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Artykuły w czasopismach na temat „Autonomous and highly oscillatory differential equations”

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Data publikacji: 7 grudnia 2024

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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 motio

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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

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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 oscillat

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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, demonstrat

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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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11

Lanets, O. S., V. T. Dmytriv, V. M. Borovets, I. A. Derevenko, and I. M. Horodetskyy. "Analytical Model of the Two-Mass Above Resonance System of the Eccentric-Pendulum Type Vibration Table." International Journal of Applied Mechanics and Engineering 25, no. 4 (2020): 116–29. http://dx.doi.org/10.2478/ijame-2020-0053.

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AbstractThe article deals with atwo-mass above resonant oscillatory system of an eccentric-pendulum type vibrating table. Based on the model of a vibrating oscillatory system with three masses, the system of differential equations of motion of oscillating masses with five degrees of freedom is compiled using generalized Lagrange equations of the second kind. For given values of mechanical parameters of the oscillatory system and initial conditions, the autonomous system of differential equations of motion of oscillating masses is solved by the numerical Rosenbrock method. The results of analyt

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12

Condon, Marissa, Alfredo Deaño, Arieh Iserles, and Karolina Kropielnicka. "Efficient computation of delay differential equations with highly oscillatory terms." ESAIM: Mathematical Modelling and Numerical Analysis 46, no. 6 (2012): 1407–20. http://dx.doi.org/10.1051/m2an/2012004.

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13

Mahdavi, Ashkan, Sheng-Wei Chi, and Negar Kamali. "Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations." Journal of Engineering Mechanics 146, no. 4 (2020): 04020014. http://dx.doi.org/10.1061/(asce)em.1943-7889.0001727.

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14

Iserles, Arieh. "Think globally, act locally: Solving highly-oscillatory ordinary differential equations." Applied Numerical Mathematics 43, no. 1-2 (2002): 145–60. http://dx.doi.org/10.1016/s0168-9274(02)00122-8.

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15

Liu, Zhongli, Tianhai Tian, and Hongjiong Tian. "Asymptotic-numerical solvers for highly oscillatory second-order differential equations." Applied Numerical Mathematics 137 (March 2019): 184–202. http://dx.doi.org/10.1016/j.apnum.2018.11.004.

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16

Sanz-Serna, J. M., and Beibei Zhu. "Word series high-order averaging of highly oscillatory differential equations with delay." Applied Mathematics and Nonlinear Sciences 4, no. 2 (2019): 445–54. http://dx.doi.org/10.2478/amns.2019.2.00042.

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AbstractWe show that, when the delay is an integer multiple of the forcing period, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word series techniques to obtain high-order averaged equations for differential equations without delay.

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17

Ariel, Gil, Bjorn Engquist, and Richard Tsai. "A multiscale method for highly oscillatory ordinary differential equations with resonance." Mathematics of Computation 78, no. 266 (2008): 929–56. http://dx.doi.org/10.1090/s0025-5718-08-02139-x.

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18

Liu, Wensheng. "Averaging Theorems for Highly Oscillatory Differential Equations and Iterated Lie Brackets." SIAM Journal on Control and Optimization 35, no. 6 (1997): 1989–2020. http://dx.doi.org/10.1137/s0363012994268667.

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19

John, Sabo, and Pius Tumba. "The Efficiency of Block Hybrid Method for Solving Malthusian Growth Model and Prothero-Robinson Oscillatory Differential Equations." International Journal of Development Mathematics (IJDM) 1, no. 3 (2024): 008–22. http://dx.doi.org/10.62054/ijdm/0103.02.

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The efficiency of block hybrid method for solving Malthusian Growth Model, Prothero-Robinson equation and highly stiff oscillatory differential equations was proposed using a power series polynomial through interpolation and collocation. The new method's basic properties, including order, error constant, consistency, zero-stability, and stability regions, were comprehensively analyzed and satisfied all necessary conditions for analysis. Tested on various real-life problems, the new method demonstrated superior performance compared to existing techniques. The study highlights the innovative app

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20

SAIRA and Wen-Xiu Ma. "An Approximation Method to Compute Highly Oscillatory Singular Fredholm Integro-Differential Equations." Mathematics 10, no. 19 (2022): 3628. http://dx.doi.org/10.3390/math10193628.

