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Journal articles on the topic 'Equation-free approach'

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Author: Grafiati

Published: 10 December 2022

Last updated: 29 January 2023

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1

Dongbin Xiu, I. G. Kevrekidis, and R. Ghanem. "An Equation-Free, Multiscale Approach to Uncertainty Quantification." Computing in Science and Engineering 7, no. 3 (May 2005): 16–23. http://dx.doi.org/10.1109/mcse.2005.46.

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2

Williams, Matthew O., Joshua L. Proctor, and J. Nathan Kutz. "Modeling disease transmission near eradication: An equation free approach." Physica D: Nonlinear Phenomena 290 (January 2015): 44–56. http://dx.doi.org/10.1016/j.physd.2014.09.006.

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3

Bold, Katherine A., Yu Zou, Ioannis G. Kevrekidis, and Michael A. Henson. "An equation-free approach to analyzing heterogeneous cell population dynamics." Journal of Mathematical Biology 55, no. 3 (April 11, 2007): 331–52. http://dx.doi.org/10.1007/s00285-007-0086-6.

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4

Chen, C. S., A. S. Muleshkov, M. A. Golberg, and R. M. M. Mattheij. "A mesh-free approach to solving the axisymmetric Poisson's equation." Numerical Methods for Partial Differential Equations 21, no. 2 (2005): 349–67. http://dx.doi.org/10.1002/num.20040.

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5

Francken, P., and C. J. Joachain. "Multiphoton Free-Free Transitions in Stochastic Laser Fields: The Kubo-Anderson Master Equation Approach." Europhysics Letters (EPL) 9, no. 6 (July 15, 1989): 517–22. http://dx.doi.org/10.1209/0295-5075/9/6/004.

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6

McDONALD, N. R. "Generalised Hele-Shaw flow: A Schwarz function approach." European Journal of Applied Mathematics 22, no. 6 (May 16, 2011): 517–32. http://dx.doi.org/10.1017/s0956792511000210.

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An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz functiong(z,t) of the free boundary. This generalises the well-known equation ∂g/∂t= 2∂w/∂z, wherewis the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

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7

Jarzynski, C. "Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach." Physical Review E 56, no. 5 (November 1, 1997): 5018–35. http://dx.doi.org/10.1103/physreve.56.5018.

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8

Zou, Y., I. G. Kevrekidis, and D. Armbruster. "Multiscale analysis of re-entrant production lines: An equation-free approach." Physica A: Statistical Mechanics and its Applications 363, no. 1 (April 2006): 1–13. http://dx.doi.org/10.1016/j.physa.2006.01.043.

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9

A. Bold, Katherine, Karthikeyan Rajendran, Balázs Ráth, and Ioannis G. Kevrekidis. "An equation-free approach to coarse-graining the dynamics of networks." Journal of Computational Dynamics 1, no. 1 (2014): 111–34. http://dx.doi.org/10.3934/jcd.2014.1.111.

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10

Huang, Dong-mei, Guo-liang Zou, and L. W. Zhang. "Numerical Approximation of Nonlinear Klein-Gordon Equation Using an Element-Free Approach." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/548905.

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Numerical approximation of nonlinear Klein-Gordon (KG) equation with quadratic and cubic nonlinearity is performed using the element-free improved moving least squares Ritz (IMLS-Ritz) method. A regular arrangement of nodes is employed in this study for the numerical integration to compute the system equation. A functional formulation for the KG equation is established and discretized by the Ritz minimization procedure. Newmark’s integration scheme combined with an iterative technique is applied to the resulting nonlinear system equations. The effectiveness and efficiency of the IMLS-Ritz method for the KG equation have been testified through convergence analyses and comparison study between the present results and the exact solutions.

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11

HU, ZHAN-NING. "NEW APPROACH ON THE GENERAL SHAPE EQUATION OF AXISYMMETRIC VESICLES." Modern Physics Letters B 13, no. 01 (January 10, 1999): 13–18. http://dx.doi.org/10.1142/s0217984999000038.

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The general Helfrich shape equation determined by minimizing the curvature free energy describes the equilibrium shapes of the axisymmetric lipid bilayer vesicles in different conditions. It is a nonlinear differential equation with variable coefficients. In this letter, by analyzing the unique property of the solution, we change this shape equation into a system of the two differential equations. One of them is a linear differential equation. This equation system contains all of the known rigorous solutions of the general shape equation. And the more general constraint conditions are found for the solution of the general shape equation.

