Abstract
The paper addresses diffusion approximations of magnetic field penetration of ferromagnetic materials with emphasis on fractional calculus applications and relevant approximate solutions. Examples with applications of time-fractional semi-derivatives and singular kernel models (Caputo time fractional operator) in cases of field independent and field-dependent magnetic diffusivities have been developed: Dirichlet problems and time-dependent boundary condition (power-law ramp). Approximate solutions in all theses case have been developed by applications of the integral-balance method and assumed parabolic profile with unspecified exponents. Tow version of the integral method have been successfully implemented: SDIM (single integration applicable to time-fractional semi-derivative model) and DIM (double-integration model to fractionalized singular memory models). The fading memory approach in the sense of the causality concept and memory kernel effect on the model constructions have been discussed.
Highlights
There are many natural phenomena which can be modelled in diffusion approximations
Magnetic field diffusions in solid ferromagnetics is considered with attempts to apply approximate solution based on synergism of fractional calculus and the integral-balance method in different versions
The models and the solutions developed consider the magnetic material as a semi-infinite with a boundary condition at x = 0 since we are interested in the laws behind the magnetic field front propagation; before reaching the physical limit L of the medium as in solution
Summary
There are many natural phenomena which can be modelled in diffusion approximations. Magnetic field diffusions in solid ferromagnetics is considered with attempts to apply approximate solution based on synergism of fractional calculus and the integral-balance method in different versions. The main idea is to demonstrate the feasibility of both the fractional calculus approach and the integral solution. In the context of the main idea of this communication magnetic diffusion of a field with parallel lines (see Figure 1) is taken as example. Two basic cases considering filed-independent and fielddependent diffusions with fixed (Dirichlet) and time dependent (power-law) boundary conditions are chosen as test examples. The problem of magnetic field diffusion with memory is discussed with either the common time fractional operator of Caputo with singular kernel or from the more fundamental fading memory principle allowing different memory functions to be used
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