Distributed order differential equations modelling dielectric induction and diffusion.

*(English)*Zbl 1042.34028Summary: 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:

34A99 | General theory for ordinary differential equations |

26A33 | Fractional derivatives and integrals |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

35K57 | Reaction-diffusion equations |

78A55 | Technical applications of optics and electromagnetic theory |