Relevant bibliographies by topics / Autonomous and highly oscillatory differential equations

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Academic literature on the topic 'Autonomous and highly oscillatory differential equations'

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Published: 7 December 2024

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Journal articles on the topic "Autonomous and highly oscillatory differential equations"

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DAVIDSON, B. D., and D. E. STEWART. "A NUMERICAL HOMOTOPY METHOD AND INVESTIGATIONS OF A SPRING-MASS SYSTEM." Mathematical Models and Methods in Applied Sciences 03, no. 03 (June 1993): 395–416. http://dx.doi.org/10.1142/s0218202593000217.

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A numerical technique is developed to determine the behavior of periodic solutions to highly nonlinear non-autonomous systems of ordinary differential equations. The method is based on shooting in conjunction with a probability one homotopy method and an implementation of the topological index. It is shown that solutions may be characterized a priori in terms of an index and this is developed into a powerful numerical and investigative tool. This method is used to investigate the periodic solutions of a nonlinear fourth order system of differential equations. These equations describe the motion of a forced mechanical oscillator and are extremely difficult to evaluate numerically. Solutions are presented which could not be found using local methods. These include flip, saddle node and symmetry breaking pitchfork bifurcations.
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Philos, Ch G., I. K. Purnaras, and Y. G. Sficas. "ON THE BEHAVIOUR OF THE OSCILLATORY SOLUTIONS OF SECOND-ORDER LINEAR UNSTABLE TYPE DELAY DIFFERENTIAL EQUATIONS." Proceedings of the Edinburgh Mathematical Society 48, no. 2 (May 23, 2005): 485–98. http://dx.doi.org/10.1017/s0013091503000993.

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AbstractSecond-order linear (non-autonomous as well as autonomous) delay differential equations of unstable type are considered. In the non-autonomous case, sufficient conditions are given in order that all oscillatory solutions are bounded or all oscillatory solutions tend to zero at $\infty$. In the case where the equations are autonomous, necessary and sufficient conditions are established for all oscillatory solutions to be bounded or all oscillatory solutions to tend to zero at $\infty$.
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Ogorodnikova, S., and F. Sadyrbaev. "MULTIPLE SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS WITH OSCILLATORY SOLUTIONS." Mathematical Modelling and Analysis 11, no. 4 (December 31, 2006): 413–26. http://dx.doi.org/10.3846/13926292.2006.9637328.

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We consider two second order autonomous differential equations with critical points, which allow the detection of an exact number of solutions to the Dirichlet boundary value problem. Non‐autonomous equations with similar behaviour of solutions also are considered. Estimations from below of the number of solutions to the Dirichlet boundary value problem are given.
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Condon, Marissa, Alfredo Deaño, and Arieh Iserles. "On second-order differential equations with highly oscillatory forcing terms." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2118 (January 13, 2010): 1809–28. http://dx.doi.org/10.1098/rspa.2009.0481.

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We present a method to compute efficiently solutions of systems of ordinary differential equations (ODEs) that possess highly oscillatory forcing terms. This approach is based on asymptotic expansions in inverse powers of the oscillatory parameter, and features two fundamental advantages with respect to standard numerical ODE solvers: first, the construction of the numerical solution is more efficient when the system is highly oscillatory, and, second, the cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided, featuring the Van der Pol and Duffing oscillators and motivated by problems in electronic engineering.
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Sanz-Serna, J. M. "Mollified Impulse Methods for Highly Oscillatory Differential Equations." SIAM Journal on Numerical Analysis 46, no. 2 (January 2008): 1040–59. http://dx.doi.org/10.1137/070681636.

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Petzold, Linda R., Laurent O. Jay, and Jeng Yen. "Numerical solution of highly oscillatory ordinary differential equations." Acta Numerica 6 (January 1997): 437–83. http://dx.doi.org/10.1017/s0962492900002750.

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One of the most difficult problems in the numerical solution of ordinary differential equations (ODEs) and in differential-algebraic equations (DAEs) is the development of methods for dealing with highly oscillatory systems. These types of systems arise, for example, in vehicle simulation when modelling the suspension system or tyres, in models for contact and impact, in flexible body simulation from vibrations in the structural model, in molecular dynamics, in orbital mechanics, and in circuit simulation. Standard numerical methods can require a huge number of time-steps to track the oscillations, and even with small stepsizes they can alter the dynamics, unless the method is chosen very carefully.
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Cohen, David, Ernst Hairer, and Christian Lubich. "Modulated Fourier Expansions of Highly Oscillatory Differential Equations." Foundations of Computational Mathematics 3, no. 4 (October 1, 2003): 327–45. http://dx.doi.org/10.1007/s10208-002-0062-x.

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Condon, M., A. Iserles, and S. P. Nørsett. "Differential equations with general highly oscillatory forcing terms." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2161 (January 8, 2014): 20130490. http://dx.doi.org/10.1098/rspa.2013.0490.

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The concern of this paper is in expanding and computing initial-value problems of the form y ′= f ( y )+ h ω ( t ), where the function h ω oscillates rapidly for ω ≫1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators and they can be used as an organizing principle for very accurate and affordable numerical solvers. However, there is no similar theory for more general oscillators, and there are sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form e i ωg k ( t ) , it is possible to expand y ( t ) in a different manner. Each r th term in the expansion is for some ς >0 and it can be represented as an r -dimensional highly oscillatory integral. Because computation of multivariate highly oscillatory integrals is fairly well understood, this provides a powerful method for an effective discretization of a numerical solution for a large family of highly oscillatory ordinary differential equations.
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Herrmann, L. "Oscillatory Solutions of Some Autonomous Partial Differential Equations with a Parameter." Journal of Mathematical Sciences 236, no. 3 (December 1, 2018): 367–75. http://dx.doi.org/10.1007/s10958-018-4117-1.

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Chartier, Philippe, Joseba Makazaga, Ander Murua, and Gilles Vilmart. "Multi-revolution composition methods for highly oscillatory differential equations." Numerische Mathematik 128, no. 1 (January 17, 2014): 167–92. http://dx.doi.org/10.1007/s00211-013-0602-0.

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Dissertations / Theses on the topic "Autonomous and highly oscillatory differential equations"

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Bouchereau, Maxime. "Modélisation de phénomènes hautement oscillants par réseaux de neurones." Electronic Thesis or Diss., Université de Rennes (2023-....), 2024. http://www.theses.fr/2024URENS034.

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Cette thèse porte sur l’application du Machine Learning à l’étude d’équations différentielles fortement oscillantes. Plus précisément, on s’intéresse à une manière d’approcher de manière précise et avec le moins de calculs possible la solution d’une équation différentielle en s’aidant de réseaux de neurones. Tout d’abord, le cas autonome est étudié, où les propriétés de l’analyse rétrograde et des réseaux de neurones sont utilisés afin d’améliorer des méthodes numériques existantes, puis une généralisation au cas fortement oscillant est proposée afin d’améliorer un schéma numérique d’ordre un spécifique à ce cas de figure. Ensuite, les réseaux de neurones sont utilisés afin de remplacer les calculs préalables nécessaires à l’implémentation de méthodes numériques uniformément précises permettant d’approcher les solutions d’équations fortement oscillantes, que ce soit en partant des travaux mis en œuvre pour le cas autonome, ou bien en utilisant une structure de réseau de neurone intégrant directement la structure de l’équation
This thesis focuses on the application of Machine Learning to the study of highly oscillatory differential equations. More precisely, we are interested in an approach to accurately approximate the solution of a differential equation with the least amount of computations, using neural networks. First, the autonomous case is studied, where the proper- ties of backward analysis and neural networks are used to enhance existing numerical methods. Then, a generalization to the strongly oscillating case is proposed to improve a specific first-order numerical scheme tailored to this scenario. Subsequently, neural networks are employed to replace the necessary pre- computations for implementing uniformly ac- curate numerical methods to approximate so- lutions of strongly oscillating equations. This can be done either by building upon the work done for the autonomous case or by using a neural network structure that directly incorporates the equation’s structure
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Khanamiryan, Marianna. "Numerical methods for systems of highly oscillatory ordinary differential equations." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/226323.

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This thesis presents methods for efficient numerical approximation of linear and non-linear systems of highly oscillatory ordinary differential equations. Phenomena of high oscillation is considered a major computational problem occurring in Fourier analysis, computational harmonic analysis, quantum mechanics, electrodynamics and fluid dynamics. Classical methods based on Gaussian quadrature fail to approximate oscillatory integrals. In this work we introduce numerical methods which share the remarkable feature that the accuracy of approximation improves as the frequency of oscillation increases. Asymptotically, our methods depend on inverse powers of the frequency of oscillation, turning the major computational problem into an advantage. Evolving ideas from the stationary phase method, we first apply the asymptotic method to solve highly oscillatory linear systems of differential equations. The asymptotic method provides a background for our next, the Filon-type method, which is highly accurate and requires computation of moments. We also introduce two novel methods. The first method, we call it the FM method, is a combination of Magnus approach and the Filon-type method, to solve matrix exponential. The second method, we call it the WRF method, a combination of the Filon-type method and the waveform relaxation methods, for solving highly oscillatory non-linear systems. Finally, completing the theory, we show that the Filon-type method can be replaced by a less accurate but moment free Levin-type method.
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Kanat, Bengi Tanoğlu Gamze. "Numerical Solution of Highly Oscillatory Differential Equations By Magnus Series Method/." [s.l.]: [s.n.], 2006. http://library.iyte.edu.tr/tezler/master/matematik/T000572.pdf.

