On the Chandrasekhar integral equation

This is the peer reviewed version of the following article: Hernández-Verón, MA, Martínez, E, Singh, S. On the Chandrasekhar integral equation. Comp and Math Methods. 2021; 3:e1150, which has been published in final form at https://doi.org/10.1002/cmm4.1150. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] This study is devoted to solve the Chandrasekhar integral equation that it is used for modeling problems in theory of radiative transfer in a plane-parallel atmosphere, and others research areas like the kinetic theory of gases, neutron transport, traffic model, the queuing theory among others. First of all, we transform the Chandrasekhar integral equation into a nonlinear Hammerstein-type integral equation with the corresponding Nemystkii operator and the proper nonseparable kernel. Them, we approximate the kernel in order to apply an iterative scheme. This procedure it is solved in two different ways. First one, we solve a nonlinear equation with separable kernel and define an adequate nonlinear operator between Banach spaces that approximates the first problem. Second one, we introduce an approximation for the inverse of the Frechet derivative that appears in the Newton's iterative scheme for solving nonlinear equations. Finally, we perform a numerical experiment in order to compare our results with previous ones published showing that are competitive.This research was partially supported by Ministerio de Economía y Competitividad under grant PGC2018-095896-B-C21-C22.Hernández-Verón, MA.; Martínez Molada, E.; Singh, S. (2021). On the Chandrasekhar integral equation. Computational and Mathematical Methods. 3(6):1-14. https://doi.org/10.1002/cmm4.1150S1143

A multistep Steffensen-type method for solving nonlinear systems of equations

[EN] This paper is devoted to the semilocal analysis of a high-order Steffensen-type method with frozen divided differences. The methods are free of bilinear operators and derivatives, which constitutes the main limitation of the classical high-order iterative schemes. Although the methods are more demanding, a semilocal convergence analysis is presented using weaker conditions than the classical Steffensen method.This work was supported supported in part by by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI/14, by the project of Generalitat Valenciana Prometeo/2016/089 and the projects MTM2015-64382-P (MINECO/FEDER), MTM2014-52016-C2-1-P and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and InnovationAmat, S.; Argyros, IK.; Busquier, S.; Hernández-Verón, MA.; Magreñán, AA.; Martínez Molada, E. (2020). A multistep Steffensen-type method for solving nonlinear systems of equations. Mathematical Methods in the Applied Sciences. 43(13):7518-7536. https://doi.org/10.1002/mma.5599S75187536431

An algorithm derivative-free to improve the steffensen-type methods

Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton’s method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments

An improvement of the Kurchatov method by means of a parametric modification

[EN] In this work, a uniparametric generalization of the iterative method due to Kurchatov is presented. The iterative model presented is derivative-free and approximates the solution of nonlinear equations when the operator is non-differenciable. As the accessibility of the Kurchatov method is usually a problem in the application of the method, since the set of initial guesses that guarantee the convergence of the method is small, the main objective of this work is to improve the Kurchatov iterative method in its accessibility while maintaining and even increasing its speed of convergence. For this purpose, we introduce a variable parameter in the iterative function of the Kurchatov method that allows us to get a better approximation of the derivative by using a symmetric uniparameteric first-order divided difference operator. We perform a complex dynamic study that corroborate the improvements in the accessibility region. Moreover, a complete analysis of the local and semilocal convergence is established for the new uniparametric iterative method. Finally, we apply the theoretical results to solve a nonlinear integral equation showing the usefulness of the work.Ministerio de Economia y Competitividad, Grant/Award Number: EEQ/2018/000720 and PGC2018-095896-B-C21-C22; Science and Engineering Research BoardHernández-Verón, MA.; Yadav, N.; Magreñán, AA.; Martínez Molada, E.; Singh, S. (2022). An improvement of the Kurchatov method by means of a parametric modification. Mathematical Methods in the Applied Sciences. 45(11):6844-6860. https://doi.org/10.1002/mma.820968446860451

An algorithm derivative-free to improve the steffensen-type methods

[EN] Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor-corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor-corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton's method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.This research was partially supported by the project PGC2018-095896-B-C21-C22 of Spanish Ministry of Economy and Competitiveness and by the project of Generalitat Valenciana Prometeo/2016/089.Hernández-Verón, MA.; Yadav, S.; Magreñán, ÁA.; Martínez Molada, E.; Singh, S. (2022). An algorithm derivative-free to improve the steffensen-type methods. Symmetry (Basel). 14(1):1-26. https://doi.org/10.3390/sym1401000412614