Faculty Publications
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Item Ball Convergence of a Fifth-Order Method for Solving Equations Under Weak Conditions(Springer, 2021) Argyros, I.K.; George, S.; Erappa, S.M.We develop a ball convergence for a fifth-order method to find a solution for an equation. Earlier studies used conditions on the sixth derivative not present in the methods. Moreover, no error estimates are provided. That is why we used conditions up to the second derivative. Numerical experiments validate the theoretical results. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.Item Ball Convergence of Multipoint Methods for Non-linear Systems(Springer Science and Business Media Deutschland GmbH, 2021) Argyros, I.K.; George, S.; Erappa, S.M.We study Multipoint methods using only the first derivative. Earlier studies use higher than three order derivatives not on the methods. Moreover Lipschitz constants are used to find error estimates not presented in earlier papers. Numerical examples complete this paper. © 2021, Springer Nature Singapore Pte Ltd.Item Ball convergence for an eighth order efficient method under weak conditions in Banach spaces(Springer Nature, 2017) Argyros, I.K.; George, S.; Erappa, S.M.We present a local convergence analysis of an eighth order- iterative method in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fourth derivative although only the first derivative appears in the definition of these methods. In this study we only use the hypothesis on the first derivative. This way we expand the applicability of these methods. Moreover, we provide a radius of convergence, a uniqueness ball and computable error bounds based on Lipschitz constants. Numerical examples computing the radii of the convergence balls as well as examples where earlier results cannot apply to solve equations but our results can apply are also given in this study. © 2016, Sociedad Española de Matemática Aplicada.Item Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations(Springer-Verlag Italia s.r.l. springer@springer.it, 2017) Argyros, I.K.; George, S.; Erappa, S.M.For the solution of nonlinear ill-posed problems, a Two Step Newton-Tikhonov methodology is proposed. Two implementations are discussed and applied to nonlinear ill-posed Hammerstein type operator equations KF(x) = y, where K defines the integral operator and F the function of the solution x on which K operates. In the first case, the Fre´ chet derivative of F is invertible in a neighbourhood which includes the initial guess x0 and the solution x^. In the second case, F is monotone. For both cases, local cubic convergence is established and order optimal error bounds are obtained by choosing the regularization parameter according to the the balancing principle of Pereverzev and Schock (2005).We also present the results of computational experiments giving the evidence of the reliability of our approach. © 2016, Springer-Verlag Italia.Item Inexact Newton’s Method to Nonlinear Functions with Values in a Cone Using Restricted Convergence Domains(Springer, 2017) Argyros, I.K.; George, S.; Erappa, S.M.Using our new idea of restricted convergence domains, a robust convergence theorem for inexact Newton’s method is presented to find a solution of nonlinear inclusion problems in Banach space. Using this technique, we obtain tighter majorizing functions. Consequently, we get a larger convergence domain and tighter error bounds on the distances involved. Moreover, we obtain an at least as precise information on the location of the solution than in earlier studies. Furthermore, a numerical example is presented to show that our results apply to solve problems in cases earlier studies cannot. © 2017, Springer (India) Private Ltd.Item Local convergence of a novel eighth order method under hypotheses only on the first derivative(Tusi Mathematical Research Group (TMRG) moslehian@memeber.ams.org, 2019) Argyros, I.K.; George, S.; Erappa, S.M.We expand the applicability of eighth order-iterative method stud- ied by Jaiswal in order to approximate a locally unique solution of an equation in Banach space setting. We provide a local convergence analysis using only hypotheses on the first Frechet-derivative. Moreover, we provide computable convergence radii, error bounds, and uniqueness results. Numerical examples computing the radii of the convergence balls as well as examples where earlier results cannot apply to solve equations but our results can apply are also given in this study. © 2019 Khayyam Journal of Mathematics.Item Extending the applicability of Newton’s and secant methods under regular smoothness(Boletim da Sociedade Paranaense de Matematica, 2020) Argyros, I.K.; George, S.; Erappa, S.M.The concept of regular smoothness has been shown to be an appropriate and powerfull tool for the convergence of iterative procedures converging to a locally unique solution of an operator equation in a Banach space setting. Motivated by earlier works, and optimization considerations, we present a tighter semi-local convergence analysis using our new idea of restricted convergence domains. Numerical examples complete this study. © 2020 Boletim da Sociedade Paranaense de Matematica. All rights reserved.Item Extending the applicability of high-order iterative schemes under Kantorovich hypotheses and restricted convergence regions(Springer-Verlag Italia s.r.l. springer@springer.it, 2020) Argyros, I.K.; George, S.; Erappa, S.M.We use restricted convergence regions to locate a more precise set than in earlier works containing the iterates of some high-order iterative schemes involving Banach space valued operators. This way the Lipschitz conditions involve tighter constants than before leading to weaker sufficient semilocal convergence criteria, tighter bounds on the error distances and an at least as precise information on the location of the solution. These improvements are obtained under the same computational effort since computing the old Lipschitz constants also requires the computation of the new constants as special cases. The same technique can be used to extend the applicability of other iterative schemes. Numerical examples further validate the new results. © 2019, Springer-Verlag Italia S.r.l., part of Springer Nature.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 Finite dimensional realization of fractional Tikhonov regularization method in Hilbert scales(Elsevier B.V., 2022) Mekoth, C.; George, S.; Padikkal, J.; Erappa, S.M.One of the intuitive restrictions of infinite dimensional Fractional Tikhonov Regularization Method (FTRM) for ill-posed operator equations is its numerical realization. This paper addresses the issue to a considerable extent by using its finite dimensional realization in the setting of Hilbert scales. Using adaptive parameter choice strategy, we choose the regularization parameter and obtain an optimal order error estimate. Also, the proposed method is applied to the well known examples in the setting of Hilbert scales. © 2021 The Author(s)