Faculty Publications
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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 Convergence of Chun’s method in Banach spaces under weaker assumptions(Springer Science and Business Media B.V., 2025) George, S.; M, M.; Godavarma, C.In this paper, we give a modified convergence analysis for the fourth-order method studied in Cordero et al. (J Math Chem 53(1):430–449, 2015). Our analysis provides the convergence order using the derivative of the involved operator up to order two only, whereas their study needs it to be five times differentiable. Apart from this, this paper obtains the convergence ball radius and the number of iterations to reach the solution with the desired accuracy. Further, we use the general Banach space settings to get these results, while their work is done only for the space. At the end of the paper, we discuss a few numerical examples and compare them with other existing fourth-order methods. © The Author(s), under exclusive licence to The Forum D’Analystes 2025.Item A procedure for increasing the convergence order of iterative methods from p to 5p for solving nonlinear system(Academic Press Inc., 2025) George, S.; M, M.; Gopal, M.; Godavarma, C.; Argyros, I.K.In this paper, we propose a procedure to obtain an iterative method that increases its convergence order from p to 5p for solving nonlinear systems. Our analysis is given in more general Banach space settings and uses assumptions on the derivative of the involved operator only up to order max?{k,2}. Here, k is the order of the highest derivative used in the convergence analysis of the iterative method with convergence order p. A particular case of our analysis includes an existing fifth-order method and improves its applicability to more problems than the problems covered by the method's analysis in earlier study. © 2024