Mathematica Research Articles

Mathematical Language to Reveal the Geometry of Shapes

This research reflects on using mathematics for processing and interpreting data obtained from digital surveying. The experiments outlined in this study demonstrate how mathematical algorithms, which can be written in computational environments like Wolfram Mathematica, can reveal the geometric structure of an architectural artifact.

How to Characterize 4–90 nm Size Gold Nanospheres with Experimental and Simulated UV–Vis and a Single SEM Image

It is crucial nowadays to predict in a fast and simple manner physical-chemical behaviors like, the size-dependent optical properties of gold nanospheres (Au NSs). The idea behind this experiment is trying to replace (as much as possible) robust and expensive microscopy techniques with UV–vis spectrophotometry and friendly simulations. Students chemically synthesized ∼4 and ∼15 nm diameter NSs and grew larger ones to ∼32, ∼50, and ∼70 nm diameter using the previously obtained ∼15 nm as seeds. They characterized them via UV–vis spectroscopy to ultimately compare the spectra with user-friendly software including MiePlot, nanoHUB.org, and the Amendola algorithm (Wolfram Mathematica). For Au NSs ≤25 nm diam., the experimental UV–vis spectra fitted very well with the Amendola algorithm to determine the size. For Au NSs >25 nm diam., the experiment exhibited larger Au NSs than expected as noticed by the systematic red shift of the plasmon band with respect to the simulated one. A SEM image confirmed larger Au NSs; therefore, students were challenged to determine the source of discrepancy between experimental and simulated UV–vis spectra, which was later attributed to the aging of the Au NSs seeds. This experiment demonstrates an easy way to accurately determine the nanoparticle size while exploring and solving potential issues that students may face during the chemical synthesis and simulations.

Influence of a Magnetic Field on Blood Flow Through an Inclined Tapered Vessel with a Heat Source

In this study, we investigated the effect of a heat source on blood flow through a gradient-tapered vessel under the influence of a gradient magnetic field by reformulating the problem using a mathematical model representing the blood momentum and energy equations. The partial differential equations were dimensionlessly scaled using a scaling parameter and further reduced to a system of ordinary differential equations. A coupled system of regular equations was solved using the series method to obtain analytical solutions for temperature and blood velocity profiles. Numerical simulations were performed using Wolfram Mathematica version 12 and varied parameters relevant to the investigation. The results showed that the relevant parameters—magnetic field, radiation parameter, Grashof number, and tilt angle contributed to liquid temperature and blood flow velocity, respectively. The novelty of this study is the fact that heat can be introduced from a heat source for the purpose of helping blood circulation, and magnetic fields tend to accelerate blood flow velocity.

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

The study of the flows of curves is one of the most fascinating research areas in differential geometry. In this paper, we investigate the geometry of the flows of timelike curves according to the quasi-frame in Minkowski space R2,1 (In this paper, we refer to these curves as “quasi-timelike curves”). We investigate the evolution of quasi-timelike curves using the velocity functions and obtain the necessary and sufficient conditions for inextensibility. Additionally, we obtain the explicit forms of the time evolution equations for the quasi-orthonormal frames (tangent, quasi-normal, and quasi-binormal vectors) of the quasi-timelike curve as well as the time evolution equations of their quasi-curvatures. We present a new application for motion with velocities equal to the quasi-curvatures of the quasi-timelike curve. In this application, the time evolution equations of the quasi-curvatures arise as a system of partial differential equations with the form of the heat equation, and by solving this system, we visualize the evolution of quasi-curvatures and the evolution of the quasi-timelike curve. In addition, the acceleration functions are used to investigate the flows of inextensible quasi-timelike curves, and an application for accelerations equal to the quasi-curvatures is given. Through this application, the position vector of the quasi-timelike curve satisfies the one-dimensional wave equation, and the time evolution equations of the quasi-curvatures arise as a system of transport equations. We obtain the solutions and graph them using Wolfram Mathematica 12.