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12

Jeon, Junkee, and Geonwoo Kim. "An Integral Equation Approach to the Irreversible Investment Problem with a Finite Horizon." Mathematics 8, no. 11 (November 22, 2020): 2084. http://dx.doi.org/10.3390/math8112084.

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This paper studies an irreversible investment problem under a finite horizon. The firm expands its production capacity in irreversible investments by purchasing capital to increase productivity. This problem is a singular stochastic control problem and its associated Hamilton–Jacobi–Bellman equation is derived. By using a Mellin transform, we obtain the integral equation satisfied by the free boundary of this investment problem. Furthermore, we solve the integral equation numerically using the recursive integration method and present the graph for the free boundary.

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13

Alizadeh, A., M. Abbasi, A. Minuchehr, and A. Zolfaghari. "A novel adaptive mesh free approach for even- parity neutron transport equation." Annals of Nuclear Energy 175 (September 2022): 109253. http://dx.doi.org/10.1016/j.anucene.2022.109253.

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14

Li, Dong. "A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials." Discrete & Continuous Dynamical Systems 42, no. 5 (2022): 2453. http://dx.doi.org/10.3934/dcds.2021198.

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15

MÖLLER, J., O. RUNBORG, P. G. KEVREKIDIS, K. LUST, and I. G. KEVREKIDIS. "EQUATION-FREE, EFFECTIVE COMPUTATION FOR DISCRETE SYSTEMS: A TIME STEPPER BASED APPROACH." International Journal of Bifurcation and Chaos 15, no. 03 (March 2005): 975–96. http://dx.doi.org/10.1142/s0218127405012399.

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We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

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16

MOON, SUNG JOON, and IOANNIS G. KEVREKIDIS. "AN EQUATION-FREE APPROACH TO COUPLED OSCILLATOR DYNAMICS: THE KURAMOTO MODEL EXAMPLE." International Journal of Bifurcation and Chaos 16, no. 07 (July 2006): 2043–52. http://dx.doi.org/10.1142/s021812740601588x.

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We present an equation-free multiscale approach to the computational study of the collective dynamics of the Kuramoto model [Kuramoto, 1984], a prototypical model for coupled oscillator populations. Our study takes place in a reduced phase space of coarse-grained "observables" of the system: the first few moments of the oscillator phase angle distribution. We circumvent the derivation of explicit dynamical equations (approximately) governing the evolution of these coarse-grained macroscopic variables; instead we use the equation-free framework [Kevrekidis et al., 2003] to computationally solve these equations without obtaining them in closed form. In this approach, the numerical tasks for the conceptually existing but unavailable coarse-grained equations are implemented through short bursts of appropriately initialized simulations of the "fine-scale", detailed coupled oscillator model. Coarse projective integration and coarse fixed point computations are illustrated.

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17

Erban, Radek, Ioannis G. Kevrekidis, David Adalsteinsson, and Timothy C. Elston. "Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation." Journal of Chemical Physics 124, no. 8 (February 28, 2006): 084106. http://dx.doi.org/10.1063/1.2149854.

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18

Poon, Wai-Yin, and Sik-Yum Lee. "A distribution free approach for analysis of two-level structural equation model." Computational Statistics & Data Analysis 17, no. 3 (March 1994): 265–75. http://dx.doi.org/10.1016/0167-9473(92)00074-2.

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19

SIETTOS, CONSTANTINOS I., IOANNIS G. KEVREKIDIS, and NIKOLAOS KAZANTZIS. "AN EQUATION-FREE APPROACH TO NONLINEAR CONTROL: COARSE FEEDBACK LINEARIZATION WITH POLE-PLACEMENT." International Journal of Bifurcation and Chaos 16, no. 07 (July 2006): 2029–41. http://dx.doi.org/10.1142/s0218127406015878.

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We present an application of equation-free computation to the coarse-grained feedback linearization problem of nonlinear systems described by microscopic/stochastic simulators. Feedback linearization with pole placement requires the solution of a functional equation involving the macroscopic (coarse-grained) system model. In the absence of such a closed-form model, short, appropriately initialized bursts of microscopic simulation are designed and performed, and their results used to estimate on demand the quantities required for the numerical solution of the (explicitly unavailable) functional equation. Our illustrative example is a kinetic Monte Carlo realization of a simplified heterogeneous catalytic reaction scheme.

