scholarly journals STABILITY ANALYSIS OF RELATIVISTIC POLYTROPES

2021 ◽  
Vol 57 (2) ◽  
pp. 407-418
Author(s):  
Abdel-Naby S. Saad ◽  
Mohamed I. Nouh ◽  
Ashraf A. Shaker ◽  
Tarek M. Kamel
Keyword(s):  
Stability Analysis ◽  
Numerical Solutions ◽  
Critical Values ◽  
Maximum Relative Error ◽  
Full Agreement ◽  
Maximum Value ◽  
Polytropic Equation ◽  
Unstable Solutions ◽  
Relativistic Parameter ◽  
The Given

We study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state, considering structures with the polytropic indices n=1(0.5)3 and illustrate the results for the relativistic parameters σ=0−0.75. We determine the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n=1(0.5)2.5, stable relativistic polytropes occur for σ less than the critical values 0.42, 0.20, 0.10, and 0.04, respectively, while unstable relativistic polytropes are obtained when σ is greater than the same values. When n=3.0 and σ>0.5, energetically unstable solutions occur. The results of critical values are in full agreement with those evaluated by several authors. Comparisons between analytical and numerical solutions of the given relativistic functions provide a maximum relative error of order 10−3.

scholarly journals INTEGRATED SYNOPTICAL METHOD FOR CALCULATING ASSIMILATION CAPACITY OF SEA WATERS

2017 ◽  
pp. 115-120
Author(s):  
Hikmat Hamid oglu Asadov ◽  
Sima Ajdar gizi Askerova
Keyword(s):  
Sea Water ◽  
Water Masses ◽  
Optimum Regime ◽  
Optimization Task ◽  
Maximum Value ◽  
Long Time ◽  
Negative Effect ◽  
Optimum Order ◽  
Assimilation Capacity ◽  
The Given

Pollution of sea waters is one of major attributes of coastal industrial centers and the norming of such emissions is one of major countermeasures. The assimilation capacity of sea waters is a major factor relevant at norming and planning of outflows into sea waters. At present time the synoptical method has been developed, which doesn’t require carrying out long time and repeated observing of the level of pollution of sea waters. This method has formed the basis for developing the integrated synoptical method for calculating sea water assimilation capacity. The suggested method provides for division of the sea waters into separated homogenous water masses. The aim of the study is to develop an inverse integrated synoptical method allowing synthesizing of such an optimum order for loading separate water masses with pollutants upon, at which the calculated total value of assimilation capacity would reach its maximum. The article shows the possibility of utilization of known synoptical method for determining assimilation capacity of sea waters in the inverse order, i.e. for calculating the maximum value of pollutant put into the fixed zone of sea waters, upon a condition of reaching the given amount of assimilation capacity and absence of essential negative effect on ecosystem. The task of calculating an optimum regime function of discrete type, upon which the integrated value of assimilation capacity would reach the maximum value, has been formulated. The solution of analogue equivalent of the formed optimization task is carried out using the Euler equation for a non-conditional variation optimization task, taking into account the accepted limitation condition. The recommendations on optimum loading of different sea water zones with determined type of pollutant have been given.

Edge (m,k)-choosability of graphs

2021 ◽  
Author(s):  
P. Soorya ◽  
K. A. Germina
Keyword(s):  
Connected Graph ◽  
Matching Number ◽  
Graph Theoretic ◽  
Maximum Value ◽  
Choice Number ◽  
The Given ◽  
Simple Connected Graph

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

State evolution laws and earthquake nucleation

2021 ◽  
Author(s):  
Robert Viesca
Keyword(s):  
Fixed Point ◽  
Stability Analysis ◽  
Direct Effect ◽  
Numerical Solutions ◽  
Slip Rate ◽  
Instability Development ◽  
State Evolution ◽  
Earthquake Nucleation ◽  
Non Linear ◽  
Evolution Laws

<p>In models of faults as elastic continua with a frictional interface, earthquake nucleation is the initiation of a propagating dynamic fault rupture nucleated by a localized slip instability. A mechanism capturing both the weakening process leading to nucleation as well as fault healing between events, is a slip rate- and state-dependent friction, with so-called direct effect and evolution effects [Dieterich, JGR 1979; Ruina, JGR 1983]. While the constitutive representation of the direct effect is theoretically supported [e.g., Nakatani, JGR 2001; Rice et al., JMPS 2001], that of the evolution effect remains empirical and a number of state-evolution laws have been proposed to fit lab rock friction data [Ruina, JGR 1983; Kato and Tullis, GRL 2001; Bar-Sinai et al., GRL 2012; Nagata et al., JGR 2012]. These laws may share a common linearization about steady-state, such that a linear stability analysis of steady, uniform sliding yields a single critical wavelength for unstable growth of perturbations [Rice and Ruina, JAM 1983]. However, the laws’ differences are apparent at later, non-linear stages of instability development.</p><div>Previously, we showed that instability development under aging-law state evolution could be understood in terms of dynamical systems [Viesca, PR-E 2016, PRS-A 2016]: the non-linear acceleration of slip occurs as the attraction of a fault’s slip rate to a fixed point, corresponding to slip rate diverging with a fixed spatial distribution and rate of acceleration. Here we show that this framework can also be applied to understand slip instability development under all commonly used evolution laws, including the so-called slip and Nagata laws. To do so, we develop an intermediate state evolution law that transitions between the slip and aging laws with the adjustment of a single parameter. We show that, to within a variable transformation, the intermediate law is equivalent to the Nagata law and that fixed-point blow-up solutions exist for any value of the transition parameter. We assess these fixed-points’ stability via a linear stability analysis and provide an explanation for previously observed behavior in numerical solutions for slip rate and state evolution under various evolution laws [Ampuero and Rubin, JGR 2008; Kame et al., 2013; Bar-Sinai et al., PR-E 2013; Bhattacharya and Rubin, JGR 2014].</div>

scholarly journals Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 96 ◽  
Author(s):  
İbrahim Avcı ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Main Idea ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