Insinuation of Arrhenius Energy and Solar Radiation on Electrical Conducting Williamson Nano Fluids Flow with Swimming Microorganism: Completion of Buongiorno's Model

The enriched thermal mechanisms and progressive of nanomaterial has enthused scientists to give devotion to this area in current days. The versatile and synthesizing utilization of such particles embrace energy production, solar systems, heating and cooling monitoring processes, renewable energy systems, cancer treatments, hybrid-powered motors and Nano electronics. Furthermore, in this era of biotechnology and bioengineering, the bio convection of Nano fluids provides for some enthralling applications, such as enzymes, biosensors and biofuels. With such magnetic applications and attentions. A mathematical model is presented for evaluating the electrical conducting Williamson nano fluid with heat and mass transfer over a porous stretched sheet in the existence of bioconvection. The bioconvection of swimming microorganisms, thermal radiation,thermal conductivity and Arrhenius energy are new facets of this investigation. The higher order non-linear governing partial differential equations (PDEs) are solved by applying appropriate similarity variables and resulting couple of ordinary differential equations (ODEs) is produced. The developing set of ODEs is solved numerically by utilizing well known shooting technique with ND solve command in Wolfram MATHEMATICA and compare the result with pvb4c code in MATLAB. The graphs for different physical quantities of interest together with non-dimension velocity, temperature, concentration and density of micro-organisms profiles are discovered for involving parameters like .magnetic parameter, Brownian motion, Rayleigh number, Peclet number, Bioconvective Lewis number, parameter of thermophoresis and buoyancy ratio parameter. The influence of numerous parameters on flow and heat transfer characteristics are debated.

Comparison of moroccan argain nut shell and olive cake combustion to determine the best combustible for CHP system and for the thermodynamic cycle

The current research aims to valorize Moroccan agricultural waste by using it as a combustible fuel. The physicochemical properties of argan cake were determined and the results were compared with other studies of argain nut shell and olive cake. A comparison of argain nuts shell, argain cake, and olive cake was carried out in order to determine the best combustible in terms of energy, emissions and thermal efficiency cycle. The CFD modeling of their combustion was presented using Ansys fluent software, The Reynolds averaged Navier-Stokes (RANS) method is the foundation of the numerical approach, which models the turbulent flow using realizable.k-ε, A non-premixed combustion model is used for the gas phase, and a Lagrangian approach is chosen for the discrete second phase, there was a good agreement between the numerical and experimental data, as well as Wolfram Mathematica 13.1 is used to predict the mechanical work generated by the Stirling engine, and the result encourages the use of studied biomasses as combustible in order to generate heat and power.

Temperature effect on drug diffusion in the stomach and bloodstream compartments

In this paper, we explain the effect of temperature on the distribution of drugs between the stomach and the bloodstream compartments of the human body. The study demonstrates that the behavior of drug distribution is linked with high or low surrounding temperatures in the compartments (the stomach and bloodstream) under study. In the study, we converted real-world problems into mathematical equations and then solved the resulting remodeled system and the mathematical models found in previous research by introducing a temperature component in an attempt to understand the behavior of a drug administered in the human body over time. The analytical solutions for the drug concentration in the stomach and bloodstream were obtained using separation of variables and variation of parameters techniques of solving ODEs, while the numerical solutions and the graphical results were obtained using Wolfram Mathematica. The results show that as the temperature rises, the drug administered into the system diffuses faster between the compartments and diffuses slower as the temperature falls, which shows the novelty of this research project.

Inventory Management in Supply Chains Based on a Linear Discrete System with a Quadratic Quality Criterion

Purpose of research. To analyze theoretical studies of models and methods used in the theory of inventory management and directions of their practical implementation. To present the problem of optimal inventory management as a discrete control problem. Tasks of this type are formulated in cases where there is a need to form strategies of management of material and production inventories in supply chains, taking into account their organization at a certain time interval. To propose a method for solving the corresponding management problem.Methods. As a basic model, we consider the process of inventory management in the warehouse, in which a certain balance between the products released from the warehouse into production, released for sale, as well as its residues, which form the inventory of the next period must be fulfilled. The process of inventory management in supply chains during the planning period is based on the procedure of finding the appropriate control action for a linear discrete controlled system with a quadratic quality index. Implementation of control uses the principle of feedback. The computer algebra system Wolfram Mathematica is chosen as a tool for implementing the control.Results. Within the framework of studying the methods of finding the optimal control, two solution methods were singled out - the method of finding the optimal program control and the method of determining the optimal control according to the feedback principle. In the process of verification of the obtained solution, several model examples with different initial conditions were considered.Conclusion. The results of testing the proposed discrete model showed that the feedback method makes it possible to quite efficiently solve the inventory management problem with different initial conditions, which is described by a system of difference equations.