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20

Das, Ashok K., Sudhakar Panda, and J. R. L. Santos. "A path integral approach to the Langevin equation." International Journal of Modern Physics A 30, no. 07 (March 5, 2015): 1550028. http://dx.doi.org/10.1142/s0217751x15500281.

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We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck–Ornstein model). The path integral description also leads to a simple derivation of the Fokker–Planck equation for the generalized Langevin equation.

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21

Sorokin, N. A. "Earth's gravity field parameters determination by the space geodesy dynamical approach." Geodesy and Cartography 919, no. 1 (February 20, 2017): 7–12. http://dx.doi.org/10.22389/0016-7126-2017-919-1-7-12.

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The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

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22

Tan, Guojin, Wensheng Wang, and Yubo Jiao. "Flexural Free Vibrations of Multistep Nonuniform Beams." Mathematical Problems in Engineering 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/7314280.

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This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as I(x)=α11+βxr+4 and m(x)=α21+βxr was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.

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23

Zhipeng, Wang, Liu Wei, Yuan Yunbo, Shuai Zhijun, Guo Yibin, and Wang Donghua. "Free Vibration Analysis of Rings via Wave Approach." Shock and Vibration 2018 (2018): 1–14. http://dx.doi.org/10.1155/2018/6181204.

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Free vibration of rings is presented via wave approach theoretically. Firstly, based on the solutions of out-of-plane vibration, propagation, reflection, and coordination matrices are derived for the case of a fixed boundary at inner surface and a free boundary at outer surface. Then, assembling these matrices, characteristic equation of natural frequency is obtained. Wave approach is employed to study the free vibration of these ring structures. Natural frequencies calculated by wave approach are compared with those obtained by classical method and Finite Element Method (FEM). Afterwards natural frequencies of four type boundaries are calculated. Transverse vibration transmissibility of rings propagating from outer to inner and from inner to outer is investigated. Finally, the effects of structural and material parameters on free vibration are discussed in detail.

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24

M A Patel and N B Desai. "Homotopy Analysis Approach of Boussinesq Equation for Infiltration Phenomenon in Unsaturated Porous Medium." Mathematical Journal of Interdisciplinary Sciences 7, no. 1 (September 6, 2018): 21–28. http://dx.doi.org/10.15415/mjis.2018.71004.

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Boussinesq’s equation is one-dimensional nonlinear partial differential equation which represents the infiltration phenomenon. This equation is frequently used to study the infiltration phenomenon in unsaturated porous medium. Infiltration is the process in which the groundwater of the water reservoir has entered in the unsaturated soil through vertical permeable wall. An approximate analytical solution of nonlinear partial differential equation is presented by homotopy analysis method. The convergence of homotopy analysis solution is discussed by choosing proper value of convergence control parameter. The solution represents the height of free surface of infiltrated water.

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25

Dias, F., and A. Il’ichev. "Interfacial waves with free-surface boundary conditions: an approach via a model equation." Physica D: Nonlinear Phenomena 150, no. 3-4 (April 2001): 278–300. http://dx.doi.org/10.1016/s0167-2789(01)00149-x.

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26

Qiao, Bi, H. E. Ruda, M. S. Zhan, and X. H. Zeng. "Kinetic equation, non-perturbative approach and decoherence free subspace for quantum open system." Physica A: Statistical Mechanics and its Applications 322 (May 2003): 345–58. http://dx.doi.org/10.1016/s0378-4371(02)01809-5.

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27

Metzler, R., W. G. Glöckle, T. F. Nonnenmacher, and B. J. West. ""Fractional Tuning" of the Riccati Equation." Fractals 05, no. 04 (December 1997): 597–601. http://dx.doi.org/10.1142/s0218348x97000474.

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The dynamics of a free-falling body in complex materials such as a polymer fluid is phenomenologically modeled using a fractional generalization of the Riccati equation. The solution exhibits a rich behavior in its parametric dependence, and unlike normal free–fall there is no terminal velocity, instead a power–law increase in time is obtained. Within this approach the fractional order allows to tune the resulting equation.

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28

Lindgren, Jussi. "Efficient Markets and Contingent Claims Valuation: An Information Theoretic Approach." Entropy 22, no. 11 (November 12, 2020): 1283. http://dx.doi.org/10.3390/e22111283.

