Faculty Publications
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Item A New Parameter Choice Strategy for Lavrentiev Regularization Method for Nonlinear Ill-Posed Equations(MDPI, 2022) George, S.; Padikkal, J.; Remesh, K.; Argyros, I.K.In this paper, we introduced a new source condition and a new parameter-choice strategy which also gives the known best error estimate. To obtain the results we used the assumptions used in earlier studies. Further, we studied the proposed new parameter-choice strategy and applied it to the method (in the finite-dimensional setting) considered in George and Nair (2017). © 2022 by the authors.Item Extending the Applicability of Cordero Type Iterative Method(MDPI, 2022) Remesh, K.; Argyros, I.K.; Saeed, M.; George, S.; Padikkal, J.Symmetries play a vital role in the study of physical systems. For example, microworld and quantum physics problems are modeled on the principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Most of these studies reduce to solving nonlinear equations in suitable abstract spaces iteratively. In particular, the convergence of a sixth-order Cordero type iterative method for solving nonlinear equations was studied using Taylor expansion and assumptions on the derivatives of order up to six. In this study, we obtained order of convergence six for Cordero type method using assumptions only on the first derivative. Moreover, we modified Cordero’s method and obtained an eighth-order iterative scheme. Further, we considered analogous iterative methods to solve an ill-posed problem in a Hilbert space setting. © 2022 by the authors.Item EXTENDING THE APPLICABILITY OF A SEVENTH-ORDER METHOD FOR EQUATIONS UNDER GENERALIZED CONDITIONS(Institute of Mathematics. Polish Academy of Sciences, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.We extend the applicability of a seventh-order method for solving Banach space-valued equations. This is achieved by using generalized conditions on the first derivative which only appears in the method. Earlier works use conditions up to the eighth derivative to establish convergence. Our technique is very general and can be used to extend the applicability of other methods along the same lines. © Instytut Matematyczny PAN, 2023.Item Order of Convergence, Extensions of Newton–Simpson Method for Solving Nonlinear Equations and Their Dynamics(MDPI, 2023) George, S.; Kunnarath, A.; Sadananda, R.; Padikkal, J.; Argyros, I.K.Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on the first two derivatives. Moreover, the semilocal convergence of NS and some of its extensions not given before is developed. Furthermore, the dynamics are explored for these methods with many illustrations. The study contains examples verifying the theoretical conditions. © 2023 by the authors.Item New Trends in Applying LRM to Nonlinear Ill-Posed Equations(Multidisciplinary Digital Publishing Institute (MDPI), 2024) George, S.; Sadananda, R.; Padikkal, J.; Kunnarath, A.; Argyros, I.K.Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation (Formula presented.), where (Formula presented.) is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn’s paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems. © 2024 by the authors.Item Convergence Order of a Class of Jarratt-like Methods: A New Approach(Multidisciplinary Digital Publishing Institute (MDPI), 2025) Kunnarath, A.; George, S.; Padikkal, J.; Argyros, I.K.Symmetry and anti-symmetry appear naturally in the study of systems of nonlinear equations resulting from numerous fields. The solutions of such equations can be obtained in analytical form only in some special situations. Therefore, algorithms or iterative schemes are mostly studied, which approximate the solution. In particular, Jarratt-like methods were introduced with convergence order at least six in Euclidean spaces. We study the methods in the Banach-space setting. Semilocal convergence is studied to obtain the ball containing the solution. The local convergence analysis is performed without the help of the Taylor series with relaxed differentiability assumptions. Our assumptions for obtaining the convergence order are independent of the solution; earlier studies used assumptions involving the solution for local convergence analysis. We compare the methods numerically with similar-order methods and also study the dynamics. © 2024 by the authors.Item Convergence Analysis of Jarratt-like Methods for Solving Nonlinear Equations for Thrice-Differentiable Operators(Multidisciplinary Digital Publishing Institute (MDPI), 2025) Bate, I.; Senapati, K.; George, S.; Argyros, I.K.; Argyros, M.I.The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the (Formula presented.) -order convergence using the Taylor series expansion technique needed at least (Formula presented.) times differentiability of the involved operator. However, we obtain the fourth- and sixth-order for Jarratt-like methods using up to the third-order derivatives only. An upper bound for the asymptotic error constant (AEC) and a convergence ball are provided. The convergence analysis is developed in the more general setting of Banach spaces and relies on Lipschitz-type conditions, which are required to control the derivative. The results obtained are examined using numerical examples, and some dynamical system concepts are discussed for a better understanding of convergence ideas. © 2025 by the authors.