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Journal articles on the topic 'Approximation of solutions'

Author: Grafiati

Published: 14 March 2023

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1

Migda, Janusz, and Malgorzata Migda. "Approximation of Solutions to Nonautonomous Difference Equations." Tatra Mountains Mathematical Publications 71, no. 1 (December 1, 2018): 109–21. http://dx.doi.org/10.2478/tmmp-2018-0010.

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Abstract We study the asymptotic properties of solutions to nonautonomous difference equations of the form $${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$ Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.

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2

Duma, Adrian, and Cristian Vladimirescu. "Approximation structures and applications to evolution equations." Abstract and Applied Analysis 2003, no. 12 (2003): 685–96. http://dx.doi.org/10.1155/s1085337503301010.

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We discuss various properties of the nonlinearA-proper operators as well as a generalized Leray-Schauder principle. Also, a method of approximating arbitrary continuous operators byA-proper mappings is described. We construct, via appropriate Browder-Petryshyn approximation schemes, approximative solutions for linear evolution equations in Banach spaces.

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3

Salas, Alvaro H., Wedad Albalawi, M. R. Alharthi, and S. A. El-Tantawy. "Some Novel Solutions to a Quadratically Damped Pendulum Oscillator: Analytical and Numerical Approximations." Complexity 2022 (May 28, 2022): 1–14. http://dx.doi.org/10.1155/2022/7803798.

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In this paper, some novel analytical and numerical techniques are introduced for solving and analyzing nonlinear second-order ordinary differential equations (ODEs) that are associated to some strongly nonlinear oscillators such as a quadratically damped pendulum equation. Two different analytical approximations are obtained: for the first approximation, the ansatz method with the help of Chebyshev approximate polynomial is employed to derive an approximation in the form of trigonometric functions. For the second analytical approximation, a novel hybrid homotopy with Krylov–Bogoliubov–Mitropolsky method (HKBMM) is introduced for the first time for analyzing the evolution equation. For the numerical approximation, both the finite difference method (FDM) and Galerkin method (GM) are presented for analyzing the strong nonlinear quadratically damped pendulum equation that arises in real life, such as nonlinear phenomena in plasma physics, engineering, and so on. Several examples are discussed and compared to the Runge–Kutta (RK) numerical approximation to investigate and examine the accuracy of the obtained approximations. Moreover, the accuracy of all obtained approximations is checked by estimating the maximum residual and distance errors.

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4

Kuzmina, E. V. "Generalized solutions of the Riccati equation." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 58, no. 2 (July 5, 2022): 144–54. http://dx.doi.org/10.29235/1561-2430-2022-58-2-144-154.

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In this paper, we consider a nonlinear differential equation of the first order of the Riccati hierarchy. The concept of a generalized solution for such an equation cannot be introduced within the framework of the classical theory of generalized functions because the product of generalized functions is not defined. To introduce the concept of a generalized solution, two approaches are considered. In the first approach, approximation by solutions of the Cauchy problem with complex initial conditions is used, and generalized solutions are defined as limits of approximating families in the sense of convergence in D′(¡). It is shown that there are two generalized solutions of the Cauchy problem. The type of solution depends on whether the poles of the approximating solution are located in the upper or lower half-plane. The second approach uses approximation with a system of equations. It is shown that there are many approximating systems, meanwhile, generalized solutions of the Cauchy problem depend on the choice of the approximating system.

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5

Stern, Steven. "Approximate Solutions to Stochastic Dynamic Programs." Econometric Theory 13, no. 3 (June 1997): 392–405. http://dx.doi.org/10.1017/s0266466600005867.

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This paper examines the properties of various approximation methods for solving stochastic dynamic programs in structural estimation problems. The problem addressed is evaluating the expected value of the maximum of available choices. The paper shows that approximating this by the maximum of expected values frequently has poor properties. It also shows that choosing a convenient distributional assumptions for the errors and then solving exactly conditional on the distributional assumption leads to small approximation errors even if the distribution is misspecified.