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This research article shows how the pricing of derivative securities can be seen from the context of stochastic optimal control theory and information theory. The financial market is seen as an information processing system, which optimizes an information functional. An optimization problem is constructed, for which the linearized Hamilton–Jacobi–Bellman equation is the Black–Scholes pricing equation for financial derivatives. The model suggests that one can define a reasonable Hamiltonian for the financial market, which results in an optimal transport equation for the market drift. It is shown that in such a framework, which supports Black–Scholes pricing, the market drift obeys a backwards Burgers equation and that the market reaches a thermodynamical equilibrium, which minimizes the free energy and maximizes entropy.

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29

Yilbas, Bekir Sami, R. S. M. Alassar, and Ahmad Y. Al-Dweik. "Thermal Stress Development in Low Dimensional Silicon Film: An Analytical Approach." Journal of Non-Equilibrium Thermodynamics 46, no. 2 (January 7, 2021): 205–19. http://dx.doi.org/10.1515/jnet-2020-0094.

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Abstract Thermal excitation of the low dimensional silicon film is introduced and an analytical approach is adopted for the solution of the transport equation. In the analysis, the phonon radiative transport equation is converted into an integral form of the Fredholm equation of the second kind. The analytical approach is extended to include the formulation of thermal stresses for the following cases: (i) stress-free at the edges and (ii) one edge is constrained to have maximum stress while the other edge is set to be stress-free. The analytical and numerical results are evaluated for comparisons. The findings demonstrate that both results are in good agreement. The dimensionless temperature rise at the film mid-thickness becomes sharp for small thickness film. The peak value of thermal stress at the film mid-thickness becomes larger as the film thickness is reduced further. Stress waves generated initially are compressive at the film mid-thickness and they become tensile at both ends of the stress-free film, which becomes more apparent as time increases. Two consecutive compressive and tensile stresses are generated at the mid-thickness of the film as the stress boundary condition is changed to the maximum stress at one edge of the film.

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30

AGNON, Y., P. A. MADSEN, and H. A. SCHÄFFER. "A new approach to high-order Boussinesq models." Journal of Fluid Mechanics 399 (November 25, 1999): 319–33. http://dx.doi.org/10.1017/s0022112099006394.

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An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters – nonlinearity, the dispersion parameter, and the bottom slope – the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.

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31

He, Jia, Xiaoxiong Zhang, Mengchen Qi, and Bin Xu. "Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation." Applied Sciences 9, no. 2 (January 16, 2019): 306. http://dx.doi.org/10.3390/app9020306.

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Nonlinearity exists widely in civil engineering structures; for example, the initiation and growth of damage under dynamic loadings is a typical nonlinear process. To date, for the purpose of structural evaluation and a better understanding nonlinear characteristics of complicated structures, a number of parametric and nonparametric methods have been developed for the identification of nonlinear restoring force (NRF). However, due to the highly individualistic nature of nonlinear systems, it would be inefficient to attempt to express the structural NRF in a general parametric form. For many nonparametric techniques, their nonparametric models or approximations may result in undesirable results or oscillations around unsmooth points. In this paper, on the basis of extended Kalman filter (EKF), a model-free NRF identification approach is proposed to circumvent the limitations mentioned above. The NRF to be identified was treated as ‘unknown fictitious input’, and thus, no prior assumptions or approximations for the NRF model were required. With the aid of a projection matrix, a modified version of observation equation was obtained. Based on the principle of EKF, the recursive solution of the proposed approach was analytically derived. The NRFs provided by the nonlinear components were identified by means of least squares estimation (LSE) at each time step. Numerical examples, including building structures equipped with magnetorheological (MR) damper and shape memory alloy (SMA) damper, demonstrated that the proposed approach is capable of satisfactorily identifying NRF without knowledge or intuitive assumptions of any nonlinear model class in advance.

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32

ÖZUĞURLU, E. "A NOTE ON THE NUMERICAL APPROACH FOR THE REACTION–DIFFUSION PROBLEM WITH A FREE BOUNDARY CONDITION." ANZIAM Journal 51, no. 3 (January 2010): 317–30. http://dx.doi.org/10.1017/s1446181110000817.

