Multipole Moments of Fractal Distribution of Charges
نویسنده
چکیده
Integrals and derivatives of fractional order have found many applications in recent studies in science. The interest in fractals and fractional analysis has been growing continually in the last few years. Fractional derivatives and integrals have numerous applications: kinetic theories [1, 2, 3]; statistical mechanics [4, 5, 6]; dynamics in complex media [7, 8, 9, 10, 11]; electrodynamics [12, 13, 14, 15] and many others. The new type of problem has increased rapidly in areas in which the fractal features of a process or the medium imposes the necessity of using non-traditional tools in smooth physical models. In order to use fractional derivatives and fractional integrals for fractal distribution, we must use some continuous medium model [7, 8]. We propose to describe the fractal distribution by a fractional continuous medium [7, 8], where all characteristics and fields are defined everywhere in the volume but they follow some generalized equations which are derived by using fractional integrals. In many problems the real fractal structure of a medium can be disregarded and the fractal distribution can be replaced by some fractional continuous mathematical model. By smoothing of microscopic characteristics over the physically infinitesimal volume, we transform the initial fractal distribution into a fractional continuous model [7, 8] that uses the fractional integrals. The order of fractional integral is equal to the fractal dimension of distribution. The fractional integrals allow us to take into account the fractality of the media [7]. Fractional integrals are considered as approximations of integrals on fractals [16]. It was proved that integrals on net of fractals can be approximated by fractional integrals [16]. In Refs. [4, 5], we proved that fractional integrals can be considered as integrals over the space with fractional dimension up to a numerical factor. In this paper, we consider electric multipole moments of the fractal distribution of charges. Fractal distribution is described by the fractional continuous model [7, 8, 9, 10]. In the general case, the fractal distribution cannot be considered as a continuous distribution. There are domains that are not filled by particles. We suggest [7] to consider the fractal distribution as a special (fractional) continuous
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