Derivative Nonlinear Equations Research Articles

On the local convergence of efficient Newton‐type solvers with frozen derivatives for nonlinear equations

On the local convergence of efficient Newton‐type solvers with frozen derivatives for nonlinear equations

The dynamics of COVID-19 in the UAE based on fractional derivative modeling using Riesz wavelets simulation

The well-known novel virus (COVID-19) is a new strain of coronavirus family, declared by the World Health Organization (WHO) as a dangerous epidemic. More than 3.5 million positive cases and 250 thousand deaths (up to May 5, 2020) caused by COVID-19 and has affected more than 280 countries over the world. Therefore studying the prediction of this virus spreading in further attracts a major public attention. In the Arab Emirates (UAE), up to the same date, there are 14,730 positive cases and 137 deaths according to national authorities. In this work, we study a dynamical model based on the fractional derivatives of nonlinear equations that describe the outbreak of COVID-19 according to the available infection data announced and approved by the national committee in the press. We simulate the available total cases reported based on Riesz wavelets generated by some refinable functions, namely the smoothed pseudosplines of types I and II with high vanishing moments. Based on these data, we also consider the formulation of the pandemic model using the Caputo fractional derivative. Then we numerically solve the nonlinear system that describes the dynamics of COVID-19 with given resources based on the collocation Riesz wavelet system constructed. We present graphical illustrations of the numerical solutions with parameters of the model handled under different situations. We anticipate that these results will contribute to the ongoing research to reduce the spreading of the virus and infection cases.

NLD Equation of State for compressed, hot and relativistic nuclear matter

We investigate the properties of compressed and hot hadronic matter within the Non-Linear Derivative (NLD) formalism. The novel feature of the NLD model is an explicit momentum dependence of the mean-fields, which is regulated by cut-off’s of natural hadronic scale. It is covariantly and thermodynamical-consistently formulated at the basis of a field-theoretical level. We show that the NLD model describes adequately all the empirical information of cold nuclear matter as function of density (equation of state) and, in particular, as function of particle momenta (optical potential). Finally, we present predictions of the NLD approach for hot and compressed hadronic matter in terms of the Equation of state as function of density and temperature. These studies are relevant for the forthcoming experiments at FAIR@GSI. They are also important for astrophysical purposes, e.g., static neutron stars and dynamic neutron star binary systems.

ФИЗИЧЕСКАЯ МОДЕЛЬ И МАТЕМАТИЧЕСКОЕ ОПИСАНИЕ ДВИЖЕНИЯ МЕТАНА В ГОРНОМ МАССИВЕ С УЧЕТОМ КОНЕЧНОЙ СКОРОСТИ РАСПРОСТРАНЕНИЯ ДАВЛЕНИЯ

The physical model and the mathematical description of the methane movement in the rock massif are substantiated, taking into account the final pressure propagation velocity. It is proved that the mathematical model of gas filtration in coal seams and host rocks is based on an equation of hyperbolic type and it is shown that the use of parabolic type equations is physically justifiedfor long periods of time. It is proposed to use the law of resistance, which is a functional relationship between the gas flow and the gas pressure gradient and the local rate of change of the gas flow for an arbitrarily selected point in the considered region of the rock mass. The derivation of non-linear equations of the hyperbolic type for modeling methane filtration in coal seams and host rocks is presented. Variants of linearized equations of hyperbolic type for modeling methane filtration are presented and their use as approximate mathematical models is recommended.

Using symbolic computation in Mathematica for verifying the convergence of the iterative methods

In [Jisheng Kou, The improvements of modified Newton’s method, Appl. Math. Comput., 189 (2007) 602–609], the improvements of some thirdorder modifications of Newton’s method for solving nonlinear equations are presented. In this paper we point out some flaws in the results of Jisheng Kou and we correct them by using symbolic computation in Mathematica. In [M. A. Noor et al., A new modified Halley method without second derivatives for nonlinear equations, Appl. Math. Comput., 189 (2007) 1268–1273] , the error equation obtained for the new method presented is wrong. We present the correct result by using symbolic computation, too. Finally, we present two examples of very simply proofs for the convergence of iterative methods by using symbolic computation. We consider that the Mathematica programs in this paper are good examples for other authors to prove the convergence of the iterative methods or to verify their results.