The Solution of Lateral Heat Loss Problem Using Collocation Method with Cubic B-Splines Finite Element

2011 ◽  
Vol 110-116 ◽  
pp. 3184-3190
Author(s):  
Necdet Bildik ◽  
Duygu Dönmez Demir
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Heat Loss ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
B Splines ◽  
Stability Method ◽  
The Stability ◽  
Lateral Heat

This paper deals with the solutions of lateral heat loss equation by using collocation method with cubic B-splines finite elements. The stability analysis of this method is investigated by considering Fourier stability method. The comparison of the numerical solutions obtained by using this method with the analytic solutions is given by the tables and the figure.

scholarly journals Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
Author(s):  
A. Bolandtalat ◽  
E. Babolian ◽  
H. Jafari
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Boubaker Polynomials ◽  
Fractional Differential ◽  
The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

Highly accurate technique for studying some chaotic models described by ABC-fractional differential equations of variable-order

2020 ◽  
pp. 2150018
Author(s):  
M. M. Khader ◽  
Ibrahim Al-Dayel
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Polynomial Interpolation ◽  
Numerical Technique ◽  
Variable Order ◽  
Lagrange Polynomial ◽  
Efficient Tool ◽  
Fractional Differential ◽  
The Given

The propose of this paper is to introduce and investigate a highly accurate technique for solving the fractional Logistic and Ricatti differential equations of variable-order. We consider these models with the most common nonsingular Atangana–Baleanu–Caputo (ABC) fractional derivative which depends on the Mittag–Leffler kernel. The proposed numerical technique is based upon the fundamental theorem of the fractional calculus as well as the Lagrange polynomial interpolation. We satisfy the efficiency and the accuracy of the given procedure; and study the effect of the variation of the fractional-order [Formula: see text] on the behavior of the solutions due to the presence of ABC-operator by evaluating the solution with different values of [Formula: see text]. The results show that the given procedure is an easy and efficient tool to investigate the solution for such models. We compare the numerical solutions with the exact solution, thereby showing excellent agreement which we have found by applying the ABC-derivatives. We observe the chaotic solutions with some fractional-variable-order functions.

scholarly journals Increasing the range of ultra wideband radars based on antenna arrays

2019 ◽  
Vol 30 ◽  
pp. 05017
Author(s):  
Boris Lagovsky ◽  
Alexander Samokhin ◽  
Malay Tripathy
Keyword(s):  
Cross Section ◽  
Antenna Arrays ◽  
Numerical Solutions ◽  
Narrow Band ◽  
Variational Problems ◽  
Optimal Shape ◽  
Ultra Wideband ◽  
Maximum Value ◽  
Efficient Processing ◽  
Processing Algorithms

Analytical and numerical solutions are presented for increasing the range of ultra wideband radars based on antenna arrays by optimizing their characteristics. It is shown that the shape of UWB pulses significantly affects the maximum value of the reflected signal. Solutions of variational problems of searching for the optimal shape of probing UWB pulses are investigated. Optimization of the form allows to obtain good results when using even approximate data on the dispersion dependences of the radar cross-section. The use of new efficient processing algorithms for UWB signals that do not have direct narrow-band analogs is substantiated.

scholarly journals A General Numerical Method for Analyzing the Linear Stability of Stratified Parallel Shear Flows

2014 ◽  
Vol 31 (12) ◽  
pp. 2795-2808 ◽  
Author(s):  
Tim Rees ◽  
Adam Monahan
Keyword(s):  
Stability Analysis ◽  
Numerical Method ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Analytical Approximations ◽  
Onset Of Turbulence ◽  
Parallel Shear Flows ◽  
Critical Layers ◽  
The Stability ◽  
Stability Results

Abstract The stability analysis of stratified parallel shear flows is fundamental to investigations of the onset of turbulence in atmospheric and oceanic datasets. The stability analysis is performed by considering the behavior of small-amplitude waves, which is governed by the Taylor–Goldstein (TG) equation. The TG equation is a singular second-order eigenvalue problem, whose solutions, for all but the simplest background stratification and shear profiles, must be computed numerically. Accurate numerical solutions require that particular care be taken in the vicinity of critical layers resulting from the singular nature of the equation. Here a numerical method is presented for finding unstable modes of the TG equation, which calculates eigenvalues by combining numerical solutions with analytical approximations across critical layers. The accuracy of this method is assessed by comparison to the small number of stratification and shear profiles for which analytical solutions exist. New stability results from perturbations to some of these profiles are also obtained.

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