scholarly journals Preliminary Orbit Determination of Artificial Satellites: A Vectorial Sixth-Order Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Carlos Andreu ◽  
Noelia Cambil ◽  
Alicia Cordero ◽  
Juan R. Torregrosa
Keyword(s):  
Iterative Methods ◽  
Orbit Determination ◽  
Order Of Convergence ◽  
Classical Method ◽  
Numerical Approach ◽  
Original Method ◽  
Weight Functions ◽  
Artificial Satellites ◽  
Local Order ◽  
The Stability

A modified classical method for preliminary orbit determination is presented. In our proposal, the spread of the observations is considerably wider than in the original method, as well as the order of convergence of the iterative scheme involved. The numerical approach is made by using matricial weight functions, which will lead us to a class of iterative methods with a sixth local order of convergence. This is a process widely used in the design of iterative methods for solving nonlinear scalar equations, but rarely employed in vectorial cases. The numerical tests confirm the theoretical results, and the analysis of the dynamics of the problem shows the stability of the proposed schemes.

scholarly journals Design of High-Order Iterative Methods for Nonlinear Systems by Using Weight Function Procedure

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Santiago Artidiello ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Methods ◽  
Order Of Convergence ◽  
Nonlinear Function ◽  
Nonlinear Problems ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Systems Of Nonlinear Equations ◽  
Functional Evaluation ◽  
Numerical Tests ◽  
High Order Iterative Methods

We present two classes of iterative methods whose orders of convergence are four and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step. Moreover, we show an extension to higher order, adding only one functional evaluation of the vectorial nonlinear function. We perform numerical tests to compare the proposed methods with other schemes in the literature and test their effectiveness on specific nonlinear problems. Moreover, some real basins of attraction are analyzed in order to check the relation between the order of convergence and the set of convergent starting points.

scholarly journals Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 47
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz ◽  
U. Iturrarán-Viveros ◽  
R. Caballero-Cruz
Keyword(s):  
Fractional Derivative ◽  
Iterative Methods ◽  
Fractional Integral ◽  
Order Of Convergence ◽  
Fractional Operators ◽  
Speed Of Convergence ◽  
Newton Raphson ◽  
Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

scholarly journals On the optimal control of coronavirus (2019-nCov) mathematical model; a numerical approach

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
N. H. Sweilam ◽  
S. M. Al-Mekhlafi ◽  
A. O. Albalawi ◽  
D. Baleanu
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Weighted Average ◽  
Necessary Optimality Conditions ◽  
Numerical Approach ◽  
Population Number ◽  
Infected People ◽  
The Stability ◽  
Novel Coronavirus ◽  
Theoretical Results

Abstract In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald–Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

The Stability of an Axially–Loaded Continuous Beam

1955 ◽  
Vol 59 (535) ◽  
pp. 506-509
Author(s):  
A. M. Dobson
Keyword(s):  
Axial Load ◽  
Classical Method ◽  
Complete Solution ◽  
Critical Value ◽  
Continuous Beam ◽  
Independent Variables ◽  
A Value ◽  
Method Of Solution ◽  
Axially Loaded ◽  
The Stability

The Classical method of solution of the stability of an axially–loaded continuous beam is by means of the three moments equation, using the Berry Functions, which are functions of the axial load. As the axial load approaches a value equal to the critical value for a pin–jointed beam, the Berry Functions tend to infinity, and the use of the three moments equations —(i. e. treating the end fixing moments as the independent variables)—leads to certain difficulties in the complete solution of the problem.The major difficulty lies in the question of stability. The critical value is determined by the vanishing of the determinant of the coefficients of the fixing moments in the three moments equations. This value could be found by plotting the determinant against end load (c. f. Pippard and Pritchard). However, in a problem involving a large number of bays, the calculation necessary to do this is likely to be considerable, for there may be many branches to the curve.

scholarly journals Insights into the dynamic trajectories of protein filament division revealed by numerical investigation into the mathematical model of pure fragmentation

2021 ◽  
Vol 17 (9) ◽  
pp. e1008964
Author(s):  
Magali Tournus ◽  
Miguel Escobedo ◽  
Wei-Feng Xue ◽  
Marie Doumic
Keyword(s):  
Mathematical Theory ◽  
Length Distribution ◽  
Numerical Approach ◽  
Distribution Profile ◽  
Fragmentation Dynamics ◽  
Mathematical Equations ◽  
Fragmentation Reactions ◽  
The Stability ◽  
Fragmentation Equation ◽  
The Mathematical Model

The dynamics by which polymeric protein filaments divide in the presence of negligible growth, for example due to the depletion of free monomeric precursors, can be described by the universal mathematical equations of ‘pure fragmentation’. The rates of fragmentation reactions reflect the stability of the protein filaments towards breakage, which is of importance in biology and biomedicine for instance in governing the creation of amyloid seeds and the propagation of prions. Here, we devised from mathematical theory inversion formulae to recover the division rates and division kernel information from time dependent experimental measurements of filament size distribution. The numerical approach to systematically analyze the behaviour of pure fragmentation trajectories was also developed. We illustrate how these formulae can be used, provide some insights on their robustness, and show how they inform the design of experiments to measure fibril fragmentation dynamics. These advances are made possible by our central theoretical result on how the length distribution profile of the solution to the pure fragmentation equation aligns with a steady distribution profile for large times.

Influence The Direction Of Blowing Gas Through A Porous Surface On The Stability Of The Supersonic Boundary Layer

2015 ◽  
Vol 10 (2) ◽  
pp. 18-26
Author(s):  
Sergey Gaponov ◽  
Aleksandr Semenov
Keyword(s):  
Boundary Layer ◽  
Classical Method ◽  
Supersonic Boundary Layer ◽  
Porous Surface ◽  
Boundary Layer Stability ◽  
Injection Angle ◽  
Plane Plate ◽  
The Stability ◽  
Gas Blowing ◽  
Elementary Waves

In the paper the influence of the gas blowing direction through a porous surface on the supersonic boundary layer stability is investigated theoretically, using the classical method of elementary waves and the evolutionary method at Mach number M = 2. It was found that with decreasing of the gas injection angle to the plane plate the boundary layer stability was improved and the tangential blowing effect on the boundary layer stability is little in a comparison with the case of a boundary layer without mass exchange.

Factors affecting constancy of acetate concentration and correct determination of methanogenic activity in pH-stat experiments

2003 ◽  
Vol 48 (6) ◽  
pp. 111-118
Author(s):  
M. Mösche ◽  
U. Meyer
Keyword(s):  
Accurate Determination ◽  
Original Method ◽  
Experimental Conditions ◽  
Methanogenic Activity ◽  
Factors Affecting ◽  
Acetate Concentration ◽  
Correct Determination ◽  
Ph Stat ◽  
The Stability

The determination of methanogenic activity with a pH-stat titration bioassay is evaluated utilising a mathematical model of this system. For given kinetic parameters and experimental conditions the model calculates the development of titrant flow and acetate concentration during experiments. Simulations of experiments under various conditions are compared. They show that the original method inherently causes a strong drift of acetate concentration during the experiments and a misestimation of methanogenic activity. As a solution to these disadvantages the addition of sodium hydroxide to the titrant and a careful control of pH during flushing the reactor with gas prior to the experiment are recommended. In this way a better constancy of acetate concentration and a more accurate determination of methanogenic activity should be achievable. The accuracy of this method is limited by the stability of pH-electrode calibration parameters.

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