New graphical observations for KdV equation and KdV–Burgers equation using modified auxiliary equation method

This study is made to extract the exact solutions of Korteweg–de Vries–Burgers (KdVB) equation and Korteweg–de Vries (KdV) equation. The original idea of this work is to investigate KdV equation and KdVB equation for possible closed form solutions by employing the modified auxiliary equation (MAE) method. Exact traveling wave solutions of the considered equations are retrieved in the form of trigonometric and hyperbolic functions. Kink, periodic and singular wave patterns are obtained from the constructed solutions. The graphical illustration of the wave solutions is presented using 3D-surface plots to acquire the understanding of physical behavior of the obtained results up to possible extent.

Relativistic velocity addition from the geometry of momentum space

With the goal of developing didactic tools, we consider the geometrization of the addition of velocities in special relativity by using Minkowski diagrams in momentum space. For the case of collinear velocities, we describe two ruler-and-compass constructions that provide simple graphical solutions working with the mass-shell hyperbola in a 1+1-dimensional energy-momentum plane. In the spirit of dimensional scaffolding, we use those results to build a generalization in 1+2 dimensions for the case of non-collinear velocities, providing in particular a graphical illustration of how the velocity transverse to a boost changes while the momentum remains fixed. We supplement the discussion with a number of interactive applets that implement the diagrammatic constructions and constitute a visual tool that should be useful for students to improve their understanding of the subtleties of special relativity.

A study of an EOQ model with public-screened discounted items under cloudy fuzzy demand rate

This article deals with an economic order quantity inventory model of imperfect items under non-random uncertain demand. Here we consider the customers screen the imperfect items during the selling period. After a certain period of time, the imperfect items are sold at a discounted price. We split the model into three cases, assuming that the demand rate increases, decreases, and is constant in the discount period. Firstly, we solve the crisp model, and then the model is converted into a fuzzy environment. Here we consider the dense fuzzy, parabolic fuzzy, degree of fuzziness and cloudy fuzzy for a comparative study. The basic novelty of this paper is that a computer-based algorithm and flow chart have been given for the solution of the proposed model. Finally, sensitivity analysis and graphical illustration have been given to check the validity of the model.

A Study of an EOQ Model of Growing Items with Parabolic Dense Fuzzy Lock Demand Rate

In this article, the parabolic dense fuzzy set is defined, and its basic arithmetic operations are studied with graphical illustration. The lock set concept is incorporated in a parabolic dense fuzzy set. Then, it is applied to the problems of fishery culture via the modeling of an economic order quantity model. Here, the fingerlings are fed to reach the ideal size to fulfill the customer’s demand. The growth rate of the fingerlings is assumed as a linear function. After the sales of all fish, the pond is cleaned properly for a new cycle. Here, the model is solved in a crisp sense first. Then, we fuzzify the model considering the demand rate as a parabolic dense lock fuzzy number and obtain the result in a fuzzy environment. The main aim of our study was to find the quantity of the ordering items such that the total inventory cost gets a minimum value. Lastly, sensitivity analysis and graphical illustrations were added for better justification of our model.

Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach

It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently.

First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface

The present study provides the heat transfer analysis of a viscous fluid in the presence of bioconvection with a Caputo fractional derivative. The unsteady governing equations are solved by Laplace after using a dimensional analysis approach subject to the given constraints on the boundary. The impact of physical parameters can be seen through a graphical illustration. It is observed that the maximum decline in bioconvection and velocity can be attained for smaller values of the fractional parameter. The fractional approach can be very helpful in controlling the boundary layers of the fluid properties for different values of time. Additionally, it is observed that the model obtained with generalized constitutive laws predicts better memory than the model obtained with artificial replacement. Further, these results are compared with the existing literature to verify the validity of the present results.

2-D Analysis of Generalized Thermoelastic Porous Medium under the Effect of Laser Pulse and Microtemperature

The paper presents the analytical solutions for a generalized thermoelastic medium consisting of microtemperatures and voids subjected to a laser pulse loading the medium thermally. The 0.02 ps pulse duration of the non-Gaussian laser beam is apt for heating a homogenous isotropic elastic half-space. A method called the normal mode analysis is employed to evaluate numerically the effects of various variables such as the micro-temperature vector, variation in the fraction field of the volume, first heat flux moment tensor, temperature distribution on the stresses and displacement components of the medium. In addition, the graphical illustration of the physical response of the medium has been presented in the presence and absence of void parameters, as well as in the presence of laser pulse with two different acting periods.

A Superimposition-Based Cephalometric Method to Quantitate Craniofacial Changes

To assess the craniofacial changes related to growth and/or to orthodontic and orthognathic treatments, it is necessary to superimpose serial radiographs on stable structures. However, conventional superimposition provides only a graphical illustration of these changes. To increase the precision of growth and treatment evaluations, it is desirable to quantitate these craniofacial changes. The aims of this study were to (1) evaluate a superimposition-based cephalometric method to process numerical data for craniofacial growth changes and (2) identify a valid, reliable, and feasible method for superimposition. Forty pairs of cephalograms were analyzed at T1 and T2 (mean age 9.9 and 15.0 years, respectively). The superimposition-based cephalometric method involved relating the sagittal and vertical measurements on the T2 radiographs to the nasion and sella landmarks on the T1 radiographs. Validity and reliability were evaluated for three superimposition methods: the sella-nasion (SN); the tuberculum sella-wing (TW); and Björk’s structural. Superimposition-based cephalometrics can be used to quantify craniofacial changes digitally. The numerical data from the superimposition-based cephalometrics reflected a graphical illustration of superimposition and differed significantly from the data acquired through conventional cephalometrics. Superimposition using the TW method is recommended as it is valid, reliable, and feasible.

Replacing surface-area to volume with diffusion length

Surface-area to volume (S/V) has central place in STEM syllabuses, explaining the relation between structure and function governed by the heat equation, such as in diffusion and heat transfer by conduction. However, teaching the abstract S/V quotient faces many difficulties, due to the need for high reasoning abilities and visual-spatial skills. Surprisingly, an exploratory survey among 64 high school biology teachers revealed that 12.5% of them teach the relation between structure and function in diffusion by one dimensional, tangible, quotient free, explanation: small and non-spherical structure results in short diffusion length, which results in fast diffusion. Those teachers tend to hold higher academic degree (p= 0.002) and be more experienced (p=0.045). A plausible explanation for the limited usage in the 'diffusion length' model may be the lack of mathematical framework and graphical illustration to support it. Here I link the diffusion length, volume (V) and surface-area (S) and show that the average diffusion length = 3V/S and present an illustration of V/S. Therefore, the small, non-spherical shapes of structures adopted for fast diffusion can be equivalently explained by the short diffusion length in these structures. Having the necessary mathematical framework and graphical illustration should help other teachers adopt this simple explanation.