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6

Huang, Wentao, and Kechen Zhang. "Information-Theoretic Bounds and Approximations in Neural Population Coding." Neural Computation 30, no. 4 (April 2018): 885–944. http://dx.doi.org/10.1162/neco_a_01056.

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While Shannon's mutual information has widespread applications in many disciplines, for practical applications it is often difficult to calculate its value accurately for high-dimensional variables because of the curse of dimensionality. This article focuses on effective approximation methods for evaluating mutual information in the context of neural population coding. For large but finite neural populations, we derive several information-theoretic asymptotic bounds and approximation formulas that remain valid in high-dimensional spaces. We prove that optimizing the population density distribution based on these approximation formulas is a convex optimization problem that allows efficient numerical solutions. Numerical simulation results confirmed that our asymptotic formulas were highly accurate for approximating mutual information for large neural populations. In special cases, the approximation formulas are exactly equal to the true mutual information. We also discuss techniques of variable transformation and dimensionality reduction to facilitate computation of the approximations.

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7

Stanojević, Bogdana, and Milan Stanojević. "A computationally efficient algorithm to approximate the pareto front of multi-objective linear fractional programming problem." RAIRO - Operations Research 53, no. 4 (July 29, 2019): 1229–44. http://dx.doi.org/10.1051/ro/2018083.

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The main contribution of this paper is the procedure that constructs a good approximation to the non-dominated set of multiple objective linear fractional programming problem using the solutions to certain linear optimization problems. In our approach we propose a way to generate a discrete set of feasible solutions that are further used as starting points in any procedure for deriving efficient solutions. The efficient solutions are mapped into non-dominated points that form a 0th order approximation of the Pareto front. We report the computational results obtained by solving random generated instances, and show that the approximations obtained by running our procedure are better than those obtained by running other procedures suggested in the recent literature. We evaluated the quality of each approximation using classic metrics.

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8

Lanzara, F., V. Maz'ya, and G. Schmidt. "Approximation of solutions to multidimensional parabolic equations by approximate approximations." Applied and Computational Harmonic Analysis 41, no. 3 (November 2016): 749–67. http://dx.doi.org/10.1016/j.acha.2015.06.001.

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9

Ganji, S. S., M. G. Sfahani, S. M. Modares Tonekaboni, A. K. Moosavi, and D. D. Ganji. "Higher-Order Solutions of Coupled Systems Using the Parameter Expansion Method." Mathematical Problems in Engineering 2009 (2009): 1–20. http://dx.doi.org/10.1155/2009/327462.

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We consider periodic solution for coupled systems of mass-spring. Three practical cases of these systems are explained and introduced. An analytical technique called Parameter Expansion Method (PEM) was applied to calculate approximations to the achieved nonlinear differential oscillation equations. Comparing with exact solutions, the first approximation to the frequency of oscillation produces tolerable error 3.14% as the maximum. By the second iteration the respective error became 1/5th, as it is 0.064%. So we conclude that the first approximation of PEM is so benefit when a quick answer is required, but the higher order approximation gives a convergent precise solution when an exact solution is required.

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10

Crandall, S. H., and A. EI-Shafei. "Momentum and Energy Approximations for Elementary Squeeze-Film Damper Flows." Journal of Applied Mechanics 60, no. 3 (September 1, 1993): 728–36. http://dx.doi.org/10.1115/1.2900865.

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To provide understanding of the effects of inertia on squeeze-film damper performance, two elementary flow patterns are studied. These elementary flows each depend on a single generalized motion coordinate whereas general planar motions of a damper are described by two independent generalized coordinates. Momentum and energy approximations for the elementary flows are compared with exact solutions. It is shown that the energy approximation, not previously applied to squeeze films, is superior to the momentum approximation in that at low Reynolds number the energy approximations agree with the exact solutions to first order in the Reynolds number whereas there are 20 percent errors in the first-order terms of the momentum approximations.

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11

Berselli, Luigi C., and Stefano Spirito. "On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations." Fluids 6, no. 1 (January 13, 2021): 42. http://dx.doi.org/10.3390/fluids6010042.

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We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.