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AbstractThe equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

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33

Wu, Jieer, Yu Zhou, and Chao Hang. "A singularity free and derivative free approach for Abel integral equation in analyzing the laser-induced breakdown spectroscopy." Spectrochimica Acta Part B: Atomic Spectroscopy 167 (May 2020): 105791. http://dx.doi.org/10.1016/j.sab.2020.105791.

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34

Spengler, Felix, Davide Campagnari, and Hugo Reinhardt. "Dynamical quark mass generation in QCD3 within the Hamiltonian approach in Coulomb gauge." European Physical Journal Special Topics 229, no. 22-23 (December 2020): 3351–61. http://dx.doi.org/10.1140/epjst/e2020-000086-4.

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AbstractWe investigate the equal-time (static) quark propagator in Coulomb gauge within the Hamiltonian approach to QCD in d = 2 spatial dimensions. Although the underlying Clifford algebra is very different from its counterpart in d = 3, the gap equation for the dynamical mass function has the same form. The additional vector kernel which was introduced in d = 3 to cancel the linear divergence of the gap equation and to preserve multiplicative renormalizability of the quark propagator makes the gap equation free of divergences also in d = 2.

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35

Vincent, Stéphane, and Jean-Paul Caltagirone. "A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics." Fluids 6, no. 3 (March 1, 2021): 95. http://dx.doi.org/10.3390/fluids6030095.

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The unification of the laws of fluid and solid mechanics is achieved on the basis of the concepts of discrete mechanics and the principles of equivalence and relativity, but also the Helmholtz–Hodge decomposition where a vector is written as the sum of divergence-free and curl-free components. The derived equation of motion translates the conservation of acceleration over a segment, that of the intrinsic acceleration of the material medium and the sum of the accelerations applied to it. The scalar and vector potentials of the acceleration, which are the compression and shear energies, give the discrete equation of motion the role of conservation law for total mechanical energy. Velocity and displacement are obtained using an incremental time process from acceleration. After a description of the main stages of the derivation of the equation of motion, unique for the fluid and the solid, the cases of couplings in simple shear and uniaxial compression of two media, fluid and solid, make it possible to show the role of discrete operators and to find the theoretical results. The application of the formulation is then extended to a classical validation case in fluid–structure interaction.

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36

Heining, C., T. Pollak, and M. Sellier. "Flow domain identification from free surface velocity in thin inertial films." Journal of Fluid Mechanics 720 (February 27, 2013): 338–56. http://dx.doi.org/10.1017/jfm.2013.14.

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AbstractWe consider the flow of a viscous liquid along an unknown topography. A new strategy is presented to reconstruct the topography and the free surface shape from one component of the free surface velocity only. In contrast to the classical approach in inverse problems based on optimization theory we derive an ordinary differential equation which can be solved directly to obtain the inverse solution. This is achieved by averaging the Navier–Stokes equation and coupling the function parameterizing the flow domain with the free surface velocity. Even though we consider nonlinear systems including inertia and surface tension, the inverse problem can be solved analytically with a Fourier series approach. We test our method on a variety of benchmark problems and show that the analytical solution can be applied to reconstruct the flow domain from noisy input data. It is also demonstrated that the asymptotic approach agrees very well with numerical results of the Navier–Stokes equation. The results are finally confirmed with an experimental study where we measure the free surface velocity for the film flow over a trench and compare the reconstructed topography with the measured one.

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WANG, C. L., C. ARAGÓ, and M. I. MARQUÉS. "CHARACTERISTIC TEMPERATURES OF FIRST-ORDER FERROELECTRIC PHASE TRANSITION: EFFECTIVE FIELD APPROACH." Journal of Advanced Dielectrics 02, no. 02 (April 2012): 1241007. http://dx.doi.org/10.1142/s2010135x1241007x.

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The explicit expression of Helmholtz free energy has been obtained from the equation of state from effective field approach. From the Helmholtz free energy, four characteristic temperatures describing a first-order ferroelectric phase transitions have been determined. The physical meaning of coefficients in Landau-type free energy has been revealed by comparison with the expanding Helmholtz function. Temperature dependence of polarization under different bias, and hysteresis loops at different temperatures are presented and discussed. These results provide the basic understandings of the static properties of first-order ferroelectric phase transitions.

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38

González Díaz, D. "Inverse relationship between diffusion coefficient and mass for a free particle system: Approach by using maximum caliber principle and Monte Carlo simulations." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 12 (December 2022): 123141. http://dx.doi.org/10.1063/5.0120977.