Diverse optical solitons to nonlinear perturbed Schrödinger equation with quadratic-cubic nonlinearity via two efficient approaches

In this study, the nonlinear perturbed Schrödinger equation(NPSE) with nonlinear terms as quadratic-cubic law nonlinearity media with the beta derivative is investigated and this investigated model is considered an icon in the field of optical fibers where it describes the wave function or state function of the optical system. Numerous solutions are extracted by engaging two novel schemes known as generalized -expansion method (GEEM) and rational extended sinh-Gordon equation expansion method (REShGEEM) in distinct forms such as bright, dark, singular and combinations of these solutions. In addition, plane wave, periodic and exponential solutions are also recovered. Using the computer application Wolfram Mathematica 11, the dynamic structure and physical characterization of some observed solutions are vividly depicted by sketching different plots. Comparing our new results with well-known literature is also given which justifies the novelty of this work. Obtained results will hold a significant place in the field of nonlinear optical fibers and suggest that the proposed methods are very influential and effective tools for solving more complex nonlinear partial differential equations in mathematical physics, engineering and nonlinear optics.

Numerical Simulation of Surface Plasmon Resonance Optical Fiber Biosensor Enhanced by Using Alloys for Medical Application.

Tuberculosis is a very dangerous disease. Therefore, early and quick diagnosis of this disease can increase the chances of overcoming it. Studies show that people with tuberculosis have a lower blood plasma refractive index than healthy people. The performance of the fiber optic sensor based on surface plasmon resonance is investigated for the metal/oxide/graphene structure and for cases where the diameter of the fiber optic core is 300, 600, and 940µm while blood plasma is considered as the sensing medium. The sensor characteristics such as sensitivity, detection accuracy and figure of merit are simulated for each structure using the theory of matrix method in Wolfram Mathematica software. The simulation results show that the aluminum/lutetium oxide/graphene structure has the highest quality factor when the core diameter of the optical fiber is 940µm. In continuation of this research, the effects of using alloys with different mixture proportions to improve the quality are investigated. According to results, the structure of aluminum/copper alloy (with a ratio of 30/70)-lutetium oxide graphene is the best choice for improving the quality of the sensor.

Eccentric pairs: Analytic gravitational waves from binary black holes in elliptic orbits

Gravitational waves (GW) from eccentric binaries have intricate signals encoding important features about the location, creation and evolution of the sources. Eccentricity shortens the merger time, making the emitted GW statistically predominant in the observed data once detectors will reach the required sensitivity. We present a novel implementation of fully analytical GW templates from eccentric binary black hole (BBH) mergers within the Wolfram Mathematica software. We increase the accuracy by identifying and minimizing the possible source of errors. We start with an overview of the physics involved in eccentric mergers, then assemble the strain for the inspiral by employing up to six post-Newtonian (PN) corrections. We complete the eccentric inspiral with the quasi-circular Backwards one Body (BOB) merger model in frequency, amplitude and phase, then we build the hybrid GW strain for the whole evolution of the binary. For low eccentricity, we reach coincidence in the overlap, with no ambiguity in the time interval, a remarkable improvement from the usual matching techniques. For high-eccentricity, we compensate for the implicit quasi-circular assumption of the BOB approach, by introducing a small rescaling in amplitude. Our streamlined implementation is relevant for the new field of GW astronomy and is straightforward to understand, use and extend, offering researchers in the field a valuable open resource tool.

Dynamical solutions and quadratic resonance of nonlinear perturbed Schrödinger equation

This work investigates the perturbed nonlinear Schrödinger equation using the modified (G′G2)-expansion method. The obtained results are generalized and classified into classes of trigonometric, hyperbolic, and rational solutions. The kinematics of soliton and kink profiles are very helpful to understand the propagation of electromagnetic waves inside nonlinear optical fibers. The proposed modified method is unique, straightforward, concise, and effective in the sense that it gives more traveling wave solutions. The findings of this study can strengthen a system's nonlinear dynamic behavior and show how practical the methodology used to attempt to replicate has been. Wolfram Mathematica 11 is used for mathematical simplification and MATLAB is used for graphical simulation.

REDUCED DIFFERENTIAL TRANSFORM METHOD FOR COUPLED PROBLEM OF THE SYMMETRIC REGULARIZED LONG WAVE EQUATIONS

The aim of this research was to deal with coupled problem of the symmetricregularized long wave equations. A semi analytical approach namely, the reduced differentialtransform method was applied to find the numerical solution of the coupled problem of symmetricregularized long wave equations. Mainly this technique was formed to compute the approximatesolution in the form of convergent power series with simple and easy calculating procedures. Thenumerical results were computed by using Wolfram Mathematica 9.0 which determined the efficacyand pertinence of the proposed method. The obtained numerical results were compared with the exactsolution which verified the recommended methodology. It was concluded that the approximate resultswere very much stable and convergent to the exact solution for the coupled problem of the symmetricregularized long wave equations.