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This paper appertains the presentation of a Clenshaw–Curtis rule to evaluate highly oscillatory Fredholm integro-differential equations (FIDEs) with Cauchy and weak singularities. To calculate the singular integral, the unknown function approximated by an interpolation polynomial is rewritten as a Taylor series expansion. A system of linear equations of FIDEs obtained by using equally spaced points as collocation points is solved to obtain the unknown function. The proposed method attains higher accuracy rates, which are proven by error analysis and some numerical examples as well.

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21

Zaman, Sakhi, Latif Ullah Khan, Irshad Hussain, and Lucian Mihet-Popa. "Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits." Symmetry 14, no. 1 (2022): 115. http://dx.doi.org/10.3390/sym14010115.

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The paper demonstrates symmetric integral operator and interpolation based numerical approximations for linear and nonlinear ordinary differential equations (ODEs) with oscillatory factor x′=ψ(x)+χω(t), where the function χω(t) is an oscillatory forcing term. These equations appear in a variety of computational problems occurring in Fourier analysis, computational harmonic analysis, fluid dynamics, electromagnetics, and quantum mechanics. Classical methods such as Runge–Kutta methods etc. fail to approximate the oscillatory ODEs due the existence of oscillatory term χω(t). Two types of methods

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22

Sanz-Serna, J. M., and Beibei Zhu. "A stroboscopic averaging algorithm for highly oscillatory delay problems." IMA Journal of Numerical Analysis 39, no. 3 (2018): 1110–33. http://dx.doi.org/10.1093/imanum/dry020.

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Abstract We propose and analyse a heterogeneous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method suggested here may provide approximations with $\mathscr{O}\big (H^{2}+1/\varOmega ^{2}\big )$ errors with a computational effort that grows like $H^{-1}$ (the inverse of the step size), uniformly in the forcing frequency $\varOmega $.

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23

Dizicheh, A. Karimi, F. Ismail, M. Tavassoli Kajani, and Mohammad Maleki. "A Legendre Wavelet Spectral Collocation Method for Solving Oscillatory Initial Value Problems." Journal of Applied Mathematics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/591636.

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In this paper, we propose an iterative spectral method for solving differential equations with initial values on large intervals. In the proposed method, we first extend the Legendre wavelet suitable for large intervals, and then the Legendre-Guass collocation points of the Legendre wavelet are derived. Using this strategy, the iterative spectral method converts the differential equation to a set of algebraic equations. Solving these algebraic equations yields an approximate solution for the differential equation. The proposed method is illustrated by some numerical examples, and the result is

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24

Bao, W. "Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations." Journal of Mathematical Study 47, no. 2 (2014): 111–50. http://dx.doi.org/10.4208/jms.v47n2.14.01.

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25

Liu, Zhongli, Hongjiong Tian, and Xiong You. "Adiabatic Filon-type methods for highly oscillatory second-order ordinary differential equations." Journal of Computational and Applied Mathematics 320 (August 2017): 1–14. http://dx.doi.org/10.1016/j.cam.2017.01.028.

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26

Blanes, Sergio, Fernando Casas, and Ander Murua. "Splitting methods for differential equations." Acta Numerica 33 (July 2024): 1–161. http://dx.doi.org/10.1017/s0962492923000077.

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This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyse in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connect

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27

Bayly, Philip V., Larry A. Taber, and Anders E. Carlsson. "Damped and persistent oscillations in a simple model of cell crawling." Journal of The Royal Society Interface 9, no. 71 (2011): 1241–53. http://dx.doi.org/10.1098/rsif.2011.0627.

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A very simple, one-dimensional, discrete, autonomous model of cell crawling is proposed; the model involves only three or four coupled first-order differential equations. This form is sufficient to describe many general features of cell migration, including both steady forward motion and oscillatory progress. Closed-form expressions for crawling speeds and internal forces are obtained in terms of dimensionless parameters that characterize active intracellular processes and the passive mechanical properties of the cell. Two versions of the model are described: a basic cell model with simple ela

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28

Lovetskiy, Konstantin P., Leonid A. Sevastianov, Michal Hnatič, and Dmitry S. Kulyabov. "Numerical Integration of Highly Oscillatory Functions with and without Stationary Points." Mathematics 12, no. 2 (2024): 307. http://dx.doi.org/10.3390/math12020307.

