Faculty Publications
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Item Enlarging the convergence ball of the method of parabola for finding zero of derivatives(Elsevier Inc. usjcs@elsevier.com, 2015) Argyros, I.K.; George, S.We present a new technique for enlarging the convergence ball of the method of parabola in order to approximate a zero of derivatives. This approach also leads to more precise error estimates on the distances involved than in earlier studies such as Hua (1974), Ren and Wu (2009) and Wand (1975). These advantages are obtained under the same computational cost on the Lipschitz constants involved as in the earlier studies. Numerical examples are also given to show the advantages over the earlier work. © 2015 Elsevier Inc. All rights reserved.Item Ball convergence comparison between three iterative methods in Banach space under hypothese only on the first derivative(Elsevier Inc. usjcs@elsevier.com, 2015) Argyros, I.K.; George, S.Abstract We present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for these methods under hypotheses only on the first Fréchet derivative in contrast to earlier studies such as Adomian (1994) [1], Babajee et al. (2008) [13], Cordero and Torregrosa (2007) [17], Cordero et al. [18], Darvishi and Barati (2007) [19], using hypotheses reaching up to the fourth Fréchet derivative although only the first derivative appears in these methods. This way we expand the applicability of these methods. Numerical examples are also presented in this study. © 2015 Elsevier Inc.Item Unified convergence domains of Newton-like methods for solving operator equations(Elsevier Inc. usjcs@elsevier.com, 2016) Argyros, I.K.; George, S.We present a unified semilocal convergence analysis in order to approximate a locally unique zero of an operator equation in a Banach space setting. Using our new idea of restricted convergence domains we generate smaller Lipschitz constants than in earlier studies leading to the following advantages: weaker sufficient convergence criteria, tighter error estimates on the distances involved and an at least as precise information on the location of the zero. Hence, the applicability of these methods is extended. These advantages are obtained under the same cost on the parameters involved. Numerical examples where the old sufficient convergence criteria cannot apply to solve equations but the new criteria can apply are also provided in this study. © 2016 Elsevier Inc. All rights reserved.Item Local Convergence of Jarratt-Type Methods with Less Computation of Inversion Under Weak Conditions(Taylor and Francis Ltd., 2017) Argyros, I.K.; George, S.We present a local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Earlier studies cannot be used to solve equations using such methods. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. © 2017, © Vilnius Gediminas Technical University, 2017.Item Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method(Elsevier B.V., 2018) George, S.; Sabari, M.We present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill-posed operator equation F(x)=y. The proposed method is a modified form of the method considered by Argyros et al. (2014). The balancing principle considered by Pereverzev and Schock (2005) is used for choosing the regularization parameter. The error estimate is derived under a general source condition and is of optimal order. The provided numerical example proves the efficiency of the proposed method. © 2017 Elsevier B.V.Item Modified Minimal Error Method for Nonlinear Ill-Posed Problems(Walter de Gruyter GmbH cmam@cmam.info, 2018) Sabari, M.; George, S.An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate. © 2018 Walter de Gruyter GmbH, Berlin/Boston.Item Fractional Tikhonov regularization method in Hilbert scales(Elsevier Inc. sinfo-f@elsevier.com, 2021) Mekoth, C.; George, S.; Padikkal, P.Fractional Tikhonov regularization method (FTRM) for linear ill-posed operator equations in the setting of Hilbert scales is being studied in this paper. Using a general Holder type source condition, we obtain an error estimate. A new parameter choice strategy is being proposed for choosing the regularization parameter in FTRM in the setting of Hilbert scales. Also, the proposed method is applied to the well known examples in the setting of Hilbert scales. © 2020 Elsevier Inc.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.