Use of math software Mathematica to learn to derivate multiple variables functions online

Technological tools, teaching materials and students’ acceptance are key aspects for online education at current times. The use of computer algebra systems (CAS) may improve students’ perception about online learning as it offers an interactive environment where academic achievement could increase. Math software Wolfram Mathematica under its cloud version was studied for learning second semester multivariate calculus online. A quasi-experiment was conducted with control and experimental groups. Pre-tests were carried out to evaluate the similarity of groups prior treatment and post-tests to assess scores after students learnt derivatives in multivariable calculus. Two-sample t-test was used to determine how similar the means of the control and experimental groups were for pre and post-tests’ scores. Experimental group scored significantly better than control group demonstrating the positive effect this software had on multivariate calculus learning. Students from experimental group answered a survey to evaluate in more detail the experience. Results indicate that math software should be implemented in calculus courses during online education.

Flight paths over greenspace in major United States airports

Greenspaces refers to urban and wetlands that contain vegetation, essentially spaces that are green. Because greenspaces are beneficial to reducing pollution, we investigated whether if there is a trend where planes tend to fly over greenspace near airports. We used flight data to generate the flight paths and satellite images and then processed them to find if there was an observable pattern between the location of greenspaces and flight paths. Our hypothesis was that each airport would show a different percentage of planes that fly over greenspace; we expected airports in the suburbs to have higher values, whereas airports near the cities to have lower values. To test the hypothesis, we used image processing functions such as Binarize and SetAlphaChannel from Wolfram Mathematica and statistical analysis in Mathematica. We observed that generally near airports, around 10–15% of total flight paths go over greenspaces. From our observations, we believe that increasing greenspaces near airports would reduce noise pollution and having more trees will absorb the carbon dioxide from planes, resulting in less pollution near airports.

Study of Secular Perturbations in the Restricted Three-Body Problem of Variable Masses Using Computer Algebra

A nonstationary restricted three-body problem for variable masses is considered taking into account the reactive forces arising due to anisotropic variation of masses of the bodies. It is assumed that the bodies are spherically symmetric and interact in accordance with Newton’s law of gravitation. On the basis of the equations of motion of the bodies in the relative system of coordinates, differential equations of aperiodic motion along quasi-conic sections in terms of osculating elements are derived. Equations determining the secular perturbations of the orbital elements are derived in the case of small eccentricities and inclinations of orbits. All symbolic computations are performed using Wolfram Mathematica.

СПЕКТРАЛЬНЫЙ АНАЛИЗ ИНФОРМАЦИОННОЙ СИСТЕМЫ

To monitor physical processes in real time, a fast Fourier transform (FFT) algorithm is widely used, which allows spectral analysis of data. The article discusses the FFT algorithm, with which you can estimate the information parameters of the harmonic signal. The data is extracted using a recursive filter satisfying the Riccati equation. The algorithm is implemented in the Wolfram Mathematica software environment. Using this algorithm reduces the amount of calculations while maintaining the required accuracy. The article also presents the results of monitoring students of technical areas, during the training of which information technologies were used. The professional skills gained during the training allow graduates of technical areas to solve applied engineering problems, to develop mathematical support for projects.

Sensitive analysis of soliton solutions of nonlinear Landau-Ginzburg-Higgs equation with generalized projective Riccati method

<abstract><p>The study aims to explore the nonlinear Landau-Ginzburg-Higgs equation, which describes nonlinear waves with long-range and weak scattering interactions between tropical tropospheres and mid-latitude, as well as the exchange of mid-latitude Rossby and equatorial waves. We use the recently enhanced rising procedure to extract the important, applicable and further general solitary wave solutions to the formerly stated nonlinear wave model via the complex travelling wave transformation. Exact travelling wave solutions obtained include a singular wave, a periodic wave, bright, dark and kink-type wave peakon solutions using the generalized projective Riccati equation. The obtained findings are represented as trigonometric and hyperbolic functions. Graphical comparisons are provided for Landau-Ginzburg-Higgs equation model solutions, which are presented diagrammatically by adjusting the values of the embedded parameters in the Wolfram Mathematica program. The propagating behaviours of the obtained results display in 3-D, 2-D and contour visualization to investigate the impact of different involved parameters. The velocity of soliton has a stimulating effect on getting the desired aspects according to requirement. The sensitivity analysis is demonstrated for the designed dynamical structural system's wave profiles, where the soliton wave velocity and wave number parameters regulate the water wave singularity. This study shows that the method utilized is effective and may be used to find appropriate closed-form solitary solitons to a variety of nonlinear evolution equations (NLEEs).</p></abstract>

New algorithms for solving nonlinear mixed integral equations

<abstract><p>In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.</p></abstract>