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Zbl 1042.34028
Caputo, Michele
Distributed order differential equations modelling dielectric induction and diffusion. (English)
[J] Fract. Calc. Appl. Anal. 4, No. 4, 421-442 (2001). ISSN 1311-0454; ISSN 1314-2224/e

Summary: Distributed order differential equations are introduced in the constitutive equations of dielectric media and in the diffusion equation. To model induction phenomena, the distributed order differential operators act on the induction and on the applied electric field, to model diffusion phenomena the distributed order differential operators act on the flux and on the pressure gradient. The solution of the classic problems of dielectrics and of diffusion are first found in the frequency domain, with a discussion of their filtering properties, and then obtained, with closed form formulae, also in the time domain. For the dielectric media, the Green function is decreasing asymptotically to zero and acts on the input as a low pass filter. The Green function is found for the classic problem of diffusion, it acts also as a low pass filter. The major difference with the case when a single fractional derivative is present in the constitutive equations of dielectric media, and also in the diffusion equation, is that the solutions found here and the associated filters are potentially more flexible to represent more complex systems.

MSC 2000:: *34A99 General theory of ODE
26A33 Fractional derivatives and integrals (real functions)
34C60 Applications of qualitative theory of ODE
35K57 Reaction-diffusion equations
78A55 Technical appl. of optics and electromagnetic theory

Keywords: differential equations of distributed order; differential equation of fractional order; constitutive equations; dielectric media; diffusion

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