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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]

Nematicons became a key topic of interest in liquid crystal technology in recent years. This paper contributes in understanding the fantastic features of nematicons in optics and further disciplines. This piece of research investigates nematicons for obtaining various exact solutions for Kerr and non-Kerr law nonlinearities with the help of the Kudryashov's approach and the tanh–coth technique. The acquired outcomes involve rational, periodic and hyperbolic solutions as well as their combo-type solutions for all the four cases of nonlinearity. A comparative study is conducted to show the novelty of present results with results already existing in the literature. The constraint conditions obtained ensured that the existence of these solutions is extraordinarily favorable to further investigate the dynamics of nematicons for various kinds of nonlinearity. The dynamics of the few of the obtained solutions are also discussed by 3D plots. [ABSTRACT FROM AUTHOR]

Uses of time-fractional diffusion wave model (TFDWM) in its singular case with damping-reaction terms are widely seen in classical physics applications, e.g. for the quantitative measurement of activity diagnoses light-mechanical waves resulting from many physical experiments. The goal and importance of this paper are to predict and build accurate and convincing numerical solutions for TFDWM in its singular version by employing the collective cubic uniform B-spline approach (CUBSA) and standard finite difference approach (SFDA). The fractional Caputo time derivative (FCTD) has been estimated and broken down using SFDT, whilst the standard splines will be utilized upon realizing spatial discretization. To study the prediction error of our approach, some convergence and bound results are given under certain constraints. We demonstrate applications of our collective algorithm to a couple of fractional singular-type models appearing in fluid dynamics and electromagnetics. Detailed analysis, delegate tables, and representative graphs are displayed and offered in different dimensions to handle the crossover meaning for several order values of FCTDs. Some conclusions, observations, recommendations, and future issues were briefly raised in the final section of this paper. [ABSTRACT FROM AUTHOR]

Recently, Atangana proposed new operators by combining fractional and fractal calculus. These recently proposed operators, referred to as fractal–fractional operators, have been widely used to study complex dynamics. In this paper, the COVID-19 model is considered via Atangana–Baleanu fractal-fractional operator. The Lyapunov stability for the model is derived for first and second derivative. Numerical results have developed through Lagrangian-piecewise interpolation for the different fractal–fractional operators. We develop numerical outcomes through different differential and integral fractional operators like power-law, exponential law, and Mittag-Leffler kernel. To get a better outcome of the proposed scheme, numerical simulation is made with different kernels having the memory effects with fractional parameters. [ABSTRACT FROM AUTHOR]

In this study, we present the idea of variational fractal transform method to find the analytical solution for the fractal Kawahara equation that arises in shallow water waves in the microgravity space or with the unsmooth boundary. The fractal semi-inverse method is used to establish a fractal variational formulation of the problem, which provides the analysis of the soliton solution and its physical properties. An example is considered to show that this new scheme is straightforward and efficient, and can be generalized to other nonlinear evolution equations with fractal derivatives performance. [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]

Wireless communication has become ubiquitous in almost every sector, both in military and commercial systems. Communicating objects are getting smaller and smaller. Therefore, the antennae which are the fundamental elements of this circuit must meet the requirements of their integrating environment in terms of bandwidth, multiple frequencies, smaller size and affordability. Microstrip patch antenna (MPA), a priori, should have been the preferred candidate, but the narrowness of its bandwidth is a limitation. Therefore, many techniques have been explored including the use of fractal shapes for improving the performance of this type of antenna. Multiform fractal shapes have been used in various works to achieve the results expected in specific cases to effectively meet the requirements of wireless communication. In this paper, we will do an overview by highlighting the various kinds of fractal geometries which have been the most used in recent years by clearly specifying that the options of miniaturization of microstrip antennae by the theories of Sierpinski and Minkowski are ideal methods and therefore, will drive our future work on miniaturized antenna designs. [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]

Under a microgravity condition, the microgravity-induced motion of a rigid pendulum has many special properties, which cannot be successfully revealed by any known theory. This work suggests a new fractal model for this purpose to describe the microgravity-induced motion by using the fractal derivative, and its variational principle is successfully established according to the semi-inverse method. The variational principle is very helpful to construct conservation laws and to suggest solution structures of the new fractal model. The approximate analytical solution of the model is gained by the two-scale transform method and the homotopy perturbation method. [ABSTRACT FROM AUTHOR]

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology. [ABSTRACT FROM AUTHOR]

In this paper, porous chamber with considering nanomaterial as operating fluid has been scrutinized. The transportation of nanopowder was controlled by magnetic force and insert of porous media boosts the cooling rate. Such zone needs special model to involve the impact of porous media and in this paper, non-Darcy technique was utilized. Low fraction of hybrid nanomaterial leads to good accuracy of homogeneous model and empirical correlations have been employed to forecast the features of operating fluid. Entropy generation was studied to find the influence of each term on irreversibility of unit. Also, two significant functions were calculated, namely, Be and Nu. Influences of Ra, Da and Ha on contours plots were reported in outputs. As Ra augments, the convection becomes stronger and augmentation of Ψ proves this fact. Also, temperature of elliptic surface declines about 48% with intensifying Ra. Temperature of elliptic surface augments about 34.6% with augment of Ha while it declines about 25.7% considering greater permeability. Nu augments about 238%, 11.49% with rise of Ra, Da but it declines about 28.3% with rise of Ha. Be intensifies with rise of Ha about 7.76% while it reduces about 75.2% with augmentation of Ra. [ABSTRACT FROM AUTHOR]

A fractal modification of the telegraph equation with fractal derivatives is given, and its variational principle is established by the semi-inverse method. The two-scale transform method and He's homotopy perturbation method are successfully adopted to solve the fractal equation. [ABSTRACT FROM AUTHOR]