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 Enhancing the applicability of Chebyshev-like method(Academic Press Inc., 2024) George, S.; Bate, I.; M, M.; Godavarma, C.; Senapati, K.Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results. © 2024 Elsevier Inc.Item On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion(Elsevier B.V., 2024) M, M.; Godavarma, G.; George, S.; Bate, I.; Senapati, K.Hueso et al. (2015) studied the fourth and sixth order methods to approximate a solution of a nonlinear equation in Rn, where the convergence analysis needs the involved operator to be five times differentiable and seven times differentiable for fourth-order and sixth-order methods, respectively. Also, they found no error estimate for those methods, as the Taylor series expansion played a leading role in proving the convergence. In this paper, we extended the method in the Banach space settings and relaxed the higher order derivative of the involved operator so that the methods can be used in a bigger class of problems which were not covered by the analysis in Hueso et al. (2015). Also, we obtained an error estimate without Taylor series expansion. This error estimate helps to get the number of iterations to achieve a given accuracy. Moreover, new sixth-order method is introduced by small modification and numerical examples were discussed for all these methods to validate our theoretical results and to study the dynamics. © 2024 Elsevier B.V.Item Jarratt-type methods and their convergence analysis without using Taylor expansion(Elsevier Inc., 2025) Bate, I.; Senapati, K.; George, S.; M, M.; Godavarma, C.In this paper, we study the local convergence analysis of the Jarratt-type iterative methods for solving non-linear equations in the Banach space setting without using the Taylor expansion. Convergence analysis using Taylor series required the operator to be differentiable at least p+1 times, where p is the order of convergence. In our convergence analysis, we do not use the Taylor expansion, so we require only assumptions on the derivatives of the involved operator of order up to three only. Thus, we extended the applicability of the methods under study. Further, we obtained a six-order Jarratt-type method by utilising the method studied by Hueso et al. in 2015. Numerical examples and dynamics of the methods are presented to illustrate the theoretical results. © 2024 Elsevier Inc.Item An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices(Multidisciplinary Digital Publishing Institute (MDPI), 2025) George, S.; Gopal, M.; Bhide, S.; Argyros, I.K.A multi-step method introduced by Raziyeh and Masoud for solving nonlinear systems with convergence order five has been considered in this paper. The convergence of the method was studied using Taylor series expansion, which requires the function to be six times differentiable. However, our convergence study does not depend on the Taylor series. We use the derivative of F up to two only in our convergence analysis, which is presented in a more general Banach space setting. Semi-local analysis is also discussed, which was not given in earlier studies. Unlike in earlier studies (where two sets of assumptions were used), we used the same set of assumptions for semi-local analysis and local convergence analysis. We discussed the dynamics of the method and also gave some numerical examples to illustrate theoretical findings. © 2025 by the authors.