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A derivation of the diffusion equation is presented using the maximum caliber principle and the continuity equation for a system composed of paths traveled by a free particle in a time interval. By identifying the diffusion coefficient in the obtained diffusion equation, it is shown that there is an inverse proportionality relationship concerning the particle’s mass so that a higher mass is related to lower diffusion, and the lower mass is connected to the higher diffusion. This relationship is also shown using Monte Carlo simulations to sample the path space for a free particle system and then using the time slicing equation to obtain the probability of the particle position for each time showing the diffusion behavior for different masses.

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39

Coenders, Germà, Joan M. Batista-Foguet, and Willem E. Saris. "Simple, Efficient and Distribution-free Approach to Interaction Effects in Complex Structural Equation Models." Quality & Quantity 42, no. 3 (December 20, 2006): 369–96. http://dx.doi.org/10.1007/s11135-006-9050-6.

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40

Moon, Sung Joon, B. Nabet, Naomi E. Leonard, Simon A. Levin, and I. G. Kevrekidis. "Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach." Journal of Theoretical Biology 246, no. 1 (May 2007): 100–112. http://dx.doi.org/10.1016/j.jtbi.2006.12.018.

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Arjmand, Doghonay, and Gunilla Kreiss. "An equation-free approach for second order multiscale hyperbolic problems in non-divergence form." Communications in Mathematical Sciences 16, no. 8 (2018): 2317–43. http://dx.doi.org/10.4310/cms.2018.v16.n8.a11.

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Siettos, C. I., R. Rico-Martinez, and I. G. Kevrekidis. "A systems-based approach to multiscale computation: Equation-free detection of coarse-grained bifurcations." Computers & Chemical Engineering 30, no. 10-12 (September 2006): 1632–42. http://dx.doi.org/10.1016/j.compchemeng.2006.05.019.

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Leuridan, J. M., D. L. Brown, and R. J. Allemang. "Time Domain Parameter Identification Methods for Linear Modal Analysis: A Unifying Approach." Journal of Vibration and Acoustics 108, no. 1 (January 1, 1986): 1–8. http://dx.doi.org/10.1115/1.3269298.

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The paper describes a method that uses a multivariate model in the form of a nonhomogeneous finite difference equation to identify modal parameters of a mechanical structure. The modal parameters of this equation are estimable using a model that involves multiple input, multiple output vibration data. Thus, improved global estimates of modal parameters can be obtained, including the identification of highly coupled and pseudo-repeated modes of vibration. When the data are in the form of impulse or free decay responses, then the parameters of the homogeneous part of the equation can be estimated separately, and the method is then related to the Least Squares Complex Exponential method, the Polyreference Time Domain method and the Ibrahim Time Domain method.

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Farwig, Reinhard, and Andreas Schmidt. "An $$L^2$$ approach to viscous flow in the half space with free elastic surface." Journal of Elliptic and Parabolic Equations 7, no. 2 (September 29, 2021): 601–21. http://dx.doi.org/10.1007/s41808-021-00111-2.

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AbstractWe consider a linearized fluid-structure interaction problem, namely the flow of an incompressible viscous fluid in the half space $${\mathbb {R}}^n_+$$ R + n such that on the lower boundary a function h satisfying an undamped Kirchhoff-type plate equation is coupled to the flow field. Originally, h describes the height of an underlying nonlinear free surface problem. Since the plate equation contains no damping term, this article uses $$L^2$$ L 2 -theory yielding the existence of strong solutions on finite time intervals in the framework of homogeneous Bessel potential spaces. The proof is based on $$L^2$$ L 2 -Fourier analysis and also deals with inhomogeneous boundary data of the velocity field on the “free boundary” $$x_n=0$$ x n = 0 .

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Adah, Edward, David Onwuka, Owus Ibearugbulem, and Chinenye Okere. "LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE." Journal of Civil Engineering, Science and Technology 12, no. 1 (April 30, 2021): 15–25. http://dx.doi.org/10.33736/jcest.3338.2021.

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The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.

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Contreras-Vergara, O., N. Lucero-Azuara, N. Sánchez-Salas, and J. I. Jiménez-Aquino. "Harmonic oscillator Brownian motion: Langevin approach revisited." Revista Mexicana de Física E 18, no. 1 (January 4, 2021): 97. http://dx.doi.org/10.31349/revmexfise.18.97.

