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doi: 10.3934/dcdss.2020172

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

Krunal B. Kachhia 1,, and  Abdon Atangana 2,

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  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.

Keywords: Atangana-Koca fractional derivative, fractional operators, dielectric media, Laplace transform, Mittag-Leffler function, wave propagation.
Mathematics Subject Classification: Primary: 34A08, 97M10, 26A33; Secondary: 78A25.
Citation: Krunal B. Kachhia, Abdon Atangana. Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel. Discrete & Continuous Dynamical Systems - S, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

A. Atangana and I. Koca, New direction in fractional differentiation, Math. Nat. Sci., 1 (2017), 18-25.   Google Scholar

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[11]

J. F. Gómez-Aguilar and D. Baleanu, Fractional transmission line with losses, Zeitschrift für Naturforschung A, 69 (2015), 539-546.   Google Scholar

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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.   Google Scholar

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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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

T. R. Prabhakar, A singular integral equation with a genearlized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.   Google Scholar

[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.   Google Scholar

[18]

T. H. Stix, Waves in Plasmas, American Institute of Physics, 1992. Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

A. Atangana and I. Koca, New direction in fractional differentiation, Math. Nat. Sci., 1 (2017), 18-25.   Google Scholar

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar

[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.  Google Scholar

[8] W. C. Chew, Waves and Flelds in Inhomogenous Medias, IEEE Press, New York, 1995.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[11]

J. F. Gómez-Aguilar and D. Baleanu, Fractional transmission line with losses, Zeitschrift für Naturforschung A, 69 (2015), 539-546.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

T. R. Prabhakar, A singular integral equation with a genearlized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.   Google Scholar

[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.   Google Scholar

[18]

T. H. Stix, Waves in Plasmas, American Institute of Physics, 1992. Google Scholar

[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.  Google Scholar

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