Improved Performance of Wind-driven Isolated Induction Generators by Six-phase Operation

paper describes a new, generalized and efficient steady- state model for the performance analysis of isolated six-phase induction generators. The mathematical model is formed directly from the equivalent circuit of six-phase induction generator by nodal admittance method. The proposed model is very simple which completely avoids lengthy derivations of non linear equations. The model results in matrix form so that inclusion or elimination of any equivalent circuit elements can be easily achieved. Also, this model is flexible to find any combination of unknown quantities of the equivalent circuit. The matrix equation is solved by genetic algorithm to determine the steady-state performance of isolated six-phase induction generator (ISPIG). To validate the improvement of performance by six-phase operation, the experimental and theoretical results were compared with three-phase operation. In addition, the winding diagram of the six-phase / three- phase induction generator which is used as a prototype model for the experimental study is also presented. Keywords-phase induction generator, Steady-state analysis, Genetic

A Novel Steady State Modeling and Analysis of Six-phase Self-excited Induction Generators for Renewable Energy Generation

This paper presents a simple and generalized steady state model and analysis of six-phase self-excited induction generators. The developed matrix equation is formulated using nodal admittance method based on inspection. This model does not involve any lengthy derivations of nonlinear equations which are followed so far. Also this model is flexible such that inclusion or elimination of any equivalent circuit elements can be carried out easily. Moreover, this model can be used to find any combination of unknown quantities of the equivalent circuit. To determine the steady state performance of six-phase self-excited induction generators, applications of genetic algorithms have been proposed. In addition, the details of winding scheme of the six-phase induction generator which is used as prototype model for the experimental study is also presented. The experimental and theoretical results are found to be in close agreement, which validates the proposed method.

Third-order family of methods in Banach spaces

Recently, Parida and Gupta [P.K. Parida, D.K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math. 206 (2007) 873–877] used Rall’s recurrence relation approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li [J.-S. Kou, Y.-T. Li, Modified Chebyshev’s method free from second derivative for non-linear equations, Appl. Math. Comput. 187 (2007) 1027–1032] and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a nonlinear equation in a Banach space. We give several applications to our method.

A note on modified Householder iterative method free from second derivatives for nonlinear equations

In Noor and Gupta [M.A. Noor, V. Gupta, Modified Householder iterative method free from second derivatives for nonlinear equations, Appl. Math. Comput. 190 (2) (2007) 1701–1706.], a new two-step predictor–corrector type iterative method free from second derivatives for solving nonlinear equations has been proposed. In this note, we show that the iterative method given by the authors has fifth-order convergence, not the fourth-order one as claimed in their work.

The effect of membrane permeability on the response of a catechol biosensor

A mathematical model is developed for the simulation of the amperometric response of a biosensor for catechol using polyphenoloxidase. The model is based on transient diffusion equations containing nonlinear terms of Michaelis-Menten for two space regions: the diffusion layer and the biomembrane containing the immobilized enzyme. The set of partial derivatives of nonlinear equations and the corresponding boundary and initial conditions was solved using the implicit finite difference technique. This numerical solution was then exploited to study the effects of permeability and thickness of the biomembrane on the maximum response of the reduction current and the amplification factor corresponding to the maximum of catalytic activity of the enzyme. This amplification factor increases with the thickness of the biomembrane while permeability is weak. In the case of the low initial concentrations (10−6 to 5.10−4 mM), its value is maximal and remains independent of substrate concentration. Also, the amplification factor is more significant when the diffusion resistance is more important, i.e. for high thicknesses or weak permeabilities of the biomembranes.

