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The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ( Q 4 -scheme) with a shape parameter and then find the precise range of the shape parameter of the Q 4 -scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures. [ABSTRACT FROM AUTHOR]

In this paper, we construct a scheme of fractional-order mathematical model for the population infected by diarrhea disease by using the four compartments S, I, T and R. The fractal-fractional derivative operator (FFO) with generalized Mittag-Leffler kernel is employed to obtain the solution of the proposed system. The system is analyzed qualitatively as well as verify non-negative unique solution. The existence and uniqueness results of fractional-order model under Atangana–Baleanu fractal-fractional operator have been proved by fixed point theory. Also error analysis has been made for the proposed fractional-order model. Simulation has been carried out for derived fractional-order scheme to check the effectiveness of the results which will help, how to prevent and control such type of epidemic in society. [ABSTRACT FROM AUTHOR]

The thin film equation is very important in electroanalytical chemistry arising in diffusion–reaction process. Considering the effort of the surface morphology on the diffusion process, the thin film equation is modified by using the fractal derivative, and its fractal variational principle is successfully established according to the semi-inverse method and two-scale transform method. It is very helpful to study the structure of the approximate solution. [ABSTRACT FROM AUTHOR]

Fractal analysis is a totally new area of research based on local fractional calculus. It has interesting applications in various fields such as a complex graph, computer graphics, the music industry, picture compression and many more fields. In this paper, we present new variants of Hadamard–Mercer-type inequalities on fractal sets ℝ α (0 < α ≤ 1) by employing generalized convex function. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hadamard–Mercer-type integral inequalities for local differentiable generalized convex functions on real linear fractal space. Finally, we give applications for probability density functions and compute new generalized means. [ABSTRACT FROM AUTHOR]

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method. [ABSTRACT FROM AUTHOR]

Snow is of porous structure and good thermal insulation property. A fractal derivative model is established to reveal its thermal property, it is extremely high thermal-stable, the whole snow will not be affected much by the sudden environmental temperature change. A simple experiment is carried out to verify the theoretical finding, and the result is helpful to design advanced materials mimicking the snow structure. [ABSTRACT FROM AUTHOR]