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12

Lieberman, Offer. "Solutions: An Approximation to GARCH." Econometric Theory 12, no. 2 (June 1996): 396–401. http://dx.doi.org/10.1017/s026646660000671x.

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13

Mateos, Alfonso, and Sixto Rios-Insua. "Approximation of Value Efficient Solutions." Journal of Multi-Criteria Decision Analysis 6, no. 4 (July 1997): 227–32. http://dx.doi.org/10.1002/(sici)1099-1360(199707)6:4<227::aid-mcda149>3.0.co;2-j.

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14

Gao, H., and M. A. M. Lynch. "A NEW MODEL APPROXIMATING M/PH/1 QUEUING SYSTEMS DURING THE TRANSIENT PERIOD." Mathematical Modelling and Analysis 6, no. 2 (December 15, 2001): 199–209. http://dx.doi.org/10.3846/13926292.2001.9637159.

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This paper presents an efficient approximation for M/PH/1 queuing systems based on the replacement of the majority of the vector valued state probabilities by a diffusion approximation. The strength of the new approximation is that it gives more accurate results than the current diffusion approximations at both high and low traffic intensities and at little extra computational cost. The accuracy of the new approximation during the transient is shown by comparing it numerically with solutions to the M/PH/1 system and current approaches based on the diffusion approximation.

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15

MATTIS, DANIEL C., ANTHONY M. SZPILKA, and HUA CHEN. "CONSERVATION LAWS AND EXACT SOLUTIONS OF THE BOLTZMANN EQUATION." Modern Physics Letters B 03, no. 03 (March 10, 1989): 215–23. http://dx.doi.org/10.1142/s0217984989000364.

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The distribution function f which satisfies the time-dependent Boltzmann equation (BE) for a Lorentz model with perfectly elastic random scatterers is proved nonnegative, and is computed exactly when backscattering dominates. Joule heating and Ohm’s law are recovered, although f has no steady-state limit, contrary to the relaxation-time approximation. (The conventional approximation to the time-independent BE also yields Ohm’s law but not the Joule heating and, worse, it unphysically predicts f<0.) The exact solution is compared with various effective-temperature approximations, and is shown to remain very nearly unchanged over a wide range of times even in the presence of a small amount of inelastic scattering.

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16

MCDANIEL, SUZANNE T. "RENORMALIZING ROUGH-SURFACE SCATTER." Journal of Computational Acoustics 10, no. 01 (March 2002): 53–68. http://dx.doi.org/10.1142/s0218396x02001541.

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Exact and approximate solutions to the rough-surface scattering problem are compared to examine the predictive capability of renormalized surface scattering theory. Numerical results are presented for scattering from one-dimensional rough periodic surfaces on which the Dirichlet (acoustic pressure-release) and Neumann (acoustically rigid) boundary conditions are imposed. For scattering from Dirichlet surfaces, the predictions of renormalized scattering theory are found to provide better agreement with exact solutions than perturbation theory. For this boundary condition, many convergent approximations exist, and the small-slope approximation is found to yield an improvement to renormalization. For the Neumann boundary condition, renormalization provides good agreement with exact solutions for scattering from slightly rough surfaces. The Kirchhoff approximation, the only other convergent approximation applicable to the Neumann problem, provides agreement with exact solutions for scattering from moderately rough surfaces for angles of scatter and incidence far from grazing.

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17

Smirnova, Anna S. "Lp -approximations for solutions of parabolic differential equations on manifolds." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 34, no. 3 (September 30, 2022): 297–303. http://dx.doi.org/10.15507/2079-6900.24.202203.297-303.

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The paper considers the Cauchy problem for a parabolic partial differential equation in a Riemannian manifold of bounded geometry. A formula is given that expresses arbitrarily accurate (in the Lp-norm) approximations to the solution of the Cauchy problem in terms of parameters - the coefficients of the equation and the initial condition. The manifold is not assumed to be compact, which creates significant technical difficulties - for example, integrals over the manifold become improper in the case when the manifold has an infinite volume. The presented approximation method is based on Chernoff theorem on approximation of operator semigroups.

