A Quintic B-Spline Technique for a System of Lane-Emden Equations Arising in Theoretical Physical Applications

In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application

Hundreds of dense star clusters exist in almost all galaxies. Each cluster is composed of approximately ten thousand through ten million stars. The stars orbit in the clusters due to the clusters\u27 self-gravity. Standard stellar dynamics expects that the clusters behave like collisionless self-gravitating systems on short time scales (~ million years) and the stars travel in smooth continuous orbits. Such clusters temporally settle to dynamically stable states or quasi-stationary states (QSS). Two fundamental QSS models are the isothermal- and polytropic- spheres since they have similar structures to the actual core (central part) and halo (outskirt) of the clusters. The two QSS models are mathematically modeled by the Lane-Emden equations. On long time scales (~ billion years), the clusters experience a relaxation effect (Fokker-Planck process). This is due to the finiteness of total star number in the clusters that causes stars to deviate from their smooth orbits. This relaxation process forms a highly-dense relaxed core and sparse-collisionless halo in a self-similar fashion. The corresponding mathematical model is called the self-similar Orbit-Averaged Fokker-Planck (ss-OAFP) equation. However, any existing numerical works have never satisfactorily solved the ss-OAFP equation last decades after it was proposed. This is since the works rely on finite difference (FD) methods and their accuracies were not enough to cover the large gap in the density of the ss-OAFP model. To overcome this numerical problem, we employ a Chebyshev pseudo-spectral method. Spectral methods are known to be accurate and efficient scheme compared with FD methods. The present work proposes a new method by combining the Chebyshev spectral method with an inverse mapping of variables. Our new method provides accurate numerical solutions of the Lane-Emden equations with large density gaps on MATLAB software. The maximum density ratio of the core to halo can reach the possible numerical (graphical) limit of MATLAB. The same method provides four significant figures of a spectral solution to the ss-OAFP equation. This spectral solution infers that existing solutions have at most one significant figure. Also, our numerical results provide three new findings. (i) We report new kinds of the end-point singularities for the Chebyshev expansion of the Lane-Emden- and ss-OAFP equations. (ii) Based on the spectral solution, we discuss the thermodynamic aspects of the ss-OAFP model and detail the cause of the negative heat capacity of the system. We suggest that to hold a \u27negative\u27 heat capacity over relaxation time scales stars need to be not only in a deep potential well but also in a non-equilibrium state with the flow of heat and stars. (iii) We propose an energy-truncated ss-OAFP model that can fit the observed structural profiles of at least half of Milky Way globular clusters. The model can apply to not only normal clusters but also post collapsed-core clusters with resolved (observable) cores; those clusters can not generally be fitted by a single model. The new model is phenomenological in the sense that the energy-truncation is based on polytropic models while the truncation suggests that low-concentration globular clusters are possibly polytropic clusters

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Review of Particle Physics

The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 2,143 new measurements from 709 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Among the 120 reviews are many that are new or heavily revised, including a new review on Machine Learning, and one on Spectroscopy of Light Meson Resonances. The Review is divided into two volumes. Volume 1 includes the Summary Tables and 97 review articles. Volume 2 consists of the Particle Listings and contains also 23 reviews that address specific aspects of the data presented in the Listings