Faculty Publications
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Item Local convergence for multi-point-parametric Chebyshev-Halley-type methods of high convergence order(Elsevier, 2015) Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.We present a local convergence analysis for general multi-point-Chebyshev-Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study. © 2014 Elsevier B.V. All rights reserved.Item A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations(Elsevier Inc. usjcs@elsevier.com, 2015) Padikkal, P.; Shubha, V.S.; George, S.George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution x?? of the method of Lavrentiev regularization, i.e., F(x) + ?(x - x0) = y?, approximating the solution x of the ill-posed problem F(x) = y where F:D(F)?X?X is a nonlinear monotone operator defined on a real Hilbert space X. The convergence analysis of the method was based on a majorizing sequence. In this paper we are concerned with the problem of expanding the applicability of the method considered by George and Elmahdy (2012) by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as George and Elmahdy (2012), Mahale and Nair (2009), Mathe and Perverzev (2003), Nair and Ravishankar (2008), Semenova (2010) and Tautanhahn (2002). We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem. © 2014 Elsevier Inc. All rights reserved.Item Derivative Free Iterative Scheme for Monotone Nonlinear Ill-posed Hammerstein-Type Equations(International Association of Engineers, 2021) Erappa, S.M.; George, S.An iterative scheme which is free of derivative is employed to approximately solve nonlinear ill-posed Hammer-stein type operator equations )TG(x) = Y, where G is a non-linear monotone operator and ) is a bounded linear operator defined on Hilbert spaces X,Y,Z. The convergence analysis adapted in the paper includes weaker Lipschitz condition and adaptive choice of Perverzev and Schock(2005) is employed to choose the regularization parameter U. Furthermore, order optimal error bounds are obtained and the method is validated by a numerical example. © 2021, IAENG International Journal of Applied Mathematics. All Rights Reserved.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.Item Convergence analysis of a class of iterative methods: a unified approach(Vilnius Gediminas Technical University, 2025) Murugan, M.; Godavarma, C.; George, S.; Bate, I.; Senapati, K.In this paper, we study the convergence of a class of iterative methods for solving the system of nonlinear Banach space valued equations. We provide a unified local and semi-local convergence analysis for these methods. The convergence order of these methods are obtained using the conditions on the derivatives of the involved operator up to order 2 only. Further, we provide the number of iterations required to obtain the given accuracy of the solution. Various numerical examples including integral equations and Caputo fractional differential equations are considered to show the performance of our methods. © 2025 The Author(s). Published by Vilnius Gediminas Technical University.