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The original strategy applied by Langevin to Brownian movement problem is used to solve the case of a free particle under a harmonic potential. Such straightforward strategy consists in separating the noise termin the Langevin equation in order to solve a deterministic equation associated with the Mean Square Displacement (MSD). In this work, to achieve our goal we first calculate the variance for the stochastic harmonic oscillator and then the MSD appears immediately. We study the problem in the damped and lightly damped cases and show that, for times greater than the relaxation time, Langevin's original strategy is quite consistent with the exact theoretical solutions reported by Chandrasekhar and Lemons, these latter obtained using the statistical properties of a Gaussian white noise. Our results for the MSDs are compared with the exact theoretical solutions as well as with the numerical simulation.

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MOJAHEDI, MAHDI, HAMID MOEENFARD, and MOHAMMAD TAGHI AHMADIAN. "NONLINEAR FREE VIBRATION OF SIMPLY SUPPORTED BEAMS CONSIDERING THE EFFECTS OF SHEAR DEFORMATION AND ROTARY INERTIA, A HOMOTOPY PERTURBATION APPROACH." International Journal of Modern Physics B 25, no. 03 (January 30, 2011): 441–55. http://dx.doi.org/10.1142/s0217979211055725.

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The objective of this paper is to apply He's homotopy perturbation method (HPM) to analyze nonlinear free vibration of simply supported Timoshenko beams considering the effects of rotary inertia and shear deformation. First, the equation governing the nonlinear free vibration of a Timoshinko beam is nondimensionalized. Galerkin's projection method is utilized to reduce the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. HPM is then used to find analytic expressions for nonlinear natural frequencies of the pre-stretched beam. A parametric study has also been applied in order to investigate the effects of design parameters such as applied axial load and slenderness ratio. Comparison between presented results and numerical results which are in full agreement shows that HPM can significantly improve the accuracy of previously reported results in the literature.

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LANZ, LUDOVICO, and BASSANO VACCHINI. "SUBDYNAMICS OF RELEVANT OBSERVABLES: A FIELD THEORETICAL APPROACH." International Journal of Modern Physics A 17, no. 04 (February 10, 2002): 435–63. http://dx.doi.org/10.1142/s0217751x02005918.

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An approach to the description of subdynamics inside the nonrelativistic quantum field theory is presented, in which the notions of relevant observable, time scale and complete positivity of the time evolution are stressed. A scattering theory derivation of the subdynamics of a microsystem interacting through collisions with a macrosystem is given, leading to a master equation expressed in terms of the operator-valued dynamic structure factor, a two-point correlation function which compactly takes the statistical mechanics properties of the macrosystem into account. For the case of a free quantum gas the dynamic structure factor can be exactly calculated and in the long wavelength limit a Fokker–Planck equation for the description of quantum dissipation and in particular quantum Brownian motion is obtained, where peculiar corrections due to quantum statistics can be put into evidence.

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JACOBONI, CARLO, ROSSELLA BRUNETTI, PAOLO BORDONE, and ANDREA BERTONI. "QUANTUM TRANSPORT AND ITS SIMULATION WITH THE WIGNER-FUNCTION APPROACH." International Journal of High Speed Electronics and Systems 11, no. 02 (June 2001): 387–423. http://dx.doi.org/10.1142/s0129156401000897.

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In this paper a review of the research performed in recent years by the group of the authors is presented. The definition and basic properties of the Wigner function are first given. Several forms of its dynamical equation are then derived with the inclusion of potential and phonon scattering. For the case of a potential V(r) the effect of the classical force, for any form of V(r), is separated from quantum effects due to rapidly varying potentials. An elaboration of the dynamical equation is introduced that leads to Wigner paths formed by free flights and scattering events. These are especially suitable for a Monte Carlo solution of the transport equation for the Wigner function very similar to the semiclassical traditional Monte carlo simulation. The Monte Carlo simulation can be extended also to the momentum and frequency dependent Wigner function based on a two-time Green function. Several numerical results are presented throuhout the paper.

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Siettos, C. I., C. W. Gear, and I. G. Kevrekidis. "An equation-free approach to agent-based computation: Bifurcation analysis and control of stationary states." EPL (Europhysics Letters) 99, no. 4 (August 1, 2012): 48007. http://dx.doi.org/10.1209/0295-5075/99/48007.

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