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Lyapunov type inequality in the frame of generalized Caputo derivatives
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 |
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.
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B. T. Alkahtani,
Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 547-551.
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B. S. T. Alkahtani, I. Koca and A. Atangana,
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New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.
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A. Atangana and I. Koca,
New direction in fractional differentiation, Math. Nat. Sci., 1 (2017), 18-25.
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M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
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| [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. |
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W. C. Chew, Waves and Flelds in Inhomogenous Medias, IEEE Press, New York, 1995.
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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. |
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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. |
| [11] |
J. F. Gómez-Aguilar and D. Baleanu,
Fractional transmission line with losses, Zeitschrift für Naturforschung A, 69 (2015), 539-546.
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| [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.
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| [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.
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| [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.
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| [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. |
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