Faculty Publications
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Item On the Order of Convergence and the Dynamics of Werner-King’s Method(Universal Wiser Publisher, 2023) George, S.; Argyros, I.K.; Kunnarath, A.; Padikkal, P.In this paper, we present the local convergence analysis of Werner-King’s method to approximate the solution of a nonlinear equation in Banach spaces. We establish the local convergence theorem under conditions on the first and second Fréchet derivatives of the operator involved. The convergence analysis is not based on the Taylor expansions as in the earlier studies (which require the assumptions on the third order Fréchet derivative of the operator involved). Thus our analysis extends the applicability of Werner-King’s method. We illustrate our results with numerical examples. Moreover, the dynamics and the basins of attraction are developed and demonstrated. © 2023 Santhosh George, et al.Item Order of Convergence, Extensions of Newton–Simpson Method for Solving Nonlinear Equations and Their Dynamics(MDPI, 2023) George, S.; Kunnarath, A.; Sadananda, R.; Padikkal, J.; Argyros, I.K.Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on the first two derivatives. Moreover, the semilocal convergence of NS and some of its extensions not given before is developed. Furthermore, the dynamics are explored for these methods with many illustrations. The study contains examples verifying the theoretical conditions. © 2023 by the authors.Item Enhancing the practicality of Newton–Cotes iterative method(Institute for Ionics, 2023) Sadananda, R.; George, S.; Kunnarath, A.; Padikkal, J.; Argyros, I.K.The new Newton-type iterative method developed by Khirallah et al. (Bull Math Sci Appl 2:01–14, 2012), is shown to be of the convergence order three, without the application of Taylor series expansion. Our analysis is based on the assumptions on second order derivative of the involved operator, unlike the earlier studies. Moreover, this technique is extended to methods of higher order of convergence, five and six. This paper also verifies the theoretical approach using numerical examples and comparisons, in addition to the visualization of Julia and Fatou sets of the corresponding methods. © 2023, The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics.Item On the convergence of open Newton’s method(Springer Science and Business Media B.V., 2023) Kunnarath, A.; George, S.; Sadananda, R.; Padikkal, J.; Argyros, I.K.Cordero and Torregrosa proved the convergence of two Newton’s-like methods in 2007. Using Taylor expansion (requiring existence of derivatives of order up to four of the involved operator) they obtained the convergence order three for these methods. The convergence order three is obtained for Open Newton’s method and two extensions of it with assumptions only on first two derivatives of the operator involved. We verified the results with examples and dynamics of the results are presented. © 2023, The Author(s), under exclusive licence to The Forum D’Analystes.Item On obtaining order of convergence of Jarratt-like method without using Taylor series expansion(Springer Nature, 2024) George, S.; Kunnarath, A.; Sadananda, R.; Padikkal, J.; Argyros, I.K.In 2014, Sharma and Arora introduced two efficient Jarratt-like methods for solving systems of non-linear equations which are of convergence order four and six. To prove the respective convergence order, they used Taylor expansion which demands existence of derivative of the function up to order seven. In this paper, we obtain the respective convergence order for these methods using assumptions only on first three derivatives of the function. Other problems with this approach are: the lack of computable a priori estimates on the error distances involved as well as isolation of the solution results. These concerns constitute our motivation for this article. One extension of the fourth order method is presented which is of convergence order eight and the same is proved without any extra assumptions on the higher order derivatives. All the results are proved in a general Banach space setting. Numerical examples and dynamics of the methods are studied to analyse the performance of the method. © The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional 2024.Item Parameter Choice Strategy That Computes Regularization Parameter before Computing the Regularized Solution(Multidisciplinary Digital Publishing Institute (MDPI), 2024) George, S.; Padikkal, J.; Kunnarath, A.; Argyros, I.K.; Regmi, S.The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example. © 2024 by the authors.Item New Trends in Applying LRM to Nonlinear Ill-Posed Equations(Multidisciplinary Digital Publishing Institute (MDPI), 2024) George, S.; Sadananda, R.; Padikkal, J.; Kunnarath, A.; Argyros, I.K.Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation (Formula presented.), where (Formula presented.) is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn’s paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems. © 2024 by the authors.Item Convergence Order of a Class of Jarratt-like Methods: A New Approach(Multidisciplinary Digital Publishing Institute (MDPI), 2025) Kunnarath, A.; George, S.; Padikkal, J.; Argyros, I.K.Symmetry and anti-symmetry appear naturally in the study of systems of nonlinear equations resulting from numerous fields. The solutions of such equations can be obtained in analytical form only in some special situations. Therefore, algorithms or iterative schemes are mostly studied, which approximate the solution. In particular, Jarratt-like methods were introduced with convergence order at least six in Euclidean spaces. We study the methods in the Banach-space setting. Semilocal convergence is studied to obtain the ball containing the solution. The local convergence analysis is performed without the help of the Taylor series with relaxed differentiability assumptions. Our assumptions for obtaining the convergence order are independent of the solution; earlier studies used assumptions involving the solution for local convergence analysis. We compare the methods numerically with similar-order methods and also study the dynamics. © 2024 by the authors.Item Local and semilocal analysis of a class of fourth order methods under common set of assumptions(Elsevier Inc., 2025) Kunnarath, A.; George, S.; Padikkal, P.This study presents an efficient class of fourth-order iterative methods introduced by Ali Zein (2024) in a more abstract Banach space setting. The Convergence Order of this class is proved by bypassing the Taylor expansion. We use the mean value theorem and relax the differentiability assumptions of the involved function. At the outset, we provide a semilocal analysis, and then, using the results and the same set of assumptions, we study the local convergence. This approach has the advantage that we do not need to use any assumptions on the unknown solution to study the local convergence. This technique can be used to extend the applicability of other methods along the same lines. Examples from both the chemical and the physical sciences are studied to analyze the performance of the class. The dynamics of the class are also studied. © 2025 Elsevier Inc.