Faculty Publications
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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 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 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.