American Institute of Mathematical Sciences

Advanced
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  December 2019

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: 93A30.
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

[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

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

[1]

Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & 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 & 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 & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020171
[4]

Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & 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 & Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031
[6]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039
[7]

Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 519-537. doi: 10.3934/dcdss.2020029
[8]

Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055
[9]

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 & Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267
[10]

Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031
[11]

Ndolane Sene. Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020173
[12]

Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. doi: 10.3934/dcdss.2020042
[13]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657
[14]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843
[15]

Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041
[16]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[17]

Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021
[18]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[19]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
[20]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (42)
  • HTML views (57)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences