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Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence...
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Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: closed (30 April 2020) | Viewed by 31187

Special Issue Editors

Prof. Dr. Juan Ramón Torregrosa Sánchez

E-Mail Website
Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alicia Cordero

E-Mail Website
Guest Editor
School of Telecommunications Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: numerical analysis; iterative methods; nonlinear problems; discrete dynamics; real and complex
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Solving nonlinear equations and systems is a nontrivial task that involves many areas of science and technology. Usually, it is not affordable in a direct way, and iterative algorithms play a fundamental role in their approach. The main (but not sole) theme of this Special Issue is the design, analysis of convergence and stability, and application to practical problems of new iterative schemes for solving nonlinear problems. This includes methods with and without memory, with derivatives or derivative-free, the real or complex dynamics associated to them, and an analysis of their convergence that can be local, semilocal or global.

Selected articles dealing with these themes of the ICIAM 2019 Conference and MME&HB 2019 will be published in this Special Volume.

Prof. Dr. Juan R. Torregrosa
Prof. Dr. Alicia Cordero
Guest Editors

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Keywords

  • Nonlinear equations and systems
  • Iterative methods
  • Stability
  • Efficiency

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Published Papers (11 papers)

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Research

24 pages, 514 KiB  
Article
On Numerical Analysis of Carreau–Yasuda Nanofluid Flow over a Non-Linearly Stretching Sheet under Viscous Dissipation and Chemical Reaction Effects
by Stanford Shateyi and Hillary Muzara
Mathematics 2020, 8(7), 1148; https://doi.org/10.3390/math8071148 - 14 Jul 2020
Cited by 8 | Viewed by 3101
Abstract
This work reports the Carreau–Yasuda nanofluid flow over a non-linearly stretching sheet viscous dissipation and chemical reaction effects. The coupled system of non-linear partial differential equations are changed into a system of linear differential equations employing similarity equations. The spectral quasi-linearization method was [...] Read more.
This work reports the Carreau–Yasuda nanofluid flow over a non-linearly stretching sheet viscous dissipation and chemical reaction effects. The coupled system of non-linear partial differential equations are changed into a system of linear differential equations employing similarity equations. The spectral quasi-linearization method was used to solve the linear differential equations numerically. Error norms were used to authenticate the accuracy and convergence of the numerical method. The effects of some thermophysical parameters of interest in this current study on the non-dimensional fluid velocity, concentration and temperature, the skin friction, local Nusselt and Sherwood numbers are presented graphically. Tables were used to depict the effects of selected parameters on the skin friction and the Nusselt number. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
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12 pages, 466 KiB  
Article
Study of Local Convergence and Dynamics of a King-Like Two-Step Method with Applications
by Ioannis K. Argyros, Ángel Alberto Magreñán, Alejandro Moysi, Íñigo Sarría and Juan Antonio Sicilia Montalvo
Mathematics 2020, 8(7), 1062; https://doi.org/10.3390/math8071062 - 1 Jul 2020
Viewed by 1994
Abstract
In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. [...] Read more.
In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. As a result, the applicability of the method is expanded. We also use different convergence planes to show family behavior. Finally, the new results are used to solve some applications related to chemistry. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
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13 pages, 305 KiB  
Article
Strong Convergence Theorems for Generalized Split Feasibility Problems in Banach Spaces
by Xuejiao Zi, Zhaoli Ma and Wei-Shih Du
Mathematics 2020, 8(6), 892; https://doi.org/10.3390/math8060892 - 2 Jun 2020
Cited by 1 | Viewed by 1895
Abstract
In this paper, we establish new strong convergence theorems of proposed algorithms under suitable new conditions for the generalized split feasibility problem in Banach spaces. As applications, new strong convergence theorems for equilibrium problems, fixed point problems and split common fixed point problems [...] Read more.
In this paper, we establish new strong convergence theorems of proposed algorithms under suitable new conditions for the generalized split feasibility problem in Banach spaces. As applications, new strong convergence theorems for equilibrium problems, fixed point problems and split common fixed point problems are also studied. Our new results are distinct from recent results on the topic in the literature. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
18 pages, 4436 KiB  
Article
Numerical Solution of Nonlinear Diff. Equations for Heat Transfer in Micropolar Fluids over a Stretching Domain
by Farooq Ahmad, A. Othman Almatroud, Sajjad Hussain, Shan E. Farooq and Roman Ullah
Mathematics 2020, 8(5), 854; https://doi.org/10.3390/math8050854 - 25 May 2020
Cited by 9 | Viewed by 2936
Abstract
A numerical study based on finite difference approximation is attempted to analyze the bulk flow, micro spin flow and heat transfer phenomenon for micropolar fluids dynamics through Darcy porous medium. The fluid flow mechanism is considered over a moving permeable sheet. The heat [...] Read more.
A numerical study based on finite difference approximation is attempted to analyze the bulk flow, micro spin flow and heat transfer phenomenon for micropolar fluids dynamics through Darcy porous medium. The fluid flow mechanism is considered over a moving permeable sheet. The heat transfer is associated with two different sets of boundary conditions, the isothermal wall and isoflux boundary. On the basis of porosity of medium, similarity functions are utilized to avail a set of ordinary differential equations. The non-linear coupled ODE’s have been solved with a very stable and reliable numerical scheme that involves Simpson’s Rule and Successive over Relaxation method. The accuracy of the results is improved by making iterations on three different grid sizes and higher order accuracy in the results is achieved by Richardson extrapolation. This study provides realistic and differentiated results with due considerations of micropolar fluid theory. The micropolar material parameters demonstrated reduction in the bulk fluid speed, thermal distribution and skin friction coefficient but increase in local heat transfer rate and couple stress. The spin behavior of microstructures is also exhibited through microrotation vector N ( η ) . Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
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12 pages, 249 KiB  
Article
A Modified Ren’s Method with Memory Using a Simple Self-Accelerating Parameter
by Xiaofeng Wang and Qiannan Fan
Mathematics 2020, 8(4), 540; https://doi.org/10.3390/math8040540 - 7 Apr 2020
Cited by 3 | Viewed by 2224
Abstract
In this paper, a self-accelerating type method is proposed for solving nonlinear equations, which is a modified Ren’s method. A simple way is applied to construct a variable self-accelerating parameter of the new method, which does not increase any computational costs. The highest [...] Read more.
In this paper, a self-accelerating type method is proposed for solving nonlinear equations, which is a modified Ren’s method. A simple way is applied to construct a variable self-accelerating parameter of the new method, which does not increase any computational costs. The highest convergence order of new method is 2 + 6 ≈ 4.4495 . Numerical experiments are made to show the performance of the new method, which supports the theoretical results. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
11 pages, 777 KiB  
Article
Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations
by Sukhjit Singh, Eulalia Martínez, Abhimanyu Kumar and D. K. Gupta
Mathematics 2020, 8(3), 384; https://doi.org/10.3390/math8030384 - 9 Mar 2020
Cited by 1 | Viewed by 2403
Abstract
In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in [...] Read more.
In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
21 pages, 857 KiB  
Article
A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems
by Ramandeep Behl and Ioannis K. Argyros
Mathematics 2020, 8(2), 271; https://doi.org/10.3390/math8020271 - 18 Feb 2020
Cited by 4 | Viewed by 2466
Abstract
Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as [...] Read more.
Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
9 pages, 208 KiB  
Article
A New Newton Method with Memory for Solving Nonlinear Equations
by Xiaofeng Wang and Yuxi Tao
Mathematics 2020, 8(1), 108; https://doi.org/10.3390/math8010108 - 10 Jan 2020
Cited by 13 | Viewed by 3166
Abstract
A new Newton method with memory is proposed by using a variable self-accelerating parameter. Firstly, a modified Newton method without memory with invariant parameter is constructed for solving nonlinear equations. Substituting the invariant parameter of Newton method without memory by a variable self-accelerating [...] Read more.
A new Newton method with memory is proposed by using a variable self-accelerating parameter. Firstly, a modified Newton method without memory with invariant parameter is constructed for solving nonlinear equations. Substituting the invariant parameter of Newton method without memory by a variable self-accelerating parameter, we obtain a novel Newton method with memory. The convergence order of the new Newton method with memory is 1 + 2 . The acceleration of the convergence rate is attained without any additional function evaluations. The main innovation is that the self-accelerating parameter is constructed by a simple way. Numerical experiments show the presented method has faster convergence speed than existing methods. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
13 pages, 276 KiB  
Article
Generalized Inverses Estimations by Means of Iterative Methods with Memory
by Santiago Artidiello, Alicia Cordero, Juan R. Torregrosa and María P. Vassileva
Mathematics 2020, 8(1), 2; https://doi.org/10.3390/math8010002 - 18 Dec 2019
Cited by 15 | Viewed by 2686
Abstract
A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the [...] Read more.
A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
14 pages, 416 KiB  
Article
A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems
by Raudys R. Capdevila, Alicia Cordero and Juan R. Torregrosa
Mathematics 2019, 7(12), 1221; https://doi.org/10.3390/math7121221 - 11 Dec 2019
Cited by 8 | Viewed by 2939
Abstract
In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in [...] Read more.
In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in both cases. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
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12 pages, 1515 KiB  
Article
Iterative Methods with Memory for Solving Systems of Nonlinear Equations Using a Second Order Approximation
by Alicia Cordero, Javier G. Maimó, Juan R. Torregrosa and María P. Vassileva
Mathematics 2019, 7(11), 1069; https://doi.org/10.3390/math7111069 - 7 Nov 2019
Cited by 13 | Viewed by 4171
Abstract
Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of [...] Read more.
Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability. Full article
(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)
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