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This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active u

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29

Banshchikov, A. V., A. V. Lakeev, and V. A. Rusanov. "On polylinear differential realization of the determined dynamic chaos in the class of higher order equations with delay." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 10 (October 26, 2023): 3–21. http://dx.doi.org/10.26907/0021-3446-2023-10-3-21.

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The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The construct

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30

Lorenz, Katina, Tobias Jahnke, and Christian Lubich. "Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition." BIT Numerical Mathematics 45, no. 1 (2005): 91–115. http://dx.doi.org/10.1007/s10543-005-2637-9.

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31

Wang, Bin, and Xinyuan Wu. "Improved Filon-type asymptotic methods for highly oscillatory differential equations with multiple time scales." Journal of Computational Physics 276 (November 2014): 62–73. http://dx.doi.org/10.1016/j.jcp.2014.07.035.

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32

Buchholz, Simone, Ludwig Gauckler, Volker Grimm, Marlis Hochbruck, and Tobias Jahnke. "Closing the gap between trigonometric integrators and splitting methods for highly oscillatory differential equations." IMA Journal of Numerical Analysis 38, no. 1 (2017): 57–74. http://dx.doi.org/10.1093/imanum/drx007.

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33

Fox, B., L. S. Jennings, and A. Y. Zomaya. "Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces." Journal of Mechanical Design 123, no. 2 (1999): 272–81. http://dx.doi.org/10.1115/1.1353587.

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The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order

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34

Philos, Ch G., I. K. Purnaras, and Y. G. Sficas. "Asymptotic Decay of the Oscillatory Solutions to First Order Non-Autonomous Linear Unstable Type Delay Differential Equations." Funkcialaj Ekvacioj 49, no. 3 (2006): 385–413. http://dx.doi.org/10.1619/fesi.49.385.

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35

Crouseilles, Nicolas, Shi Jin, and Mohammed Lemou. "Nonlinear geometric optics method-based multi-scale numerical schemes for a class of highly oscillatory transport equations." Mathematical Models and Methods in Applied Sciences 27, no. 11 (2017): 2031–70. http://dx.doi.org/10.1142/s0218202517500385.

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We introduce a new numerical strategy to solve a class of oscillatory transport partial differential equation (PDE) models which is able to capture accurately the solutions without numerically resolving the high frequency oscillations in both space and time. Such PDE models arise in semiclassical modeling of quantum dynamics with band-crossings, and other highly oscillatory waves. Our first main idea is to use the geometric optics ansatz, which builds the oscillatory phase into an independent variable. We then choose suitable initial data, based on the Chapman–Enskog expansion, for the new mod

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36

O’NEALE, DION R. J., and ROBERT I. MCLACHLAN. "RECONSIDERING TRIGONOMETRIC INTEGRATORS." ANZIAM Journal 50, no. 3 (2009): 320–32. http://dx.doi.org/10.1017/s1446181109000042.

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AbstractIn this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi–Pasta–Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but a

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37

Han, Houde, and Zhongyi Huang. "The Tailored Finite Point Method." Computational Methods in Applied Mathematics 14, no. 3 (2014): 321–45. http://dx.doi.org/10.1515/cmam-2014-0012.

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Abstract.In this paper, a brief review of tailored finite point methods (TFPM) is given. The TFPM is a new approach to construct the numerical solutions of partial differential equations. The TFPM has been tailored based on the local properties of the solution for each given problem. Especially, the TFPM is very efficient for solutions which are not smooth enough, e.g., for solutions possessing boundary/interior layers or solutions being highly oscillated. Recently, the TFPM has been applied to singular perturbation problems, the Helmholtz equation with high wave numbers, the first-order wave

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38

Brunner, Hermann, Yunyun Ma, and Yuesheng Xu. "The oscillation of solutions of Volterra integral and integro-differential equations with highly oscillatory kernels." Journal of Integral Equations and Applications 27, no. 4 (2015): 455–87. http://dx.doi.org/10.1216/jie-2015-27-4-455.

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Khanamiryan, M. "Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: part I." BIT Numerical Mathematics 48, no. 4 (2008): 743–61. http://dx.doi.org/10.1007/s10543-008-0201-0.

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Denk, G. "A new numerical method for the integration of highly oscillatory second-order ordinary differential equations." Applied Numerical Mathematics 13, no. 1-3 (1993): 57–67. http://dx.doi.org/10.1016/0168-9274(93)90131-a.

