Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches

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.

Non-classical heat equation with singular memory term

In this paper, we consider the non-classical heat equation with singular memory term. This equation has many applications in various fields, for example liquids mechanics, solid mechanics, and heat conduction theory first, we prove that the solution exists locally in time. Then we investigate the converegence of the mild solution of non-classical heat equation, and the mild solution of classical heat equation.

Response functions in linear viscoelastic constitutive equations and related fractional operators

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.