Modifying Maxwell’s equations for dielectric materials based on techniques from viscoelasticity and concepts from fractional calculus. (English) Zbl 1423.78009
Summary: A mathematical model of viscoelasticity employing fractional order derivatives is adapted and applied to model the dielectric behavior of materials while remaining consistent with thermodynamic principles. The model is then incorporated into Maxwell’s equations using techniques from viscoelasticity. The modified Maxwell’s equations are found to yield a fractional order wave equation that is solved analytically and is found to remain consistent with dissipative and dispersive phenomena.
MSC:
| 78A30 | Electro- and magnetostatics |
| 35R11 | Fractional partial differential equations |
| 35Q61 | Maxwell equations |
Keywords:
Maxwell’s equations; viscoelasticity; dielectrics; fractional calculus; generalized derivatives; wave equationReferences:
| [1] | Bagley, R. L.; Torvik, P. J., On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30, 133-155, (1986) · Zbl 0613.73034 |
| [2] | Caputo, M., Free modes splitting and alterations of electrochemically polarizable media, Rendiconti Lincei, 4, 2, 89-98, (1993) |
| [3] | Caputo, M., The set valued unified model of dispersion and attenuation for wave propagation in dielectric (and anelastic) media, Annals of Geophysics, 41, 5-6, (1998) |
| [4] | Caputo, M.; Mainardi, F., Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento (1971-1977), 1, 2, 161-198, (1971) |
| [5] | Carew, E. O., Doehring, T. C., Barber, J. E., Freed, A. D. & Vesley, I. (2003). Fractional-order viscoelasticity applied to heart valve tissues. In Proc summer bioeng conf, key biscayne, FL, Jun 25-29 2003. |
| [6] | Christensen, R. M., Theory of viscoelasticity, (1982), Dover Publications Inc |
| [7] | Cole, K. S.; Cole, R. H., Dispersion and absorption in dielectrics I. alternating current characteristics, The Journal of Chemical Physics, 9, 341-351, (1941) |
| [8] | Fabry, B.; Maksym, G. N.; Butler, J. P.; Glogauer, M.; Navajas, D.; Fredberg, J. J., Scaling the microrheology of living cells, Physical Review Letters, 87, 148102-1-148102-4, (2001) |
| [9] | Glöckle, W. G.; Nonnenmacher, T. F., A fractional calculus approach to self-similar protein dynamics, Biophysical Journal, 68, 1, 46-53, (1995) |
| [10] | Hilfer, R., Experimental evidence for fractional time evolution in Glass forming materials, Chemical Physics, 284, 1, 399-408, (2002) |
| [11] | Mainardi, F., Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, (2010), World Scientific · Zbl 1210.26004 |
| [12] | Metzler, R.; Schick, W.; Kilian, H.-G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, The Journal of Chemical Physics, 103, 7180, (1995) |
| [13] | Oldham, K. B.; Spanier, J., The fractional calculus, (1974), Academic Press · Zbl 0428.26004 |
| [14] | Podlubny, I., Fractional differential equations: An introduction to fractional derivatives, Fractional differential equations, to methods of their solution and some of their applications, (1999), Academic Press · Zbl 0924.34008 |
| [15] | Von Hippel, A. R., Dielectric materials and applications, (1966), MIT Press |
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