A note on “New classes of iterative methods for nonlinear equations” and “Some iterative methods free from second derivatives for nonlinear equations”

In [M.A. Noor, New classes of iterative methods for nonlinear equations, Appl. Math. Comput., in press; M.A. Noor, Some iterative methods free from second derivatives for nonlinear equations, Appl. Math. Comput., in press], Noor introduced a generalized one parameter Halley’s method x n + 1 = x n - f ( x n ) f ′ 2 ( x n ) f ′ 3 ( x n ) - α f ( x n ) f ″ ( x n ) for solving the nonlinear equation f( x) = 0. Noor further showed that for α = 1 2 f ′ ( x n ) , the above method reduces to the Halley’s method [E. Halley, A new exact and easy method for finding the roots of equations generally and without any previous reduction, Philos. Roy. Soc. London 18 (1964) 136–147]. It is interesting to note that for α = f ′ 3 ( x n ) f ( x n ) f ′ ′ ( x n ) , the above method fails. In this note, we point out some major bugs in the results of Noor (in press).

Some iterative methods free from second derivatives for nonlinear equations

In a recent paper, Noor [M. Aslam Noor, New classes of iterative methods for nonlinear equations, Appl. Math. Comput., 2007, doi:10.1016/j.amc:2007], suggested and analyzed a generalized one parameter Halley method for solving nonlinear equations using. In this paper, we modified this method which has fourth order convergence. As special cases, we obtain a family of third-order iterative methods for appropriate and suitable choice of the parameter. We have compared this modified Noor method with some other iterative methods which shows that this new iterative method is robust and efficient one. Several examples are given to illustrate the efficiency and the performance of this new method.

Modified Householder iterative method free from second derivatives for nonlinear equations

In this paper, we suggest and analyze a new two-step predictor–corrector type iterative method free from second derivatives for solving nonlinear equations of the type f ( x ) = 0 . This new method includes the two-step Newton method as a special case. We prove that the new iterative method is of fourth-order. Several examples are given to illustrate the efficiency of this new method and its comparison with other iterative methods. This method can be considered as a significant improvement of the Newton method and its variant forms.

Steady State Modeling and Analysis of Single-phase Self-excited Induction Generators

Two mathematical models, including core loss component, are developed for the steady state analysis of single phase self excited induction generators (SEIG) using nodal admittance and loop impedance method based on graph theory. These models do not involve any lengthy derivations of nonlinear equations, and inclusion or elimination of any equivalent circuit elements can be done easily. It can also be used to find any combination of unknown quantities of the equivalent circuit. To predetermine the steady state performance of single-phase SEIG, application of genetic algorithm (GA) has been proposed. The experimental and theoretical results are found to be in close agreement, which validates the proposed method.

Modified Chebyshev’s method free from second derivative for non-linear equations

In this paper, we present some new modifications of Chebyshev’s method free from second derivative. The order of convergence of the new methods is three. Per iteration the methods require two evaluations of the function and one evaluation of its first derivative. The convergence of the new methods for the case of multiple roots is also discussed. Numerical examples show that the new methods have definite practical utility.

Analysis and control of three-phase self-excited induction generators supplying single-phase AC and DC loads

An approach employing genetic algorithm (GA) has been evolved for the analysis of single-phase operation of three-phase self-excited induction generators. Using symmetrical components, the circuit equations have been written and an impedance matrix has been derived, which can be straightaway solved using this method. The lengthy derivations of nonlinear equations required for the unknown variables, in the earlier methods, are thus not required in the present method. Apart from single-phase AC loads, feeding of DC loads through a controlled rectifier has also been considered and the analysis has been extended for this combined loading. For maintaining a desired DC voltage on the rectifier output side, symmetrical and extinction angle control techniques have been adopted. These techniques are shown to be better compared to the conventional phase angle control, in improving voltage regulation of the generator. The suitability of each of these control schemes for specific load conditions is discussed with examples. A biomass gasifier-based diesel engine has also been used as a prime-mover and the experimental results obtained on two generators with AC as well as DC loads, closely agree with the values predetermined using the GA approach.