On Fractional Spherically Restricted Hyperbolic Diffusion Random Field

The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial-temporal homogeneous and isotropic random fields and their spherical restrictions are studied. The spectral representations of these fields are derived and the associated angular spectrum is analysed. The obtained mathematical results are illustrated by numerical examples. In addition, the numerical investigations assess the dependence of the covariance structure and other properties of these fields on the orders of fractional derivatives.Comment: 32 pages, 18 figure

Some inverse source problems of determining a space dependent source in fractional-dual-phase-lag type equations

The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the formh(t)f(x)with a known functionh(t). The unknown space dependent sourcef(x)is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions onh(t)(monotonically increasing character ofh(t)was removed) in a case ofdominant parabolicbehavior. The proof technique was based on spectral analysis. Section Modified Model for tau q>tau Tshows that an analogy of Theorem 2 fordominant hyperbolicbehavior (fractional Cattaneo-Vernotte equation) is not possible

The Zoo of Non-Fourier Heat Conduction Models

The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or time lags become dominant and the memory or/and spatial non-local effects significant -- in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid state laser medium at the high pump density or under the ultra-short pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from $10^{-8}$ to $10^{-14}$ sec, however, it could be as high as 1 sec in the degenerate cores of aged stars and its reported values in granular and biological objects varies up to 30 sec. The paper considers numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory (hereditary materials, including fractional hereditary materials) or/and spatial non-locality, i.e. materials with non-homogeneous inner structure

Magnetic helicity in stellar dynamos: new numerical experiments

The theory of large scale dynamos is reviewed with particular emphasis on the magnetic helicity constraint in the presence of closed and open boundaries. In the presence of closed or periodic boundaries, helical dynamos respond to the helicity constraint by developing small scale separation in the kinematic regime, and by showing long time scales in the nonlinear regime where the scale separation has grown to the maximum possible value. A resistively limited evolution towards saturation is also found at intermediate scales before the largest scale of the system is reached. Larger aspect ratios can give rise to different structures of the mean field which are obtained at early times, but the final saturation field strength is still decreasing with decreasing resistivity. In the presence of shear, cyclic magnetic fields are found whose period is increasing with decreasing resistivity, but the saturation energy of the mean field is in strong super-equipartition with the turbulent energy. It is shown that artificially induced losses of small scale field of opposite sign of magnetic helicity as the large scale field can, at least in principle, accelerate the production of large scale (poloidal) field. Based on mean field models with an outer potential field boundary condition in spherical geometry, we verify that the sign of the magnetic helicity flux from the large scale field agrees with the sign of alpha. For solar parameters, typical magnetic helicity fluxes lie around 10^{47} Mx^2 per cycle.Comment: 23 pages, 27 figures, Astron. Nach

Kernel Functions and New Applications of an Accurate Technique

In this article, some general reproducing kernel Sobolev spaces was constructed. We find the general functions in these reproducing kernel Sobolev spaces. Many higher order boundary value problems can be investigated by these special functions