Faculty Publications
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Item A derivative free iterative method for the implementation of Lavrentiev regularization method for ill-posed equations(Kluwer Academic Publishers, 2015) Shubha, V.S.; George, S.; Padikkal, P.In this work, we develop a derivative free iterative method for the implementation of Lavrentiev regularization for approximately solving the nonlinear ill-posed operator equation F(x) = y. Convergence analysis shows that the method converges quadratically. Apart from being totally free of derivatives, the method, under a general source condition provide an optimal order error estimate. We use the adaptive method introduced in Pereverzyev and Schock (SIAM J. Numer. Anal. 43, 2060–2076, 2005) for choosing the regularization parameter. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem. © 2014, Springer Science+Business Media New York.Item Finite dimensional realization of a Tikhonov gradient type-method under weak conditions(Springer-Verlag Italia s.r.l. springer@springer.it, 2016) Shubha, V.S.; George, S.; Padikkal, P.In this paper we consider projection techniques to obtain the finite dimensional realization of a Tikhonov gradient type-method considered in George et al. (Local convergence of a Tikhonov gradient type-method under weak conditions, communicated, 2016) for approximating a solution x^ of the nonlinear ill-posed operator equation F(x) = y. The main advantage of the proposed method is that the inverse of the operator F is not involved in the method. The regularization parameter is chosen according to the adaptive method considered by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005). We also derive optimal stopping conditions on the number of iterations necessary for obtaining the optimal order of convergence. Using two numerical examples we compare our results with an existing method to justify the theoretical results. © 2016, Springer-Verlag Italia.Item Convergence of a Tikhonov Gradient Type-Method for Nonlinear Ill-Posed Equations(Springer, 2017) George, S.; Shubha, V.S.; Padikkal, P.In this study Tikhonov Gradient type-method is considered for nonlinear ill-posed operator equations. In our convergence analysis, we use hypotheses only on the first Frec?het derivative of F in contrast to the higher order Frec?het derivatives used in the earlier studies. We obtained ‘optimal’ order error estimate by choosing the regularization parameter according to the adaptive method proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005). © 2017, Springer (India) Private Ltd.Item Convergence Analysis of a Fifth-Order Iterative Method Using Recurrence Relations and Conditions on the First Derivative(Birkhauser, 2021) George, S.; Argyros, I.K.; Padikkal, P.; Mahapatra, M.; Saeed, M.Using conditions on the second Fréchet derivative, fifth order of convergence was established in Singh et al. (Mediterr J Math 13:4219–4235, 2016) for an iterative method. In this paper, we establish fifth order of convergence of the method using assumptions only on the first Fréchet derivative of the involved operator. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.Item Advances in Nonlinear Variational Inequalities Volume 25 (2022), Number 1, 49-58 Comparing and Extending Two Fourth Order Methods Under the Same Hypotheses for Equations(International Publications, 2022) Argyros, I.K.; George, S.; Argyros, C.I.We compare and extend two fourth order methods for nonlinear equations. Our convergence analysis used assumptions only on the first derivative. Earlier studies have used hypotheses up to the fifth derivative, limiting the applicability of the method. Numerical examples complete the article. © 2022, International Publications. All rights reserved.Item Local convergence analysis of two iterative methods(Springer Science and Business Media B.V., 2022) George, S.; Argyros, I.K.; Senapati, K.; Kanagaraj, K.In this paper we consider two three-step iterative methods with common first two steps. The convergence order five and six, respectively of these methods are proved using assumptions on the first derivative of the operator involved. We also provide dynamics of these methods © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.Item On the convergence of the sixth order Homeier like method in Banach spaces(Erdal Karapinar, 2022) Suma, P.B.; Erappa, S.M.; George, S.A sixth order Homeier-like method is introduced for approximating a solution of the non-linear equation in Banach space. Assumptions only on first and second derivatives are used to obtain a sixth order convergence. Our proof does not depend on Taylor series expansions as in the earlier studies for the similar methods. © 2022, Erdal Karapinar. All rights reserved.Item On the semilocal convergence analysis of a seventh order four step method for solving nonlinear equations(Ptolemy Scientific Research Press, 2024) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.We provide a semi-local convergence analysis of a seventh order four step method for solving nonlinear problems. Using majorizing sequences and under conditions on the first derivative, we provide sufficient convergence criteria, error bounds on the distances involved and uniqueness. Earlier convergence results have used the eighth derivative not on this method to show convergence. Hence, limiting its applicability. © 2024 by the authors; licensee PSRP, Lahore, Pakistan.Item Extended convergence for two-step methods with non-differentiable parts in Banach spaces(Springer Science and Business Media B.V., 2024) Argyros, I.K.; George, S.; Senapati, K.In this study, we have extended the applicability of two-step methods with non-differentiable parts for solving nonlinear equations defined in Banach spaces. The convergence analysis uses conditions weaker than the ones in earlier studies. Other advantages include computable a priori error distances based on generalized conditions, an extended region of convergence as well as a better knowledge of the isolation for the solutions. By setting the divided differences equal to zero the results can be used to solve equations with differentiable part too. © The Author(s), under exclusive licence to The Forum D’Analystes 2023.