Faculty Publications
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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.