Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed.

Dimension or scale is everything. When a thing is observed by different scales, different results can be obtained. Two scales are enough for most of practical problems, and a new definition of a two-scale dimension instead of the fractal dimension is given to deal with discontinuous problems. Fractal theory considers a self-similarity pattern, which cannot be found in any a real problem, while the two-scale theory observes each problem with two scales, the large scale is for an approximate continuous problem, where the classic calculus can be fully applied, and on the smaller scale, the effect of the porous structure on the properties can be easily elucidated. This paper sheds a new light on applications of fractal theory to real problems.

This paper highlights Li-He’s approach in which the enhanced perturbation method is linked with the parameter expansion technology in order to obtain frequency amplitude formulation of electrically actuated microbeams-based microelectromechanical system (MEMS). The governing equation is a second-order nonlinear ordinary differential equation. The obtained results are compared with the solution achieved numerically by the Runge-Kutta (RK) method that shows the effectiveness of this variation in the homotopy perturbation method (HPM).

The current paper deals with the transmission of MERS-CoV model between the humans populace and the camels, which are suspected to be the primary source for the infection. The effect of time MERS-CoV disease transmission is explored using a non-linear fractional order in the sense of Caputo operator in this paper. The considered model is analyzed for the qualitative theory, uniqueness of the solution are discussed by using the Banach contraction principle. Stability analysis is investigated by the aid of Ulam-Hyres (UH) and its generalized version. Finally, we show the numerical results with the help of generalized Adams-Bashforth-Moulton Method (GABMM) are used for the proposed model, for supporting our analytical work.