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41

Spigler, Renato. "Asymptotic-numerical approximations for highly oscillatory second-order differential equations by the phase function method." Journal of Mathematical Analysis and Applications 463, no. 1 (2018): 318–44. http://dx.doi.org/10.1016/j.jmaa.2018.03.027.

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Bayly, P. V., and S. K. Dutcher. "Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella." Journal of The Royal Society Interface 13, no. 123 (2016): 20160523. http://dx.doi.org/10.1098/rsif.2016.0523.

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Cilia and flagella are highly conserved organelles that beat rhythmically with propulsive, oscillatory waveforms. The mechanism that produces these autonomous oscillations remains a mystery. It is widely believed that dynein activity must be dynamically regulated (switched on and off, or modulated) on opposite sides of the axoneme to produce oscillations. A variety of regulation mechanisms have been proposed based on feedback from mechanical deformation to dynein force. In this paper, we show that a much simpler interaction between dynein and the passive components of the axoneme can produce c

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43

Bissembayev, Jomartov, Tuleshov, and Dikambay. "Analysis of the Oscillating Motion of a Solid Body on Vibrating Bearers." Machines 7, no. 3 (2019): 58. http://dx.doi.org/10.3390/machines7030058.

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This article considers the oscillation of a solid body on kinematic foundations, the main elements of which are rolling bearers bounded by high-order surfaces of rotation at horizontal displacement of the foundation. Equations of motion of the vibro-protected body have been obtained. It is ascertained that the obtained equations of motion are highly nonlinear differential equations. Stationary and transitional modes of the oscillatory process of the system have been investigated. It is determined that several stationary regimes of the oscillatory process exist. Equations of motion have been in

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44

Gong, Ya Qi, Qin Chen, and Yong Feng Qi. "Solving of Partial Differential Equations by Numerical Manifold Method with Partially Overlapping Covers." Applied Mechanics and Materials 638-640 (September 2014): 1737–40. http://dx.doi.org/10.4028/www.scientific.net/amm.638-640.1737.

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Governing equations of 1D high-order numerical manifold method with partially overlapping covers have been deduced by the general method of weighted residuals. By using the proposed method, a highly oscillatory differential equation has been solved. In addition, a posteriori error method is adopted for evaluating the accuracy of the algorithm. Meanwhile, several factors affect the accuracy are also discussed. The results indicate that accuracy of solution increase with the decrease of overlapping ratio and the order of cover function. When higher order cover function such as 6th is used, highe

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Chartier, Philippe, Florian Méhats, Mechthild Thalhammer, and Yong Zhang. "Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 5 (2017): 1859–82. http://dx.doi.org/10.1051/m2an/2017010.

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Khanamiryan, Marianna. "Quadrature methods for highly oscillatory linear and non-linear systems of ordinary differential equations: part II." BIT Numerical Mathematics 52, no. 2 (2011): 383–405. http://dx.doi.org/10.1007/s10543-011-0355-z.

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Philos, Ch G., I. K. Purnaras, and Y. G. Sficas. "Asymptotic behavior of the oscillatory solutions to first order non-autonomous linear neutral delay differential equations of unstable type." Mathematical and Computer Modelling 46, no. 3-4 (2007): 422–38. http://dx.doi.org/10.1016/j.mcm.2006.11.012.

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Marszalek, Wieslaw, Jan Sadecki, and Maciej Walczak. "Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams." Entropy 23, no. 7 (2021): 876. http://dx.doi.org/10.3390/e23070876.

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Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular a

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Vilmart, Gilles. "Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise." SIAM Journal on Scientific Computing 36, no. 4 (2014): A1770—A1796. http://dx.doi.org/10.1137/130935331.

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Romanchuk, Yaroslav, Mariia Sokil, and Leonid Polishchuk. "PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES." Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska 14, no. 3 (2024): 15–20. http://dx.doi.org/10.35784/iapgos.6377.

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In the process of operation, the simplest elements (hereinafter elastic bodies) of machines and mechanisms under the influence of external and internal factors carry out complex oscillations ‒ a combination of longitudinal, bending and torsion combinations in various combinations. In general, mathematical models of the process of such complex phenomena in elastic bodies, even for one-dimensional calculation models, are boundary value problems for systems of partial differential equations. A two-dimensional mathematical model of oscillatory processes in a nonlinear elastic body is considered. A

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