Nonlinear Lane Research Articles

The modified decomposition method for analytic treatment of differential equations

The modified decomposition method is applied for analytic treatment of nonlinear differential equations. The modified method accelerates the rapid convergence of the series solution, dramatically reduces the size of work, and provides the exact solution by using few iterations only without any need to the so-called Adomian polynomials. Numerical illustrations that include nonlinear PDEs, nonlinear Klein–Gordon equations, and nonlinear Lane–Emden equation, are investigated to show the pertinent features of the technique.

Linearization techniques for singular initial-value problems of ordinary differential equations

Linearization methods for singular initial-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions and globally smooth solutions. The accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non-homogeneous, linear and nonlinear Lane–Emden equations. It is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution. In fact, it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods. It is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size.

Differential Equations: Linearity vs. Nonlinearity

Previous article Next article Differential Equations: Linearity vs. NonlinearityJack K. Hale and Joseph P. LaSalleJack K. Hale and Joseph P. LaSallehttps://doi.org/10.1137/1005068PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited byDetails A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model29 June 2022 | Physica Scripta, Vol. 97, No. 8 Cross Ref Solving a novel designed second order nonlinear Lane–Emden delay differential model using the heuristic techniquesApplied Soft Computing, Vol. 102 Cross Ref Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method16 September 2020 | International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 22, No. 1 Cross Ref Introduction15 July 2021 Cross Ref Computational modelling and simulations in tourism: A primerAnnals of Tourism Research Empirical Insights, Vol. 1, No. 1 Cross Ref Design of a Novel Second-Order Prediction Differential Model Solved by Using Adams and Explicit Runge–Kutta Numerical MethodsMathematical Problems in Engineering, Vol. 2020 Cross Ref A Bayesian approach to evaluation of soil biogeochemical models10 August 2020 | Biogeosciences, Vol. 17, No. 15 Cross Ref Vector Barrier Certificates and Comparison Systems12 July 2018 Cross Ref Legendre Wavelet Collocation Solution for System of Linear and Nonlinear Delay Differential Equations29 April 2017 | International Journal of Applied and Computational Mathematics, Vol. 3, No. S1 Cross Ref Operational Models for Piecewise-Smooth SystemsACM Transactions on Embedded Computing Systems, Vol. 16, No. 5s Cross Ref Decoupling Abstractions of Non-linear Ordinary Differential Equations8 November 2016 Cross Ref A Method for Invariant Generation for Polynomial Continuous Systems25 December 2015 Cross Ref Obituary of Jack K. Hale23 June 2010 | Journal of Dynamics and Differential Equations, Vol. 22, No. 2 Cross Ref Jack K. Hale: A Brief BiographyJournal of Differential Equations, Vol. 168, No. 1 Cross Ref The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise modelsComposite Structures, Vol. 29, No. 4 Cross Ref ΜΑΘΗΜΑΤΙΚΑ. — Characteristic Properties of Linear and Nonlinear Systems Cross Ref Joseph Pierre LaSalle May 28, 1916–July 7, 1983Journal of Differential Equations, Vol. 52, No. 3 Cross Ref Questions of Stability Cross Ref Theory of Elastic Stability Cross Ref 5 Computational Aspects of the Calculus of Variations Cross Ref On the Existence of Optimal Feedback Controls. II.T. F. Bridgland, Jr.18 July 2006 | Journal of the Society for Industrial and Applied Mathematics Series A Control, Vol. 2, No. 2AbstractPDF (1026 KB) Volume 5, Issue 3| 1963SIAM Review History Submitted:10 April 1963Published online:18 July 2006 InformationCopyright © 1963 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1005068Article page range:pp. 249-272ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics