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Relevant bibliographies by topics / Autonomous and highly oscillatory differential equations

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Journal articles

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

Author: Grafiati

Published: 7 December 2024

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

1

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 (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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2

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 (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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3

Ogorodnikova, S., and F. Sadyrbaev. "MULTIPLE SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS WITH OSCILLATORY SOLUTIONS." Mathematical Modelling and Analysis 11, no. 4 (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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4

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 (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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5

Sanz-Serna, J. M. "Mollified Impulse Methods for Highly Oscillatory Differential Equations." SIAM Journal on Numerical Analysis 46, no. 2 (2008): 1040–59. http://dx.doi.org/10.1137/070681636.

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6

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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7

Cohen, David, Ernst Hairer, and Christian Lubich. "Modulated Fourier Expansions of Highly Oscillatory Differential Equations." Foundations of Computational Mathematics 3, no. 4 (2003): 327–45. http://dx.doi.org/10.1007/s10208-002-0062-x.

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8

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 (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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9

Herrmann, L. "Oscillatory Solutions of Some Autonomous Partial Differential Equations with a Parameter." Journal of Mathematical Sciences 236, no. 3 (2018): 367–75. http://dx.doi.org/10.1007/s10958-018-4117-1.

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10

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

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