Faculty Publications
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Item On the Order of Convergence of the Noor–Waseem Method(MDPI, 2022) George, S.; Sadananda, R.; Padikkal, J.; Argyros, I.K.In 2009, Noor and Waseem studied an important third-order iterative method. The convergenceorder is obtained using Taylor expansion and assumptions on the derivatives of order up tofour. In this paper, we have obtained convergence order three for this method using assumptionson the first and second derivatives of the involved operator. Further, we have extended the methodto obtain a fifth- and a sixth-order methods. The dynamics of the methods are also provided in thisstudy. Numerical examples are included. The same technique can be used to extend the utilization ofother single or multistep methods. © 2022 by the authors.Item Extending the Applicability of Cordero Type Iterative Method(MDPI, 2022) Remesh, K.; Argyros, I.K.; Saeed, M.; George, S.; Padikkal, J.Symmetries play a vital role in the study of physical systems. For example, microworld and quantum physics problems are modeled on the principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Most of these studies reduce to solving nonlinear equations in suitable abstract spaces iteratively. In particular, the convergence of a sixth-order Cordero type iterative method for solving nonlinear equations was studied using Taylor expansion and assumptions on the derivatives of order up to six. In this study, we obtained order of convergence six for Cordero type method using assumptions only on the first derivative. Moreover, we modified Cordero’s method and obtained an eighth-order iterative scheme. Further, we considered analogous iterative methods to solve an ill-posed problem in a Hilbert space setting. © 2022 by the authors.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 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.