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Bréhier, Charles-Edouard. "Numerical analysis of highly oscillatory Stochastic PDEs." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00824693.

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In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.
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Books on the topic "Autonomous and highly oscillatory differential equations"

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Wu, Xinyuan, and Bin Wang. Geometric Integrators for Differential Equations with Highly Oscillatory Solutions. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-0147-7.

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Schütte, Christof. A quasiresonant smoothing algorithm for solving large highly oscillatory differential equations from quantum chemistry. Aachen: Verlag Shaker, 1994.

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Bin, Wang, and Xinyuan Wu. Geometric Integrators for Differential Equations with Highly Oscillatory Solutions. Springer Singapore Pte. Limited, 2021.

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Bin, Wang, and Xinyuan Wu. Geometric Integrators for Differential Equations with Highly Oscillatory Solutions. Springer, 2022.

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Book chapters on the topic "Autonomous and highly oscillatory differential equations"

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Hairer, Ernst, Gerhard Wanner, and Christian Lubich. "Highly Oscillatory Differential Equations." In Springer Series in Computational Mathematics, 407–53. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/978-3-662-05018-7_13.

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Wu, Xinyuan, Xiong You, and Bin Wang. "Effective Methods for Highly Oscillatory Second-Order Nonlinear Differential Equations." In Structure-Preserving Algorithms for Oscillatory Differential Equations, 185–96. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35338-3_8.

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Le Bris, Claude, Frédéric Legoll, and Alexei Lozinski. "MsFEM à la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems." In Partial Differential Equations: Theory, Control and Approximation, 265–94. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-41401-5_11.

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Wu, Xinyuan, Kai Liu, and Wei Shi. "Improved Filon-Type Asymptotic Methods for Highly Oscillatory Differential Equations." In Structure-Preserving Algorithms for Oscillatory Differential Equations II, 53–68. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-48156-1_3.

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Wu, Xinyuan, Kai Liu, and Wei Shi. "Error Analysis of Explicit TSERKN Methods for Highly Oscillatory Systems." In Structure-Preserving Algorithms for Oscillatory Differential Equations II, 175–92. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-48156-1_8.

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Wu, Xinyuan, and Bin Wang. "Symplectic Approximations for Efficiently Solving Semilinear KG Equations." In Geometric Integrators for Differential Equations with Highly Oscillatory Solutions, 351–91. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-0147-7_11.

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Wu, Xinyuan, Kai Liu, and Wei Shi. "Highly Accurate Explicit Symplectic ERKN Methods for Multi-frequency Oscillatory Hamiltonian Systems." In Structure-Preserving Algorithms for Oscillatory Differential Equations II, 193–209. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-48156-1_9.

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Wu, Xinyuan, and Bin Wang. "Energy-Preserving Schemes for High-Dimensional Nonlinear KG Equations." In Geometric Integrators for Differential Equations with Highly Oscillatory Solutions, 263–97. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-0147-7_9.

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Wu, Xinyuan, and Bin Wang. "Linearly-Fitted Conservative (Dissipative) Schemes for Nonlinear Wave Equations." In Geometric Integrators for Differential Equations with Highly Oscillatory Solutions, 235–61. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-0147-7_8.

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Bensoussan, Alain. "Homogenization for Non Linear Elliptic Equations with Random Highly Oscillatory Coefficients." In Partial Differential Equations and the Calculus of Variations, 93–133. Boston, MA: Birkhäuser Boston, 1989. http://dx.doi.org/10.1007/978-1-4684-9196-8_5.

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Conference papers on the topic "Autonomous and highly oscillatory differential equations"

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Kuo, Chi-Wei, and C. Steve Suh. "On Controlling Non-Autonomous Time-Delay Feedback Systems." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-51128.

