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 Order of Convergence and Dynamics of Newton–Gauss-Type Methods(MDPI, 2023) Sadananda, R.; George, S.; Argyros, I.K.; Padikkal, J.On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in this paper. The study of high-convergence-order iterative methods under weak conditions is of extreme importance, because higher order means that fewer iterations are carried out to achieve a predetermined error tolerance. In order to enhance the practicality of these methods by Zhongli Liu, the convergence analysis is carried out without the application of Taylor expansion and requires the operator to be only two times differentiable, unlike the earlier studies. A semilocal convergence analysis is provided. Furthermore, numerical experiments verifying the convergence criteria, comparative studies and the dynamics are discussed for better interpretation. © 2023 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 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 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 Unified Convergence Analysis of Certain At Least Fifth Order Methods(SINUS Association, 2025) Sadananda, R.; Gopal, M.; George, S.; Argyros, I.K.A class of iterative methods was developed by Xiao and Yin in 2015 and obtained convergence order five using Taylor expansion. They had imposed the conditions on the derivatives of the involved operator of order at least up to four. In this paper, the order of convergence is achieved by imposing conditions only on the first two derivatives of the operator involved. The assumptions under consideration are weaker and the analysis is done in the more general setting of Banach spaces without using Taylor series expansion. The semi-local convergence analysis is also given. Further, the theory is justified by numerical examples. © 2024, SINUS Association. All rights reserved.