Higher Order Methods of the Basic Family of Iterations via S-Iteration Scheme with s-Convexity

There are many methods for solving a polynomial equation and many different modifications of those methods have been proposed in the literature. One of such modifications is the use of various iteration processes taken from the fixed point theory. In this paper, we propose a modification of the iteration processes used in the Basic Family of iterations by replacing the convex combination with an s-convex one. In our study, we concentrate only on the S-iteration with s-convexity. We present some graphical examples, the so-called polynomiographs, and numerical experiments showing the dependency of polynomiograph’s generation time on the value of the s parameter in the s-convex combination

Mandelbrot and Julia Sets via Jungck-CR Iteration with s-convexity

In today’s world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of s-convex combination. The Jungck-CR iteration process with s-convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria

The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets

In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the (k + 1)st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration

Augmentation of Fatigue and Tensile Strength of AA-6061 Processed through Equal Channel Angular Pressing

ECAP (Equal Channel Angular Pressing) is a technique used to enhance the strength of material by grain refinement. In this research, an aerospace grade aluminum alloy-6061 is investigated. The specimens were pressed through ECAP die channels, intersecting each other at an angle of 90oC where a shear plane of 45oC was developed, that results grains refinement. Fatigue strengths and CGR (Crack Growth Rate) for the stress ratio R 0.7 and 0.1 are found and compared with the as-received material.It was observed that the CGR is slower at stress ratio R=0.1, as compared to stress ration R=0.7. An electric furnace was embedded with ECAP die to regulate the material flow through this die. The temperature of the die was maintained at 450oC during ECAP pressing and the specimens were also preheated at this temperature using another furnace. The ECAP die consistsof two channels intersecting at 90o provided with safe inner and outer corner radius to avoid scaling.The microstructural observations revealed that the deformation was perfectly plastic. The ECAPed and as-received materials were also characterized by tensile tests, micro-hardness tests, and 3-point bend fatigue tests

Fractal generation via CR iteration scheme with s-convexity

The visual beauty, self-similarity, and complexity of Mandelbrot sets and Julia sets have made an attractive eld of research. One can nd many generalizations of these sets in the literature. One such generalization is the use of results from xed-point theory. The aim of this paper is to provide escape criterion and generate fractals (Julia sets and Mandelbrot sets) via CR iteration scheme with s-convexity. Many graphics of Mandelbrot sets and Julia sets of the proposed three-step iterative process with s-convexity are presented. We think that the results of this paper can inspire those who are interested in generating automatically aesthetic patterns.Dong-A University Funds, Busan, South Koreahttp://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639am2020Mathematics and Applied Mathematic

Rheological effects of micropolar slime on the gliding motility of bacteria with slip boundary condition

The gliding organisms are phylogenetically diverse with their hundreds of types, different shapes and several mechanism of motility. Gliding bacteria are rod-shaped bacteria without any flagella on their surface. They exhibit a creeping type of self-powered motion when nearly in contact with a solid surface. These bacteria leave an adhesive trail of slime and propel themselves by producing undulating waves in their body, which is one possible mode of motility for gliding bacteria. In the present study an undulating surface model is considered to discuss this type of bacterial locomotion. The classical Navier-Stokes equations are incapable of explaining the slime rheology at the microscopic level. Micropolar fluid dynamics however provides a solid framework for mimicking bacterial physical phenomena at both micro and nano-scales, and therefore in the present study, the constitutive equations of micropolar fluid are implemented to characterize the rheology of the slime. The flow equations are formulated under long wavelength and low Reynolds number assumptions. Exact expressions for stream function and pressure gradient are obtained. The speed of the gliding bacteria is numerically calculated by using a modified Newton-Raphson method. In addition, when the glider is fixed, the effects of micropolar slime parameters on the velocity, micro-rotation (angular velocity) of spherical slime particles, pressure rise per wavelength, pumping and trapping phenomena are also shown graphically and discussed in detail. The study is relevant to emerging biofuel cell technologies and also bacterial biophysics

Fractals via Generalized Jungck–S Iterative Scheme

The purpose of this research is to introduce a Jungck–S iterative method with m,h1,h2–convexity and hence unify different comparable iterative schemes existing in the literature. A Jungck–S orbit is constructed, and escape radius is derived with our scheme. A new escape radius is also obtained for generating the fractals. Julia and Mandelbrot set are visualized with the help of proposed algorithms based on our iterative scheme. Moreover, we present some complex graphs of Julia and Mandelbrot sets using the derived orbit and discuss their nature in detail