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Time-delay feedback oscillators of non-autonomous type are considered in the paper. These oscillators have been studied extensively for many decades in a broad set of fields such as sensor design, manufacturing, and machine dynamics. A time-delay model system having one time-delay constant and several nonlinear feedback terms in the governing differential equation is first studied. Many researches have demonstrated that a time-delay feedback even in the form of a small perturbation is able to perturb the oscillator to exhibit complex dynamical responses including bifurcation and route-to-chaos. These motions are harmful as they have a very negative impact on the stability, and thus output quality, of the system. For example, manufacturing processes that are characterized by time-delay feedback all have an operation limit on speed because the chaotic behaviors which are unpredictable and extremely unstable are difficult to control. With a viable control solution, the performance, quality, and capacity of manufacturing can be improved enormously. A novel concept capable of simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain has been demonstrated to be feasible for the control of dynamic instability including bifurcation and route-to-chaos in many nonlinear systems. The concept is followed to create a control configuration that is feasible for the mitigation of non-autonomous time-delay feedback oscillators. Featuring wavelet adaptive filters for simultaneous time-frequency resolution and filtered-x least mean square algorithm for online identification, the controller design is shown to successfully moderate the dynamic instability of the time-delay feedback oscillator and unconditionally warrant a limit cycle. The controller design that integrates all these features is able to mitigate dynamical deterioration in both the time and frequency domains and properly regulate the responses with the desired reference signal. Specifically the qualitative behavior of the controlled oscillator output follows a definitive fractal topology before settling into a stable manifold. The controlled response is categorically quasi-periodic and of the prescribed vibration amplitude and frequency spectrum. The control scheme is novel and requires no linearization. By applying wavelet domain analysis approach to the nonlinear control of instability, the true dynamics of the time-delay feedback system as delineated by both the time and frequency information are faithfully retained without being distorted or misinterpreted. Through employing adaptive technique, the high sensitivity of the time-delay feedback system to external disturbances is also properly addressed.
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Feng, Dehua, Frederick Ferguson, Yang Gao, and Xinru Niu. "Investigating the Start-Up Structures and Their Evolution Within an Under-Expanded Jet Flows." In ASME 2023 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2023. http://dx.doi.org/10.1115/imece2023-113767.

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Abstract The initialization and evolution of the unsteady flow structures produced by a supersonic under-expanded planar jet are numerically simulated and analyzed. Two sets of engineering tools were used in this study: namely, the Integro-Differential Scheme (IDS) and the Weighted Essentially Non-Oscillatory (WENO) Scheme. The IDS is based on solving the system of Navier-Stokes equations through integration principles, Ref. [1]. Whereas the 5th order WENO-Z with Strong Stability Preserving Runge Kutta based on Finite Volume Method has been conducted Ref. [2–3]. The intent of this study is to understand the evolving nature of the flow phenomena and to identify the mechanisms that are influential in determining the nature of the fully developed flow field. Such understandings are of great engineering importance to mixing and turbulence. As a target of engineering analysis, the under-expanded free jet is perhaps one of the simplest flows. Its flow field physics closely matches the realities of the intended industrial applications, comprising of the many complex unsteady flow phenomena and their interactions, Ref. [4]. During its initial phase, the profile of the emerging jet is primarily dictated by the pressure ratio between the reservoir stagnation conditions and the ambient pressure. However, the genesis and evolution of the under expanded jet is even more interesting, as this developing flow is highly unsteady. In this case, the pressure of the supersonic jet is expected to reach the ambient pressure through a series of oblique shock and expansion waves, and their interactions with the emerging shear layers originating from the nozzle lip. In this analysis, a Mach 1.4 jet steam enters an ambient chamber of equal pressure at a temperature of 1.4 atm and a temperature of 300K. The developing under-expanded jet is modeled using the two sets of CFD tools described earlier, starting with the same initial and boundary conditions. The simulated IDS and Co-WENO results of the unsteady jet streams were analyzed and compared at specified times. This report compares the two simulated solutions and discusses the flow physics revealed by these tools. It was observed that both sets of tools revealed the interactions create within the shear layers instability mechanisms. In both cases, the instability waves grow spatially to create large scale coherent structures that travel downstream. These coherent structures grow as they interact with the surrounding ambient flow field, and with time, they interact with each other, merging to form the larger so-called main vortex. It is of interest to note that the two simulations were not identical, even though they delivered more commonalities and a few less striking differences. No doubt, the flow regimes of interest, the instabilities and the coherent structures are slight functions of the numerical parameters associated with these two schemes. Since at the moment no exact solution exists, the authors of this study will let the interested readers derive which of the two CFD tools best capture the expected flow field physics.
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