Faculty Publications
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Item Solving a fourth order partial differential equations using deep neural networks(American Institute of Physics, 2024) Francis, J.M.; Godavarma, C.In this study, we compare the effect of different hyperparameters of the deep neural network while solving partial differential equations. We consider feed-forward neural networks, and the hyperparameters such as the number of hidden layers, number of neurons, activation function, optimization methods, and learning rate are studied in this paper. Numerical results for the effect of all the hyperparameters while solving fourth-order partial differential equations are mentioned in this study. © 2024 Author(s).Item A radial basis function method for fractional Darboux problems(Elsevier Ltd, 2018) Godavarma, C.; Prashanthi, P.; Vijesh, V.A.In this paper, a radial basis function (RBF) collocation known as Kansa's method has been extended to solve fractional Darboux problems. The fractional derivatives are described in the Caputo sense. Integration of radial functions that appears due to fractional derivatives have been dealt using Gauss–Jacobi quadrature method. The equation has been linearized using successive approximation. A few test problems have been solved and compared with available solutions. The effect of RBF shape parameter on accuracy and convergence has also been discussed. © 2017 Elsevier LtdItem Mollification of Fourier spectral methods with polynomial kernels(John Wiley and Sons Ltd, 2024) Puthukkudi, M.; Godavarma, C.Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter (Formula presented.) is analogous to that in the Dirichlet mollifier, which is detailed by analyzing the numerical solution. Further, they are extended to linear scalar conservation law problems. © 2024 John Wiley & Sons Ltd.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 Convergence of Chun’s method in Banach spaces under weaker assumptions(Springer Science and Business Media B.V., 2025) George, S.; M, M.; Godavarma, C.In this paper, we give a modified convergence analysis for the fourth-order method studied in Cordero et al. (J Math Chem 53(1):430–449, 2015). Our analysis provides the convergence order using the derivative of the involved operator up to order two only, whereas their study needs it to be five times differentiable. Apart from this, this paper obtains the convergence ball radius and the number of iterations to reach the solution with the desired accuracy. Further, we use the general Banach space settings to get these results, while their work is done only for the space. At the end of the paper, we discuss a few numerical examples and compare them with other existing fourth-order methods. © The Author(s), under exclusive licence to The Forum D’Analystes 2025.Item Jarratt-type methods and their convergence analysis without using Taylor expansion(Elsevier Inc., 2025) Bate, I.; Senapati, K.; George, S.; M, M.; Godavarma, C.In this paper, we study the local convergence analysis of the Jarratt-type iterative methods for solving non-linear equations in the Banach space setting without using the Taylor expansion. Convergence analysis using Taylor series required the operator to be differentiable at least p+1 times, where p is the order of convergence. In our convergence analysis, we do not use the Taylor expansion, so we require only assumptions on the derivatives of the involved operator of order up to three only. Thus, we extended the applicability of the methods under study. Further, we obtained a six-order Jarratt-type method by utilising the method studied by Hueso et al. in 2015. Numerical examples and dynamics of the methods are presented to illustrate the theoretical results. © 2024 Elsevier Inc.Item A procedure for increasing the convergence order of iterative methods from p to 5p for solving nonlinear system(Academic Press Inc., 2025) George, S.; M, M.; Gopal, M.; Godavarma, C.; Argyros, I.K.In this paper, we propose a procedure to obtain an iterative method that increases its convergence order from p to 5p for solving nonlinear systems. Our analysis is given in more general Banach space settings and uses assumptions on the derivative of the involved operator only up to order max?{k,2}. Here, k is the order of the highest derivative used in the convergence analysis of the iterative method with convergence order p. A particular case of our analysis includes an existing fifth-order method and improves its applicability to more problems than the problems covered by the method's analysis in earlier study. © 2024Item Convergence analysis of a class of iterative methods: a unified approach(Vilnius Gediminas Technical University, 2025) Murugan, M.; Godavarma, C.; George, S.; Bate, I.; Senapati, K.In this paper, we study the convergence of a class of iterative methods for solving the system of nonlinear Banach space valued equations. We provide a unified local and semi-local convergence analysis for these methods. The convergence order of these methods are obtained using the conditions on the derivatives of the involved operator up to order 2 only. Further, we provide the number of iterations required to obtain the given accuracy of the solution. Various numerical examples including integral equations and Caputo fractional differential equations are considered to show the performance of our methods. © 2025 The Author(s). Published by Vilnius Gediminas Technical University.