomogenous Medias, IEEE Press, New York, 1995. [9] T.-C. Chiu and F. Erdogan, One-dimensional wave propagation in a functionally graded elastic medium, J. Sound vib., 222 (1999), 453-487.  doi: 10.1006/jsvi.1998.2065. [10] M. J. Gander, L. Halpern and F. Nata, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., 41 (2003), 1643-1681.  doi: 10.1137/S003614290139559X. [11] J. F. Gómez-Aguilar and D. Baleanu, Fractional transmission line with losses, Zeitschrift für Naturforschung A, 69 (2015), 539-546. [12] J. F. Gómez-Aguilar, J. J. Rosales-García, J. J. Bernal-alvarado, T. Córdova-fraga and R. Gujmán-cabrera, Fractional mechanical oscillators, Rev. Mex. Fis., 58 (2012), 348-352. [13] J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza, L. J. Morales-Mendoza and M. González-Lee, Universal character of the fractional space-time electromagnetic waves in dielectric media, J. Electromagnet. Wave, 29 (2015), 727-740. [14] J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, Curr. Dev. Math. Sci., Bentham Sci. Publ., Sharjah, 1 (2018), 269-341. [15] D. C. Labora, J. J. Nieto and R. Rodriguez-Lopez, Is it possible to construct a fractional derivative such that the index law holds?, Progr. Fract. Differ. Appl., 4 (2018), 1-3.  doi: 10.18576/pfda/040101. [16] T. R. Prabhakar, A singular integral equation with a genearlized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15. [17] M. B. Riaz, N. A. Asif, A. Atangana and M. I. Asjad, Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 645-664. [18] T. H. Stix, Waves in Plasmas, American Institute of Physics, 1992. [19] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Front. Phys., 5 (2017), 52. doi: 10.3389/fphy.2017.00052.
 [1] Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058 [2] Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 609-627. doi: 10.3934/dcdss.2020033 [3] Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171 [4] Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050 [5] Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031 [6] Ricardo Almeida, M. Luísa Morgado. Optimality conditions involving the Mittag–Leffler tempered fractional derivative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 519-534. doi: 10.3934/dcdss.2021149 [7] Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3577-3587. doi: 10.3934/dcdss.2020428 [8] Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 519-537. doi: 10.3934/dcdss.2020029 [9] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 [10] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055 [11] Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267 [12] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [13] Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173 [14] Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. doi: 10.3934/dcdss.2020042 [15] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435 [16] Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure and Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657 [17] D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843 [18] Rahmat Ali Khan, Yongjin Li, Fahd Jarad. Exact analytical solutions of fractional order telegraph equations via triple Laplace transform. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2387-2397. doi: 10.3934/dcdss.2020427 [19] Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041 [20] Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3719-3746. doi: 10.3934/dcdss.2020430

2021 Impact Factor: 1.865

## Metrics

• PDF downloads (269)
• HTML views (342)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]