Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell's equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

The local fractional Schrödinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.

Based on the core hysteresis features, the RTD-fluxgate core, while working, is repeatedly saturated with excitation field. When the fluxgate simulates, the accurate characteristic model of the core may provide a precise simulation result. As the shape of the ideal hysteresis loop model is fixed, it cannot accurately reflect the actual dynamic changing rules of the hysteresis loop. In order to improve the fluxgate simulation accuracy, a dynamic hysteresis loop model containing the parameters which have actual physical meanings is proposed based on the changing rule of the permeability parameter when the fluxgate is working. Compared with the ideal hysteresis loop model, this model has considered the dynamic features of the hysteresis loop, which makes the simulation results closer to the actual output. In addition, other hysteresis loops of different magnetic materials can be explained utilizing the described model for an example of amorphous magnetic material in this manuscript. The model has been validated by the output response comparison between experiment results and fitting results using the model.

The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.