Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches

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.

Examining of a tumor system with Caputo derivative

“…Fractional calculus is more important real-life problems [5][6][7][8][9][10][11][12][13][14]. Recently, work on this subject has increased about physic, engineering, disease etc.…”

Two-dimensional Cattaneo-Hristov heat diffusion in the half-plane

“…As a result, different types of heat conduction equations have emerged and this has led to the development of non-classical theories on heat conduction. In this sense, fractional operators with singular or non-singular kernels have played a significant role in various types of real-world problems [1][2][3][4][5][6]. For instance, in a thin rectangular plate the non-local relation between the heat flux q (t) and the temperature gradient gradT = ∂T ∂x ∂T ∂y can be given by [7] q (t) = −k t 0 K (t − τ) gradT (x, y, τ) dτ,…”