A first approach in solving initial-value problems in ODEs by elliptic fitting methods

Exponentially-fitted and trigonometrically-fitted methods have a long successful history in the solution of initial-value problems, but other functions might be considered in adapted methods. Specifically, this paper aims at the derivation of a new numerical scheme for approximating initial value problems of ordinary differential equations using elliptic functions. The example considered is the undamped Duffing equation where the forcing term is of autonomous type affected by a perturbation parameter. The new scheme is constructed by considering a suitable approximation to the theoretical solution based on elliptic functions. The proposed elliptic fitting procedure has been tested on a variety of problems, showing its good performance

A piecewise-linearized algorithm based on the Krylov subspace for solving stiff ODEs

Numerical methods for solving Ordinary Differential Equations (ODEs) have received considerable attention in recent years. In this paper a piecewise-linearized algorithm based on Krylov subspaces for solving Initial Value Problems (IVPs) is proposed. MATLAB versions for autonomous and non-autonomous ODEs of this algorithm have been implemented. These implementations have been compared with other piecewise-linearized algorithms based on Pad approximants, recently developed by the authors of this paper, comparing both precisions and computational costs in equal conditions. Four case studies have been used in the tests that come from stiff biology and chemical kinetics problems. Experimental results show the advantages of the proposed algorithms, especially when the dimension is increased in stiff problems. © 2009 Elsevier B.V. All rights reserved.This work was supported by the Spanish CICYT project CGL2007-66440-C04-03.Ibáñez González, JJ.; Hernández García, V.; Ruiz Martínez, PA.; Arias, E. (2011). A piecewise-linearized algorithm based on the Krylov subspace for solving stiff ODEs. Journal of Computational and Applied Mathematics. 235(7):1798-1804. https://doi.org/10.1016/j.cam.2010.07.012S17981804235

Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers

This article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers. Classical finite-difference schemes do not yield parameter uniform convergent results on a uniform mesh, in general (Robust Computational Techniques for Boundary Layers, Chapman & Hall, London, CRC Press, Boca Raton, FL, 2000). In order to overcome this difficulty, we propose an appropriate piecewise uniforrn (Shishkin) mesh and apply the classical finite-difference schemes on this mesh. Error estimates are derived by decomposing the solution into smooth and singular components. The present method is layer resolving as well as parameter uniform convergent. Numerical examples are presented to show the applicability and efficiency of the method