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18

Micula, Sanda. "Numerical Solution of Two-Dimensional Fredholm–Volterra Integral Equations of the Second Kind." Symmetry 13, no. 8 (July 23, 2021): 1326. http://dx.doi.org/10.3390/sym13081326.

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The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their solutions. The method proposed here uses successive approximations of the Mann type and a suitable cubature formula. Mann’s procedure is known to converge faster than the classical Picard iteration given by the contraction principle, thus yielding a better numerical method. The existence and uniqueness of the solution is derived under certain conditions. The convergence of the method is proved, and error estimates for the approximations obtained are given. At the end, several numerical examples are analyzed, showing the applicability of the proposed method and good approximation results. In the last section, concluding remarks and future research ideas are discussed.

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19

Yoshida, Shin’ichirou. "Assessment of the Roche and the Darwin–Radau Approximations for Rotating Astrophysical Objects." Research Notes of the AAS 6, no. 10 (October 19, 2022): 217. http://dx.doi.org/10.3847/2515-5172/ac9afb.

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Abstract Assessment of two of the frequently used approximations for rotating astrophysical objects, the Roche and the Darwin–Radau approximations, is performed by comparing the rotational frequencies computed by these approximations with those of self-consistent numerical hydrostatic solutions. Overall, the Darwin–Radau approximation performs better, though the error increases as the rotational oblateness increases. It should be also remarked that the Darwin–Radau approximation does not provide a closed-form formula of the frequency as a function of the oblateness.

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20

Altac¸, Zekeriya. "The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1." Journal of Heat Transfer 124, no. 4 (July 16, 2002): 674–84. http://dx.doi.org/10.1115/1.1464130.

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A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

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21

Bougoffa, Lazhar, and Abdul-Majid Wazwaz. "New approximate solutions of the Blasius equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 7 (September 7, 2015): 1590–99. http://dx.doi.org/10.1108/hff-08-2014-0263.

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Purpose – The purpose of this paper is to propose a reliable treatment for studying the Blasius equation, which arises in certain boundary layer problems in the fluid dynamics. The authors propose an algorithm of two steps that will introduce an exact solution to the equation, followed by a correction to that solution. An approximate analytic solution, which contains an auxiliary parameter, is obtained. A highly accurate approximate solution of Blasius equation is also provided by adding a third initial condition y ' ' (0) which demonstrates to be quite accurate by comparison with Howarth solutions. Design/methodology/approach – The approach consists of two steps. The first one is an assumption for an exact solution that satisfies the Blasius equation, but does not satisfy the given conditions. The second step depends mainly on using this assumption combined with the given conditions to derive an accurate approximation that improves the accuracy level. Findings – The obtained approximation shows an enhancement over some of the existing techniques. Comparing the calculated approximations confirm the enhancement that the derived approximation presents. Originality/value – In this work, a new approximate analytical solution of the Blasius problem is obtained, which demonstrates to be quite accurate by comparison with Howarth solutions.

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22

Sheng, W., N. Kalogerakis, and P. R. Bishnoi. "Explicit approximations of the mean spherical approximation model for electrolyte solutions." Journal of Physical Chemistry 97, no. 20 (May 1993): 5403–9. http://dx.doi.org/10.1021/j100122a036.

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23

Rao, Ling, and Hongquan Chen. "Fictitious Domain Technique for the Calculation of Time-Periodic Solutions of Scattering Problem." Mathematical Problems in Engineering 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/503791.

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The fictitious domain technique is coupled to the improved time-explicit asymptotic method for calculating time-periodic solution of wave equation. Conventionally, the practical implementation of fictitious domain method relies on finite difference time discretizations schemes and finite element approximation. Our new method applies finite difference approximations in space instead of conventional finite element approximation. We use the Dirac delta function to transport the variational forms of the wave equations to the differential form and then solve it by finite difference schemes. Our method is relatively easier to code and requires fewer computational operations than conventional finite element method. The numerical experiments show that the new method performs as well as the method using conventional finite element approximation.

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24

Tuzlukov, V. P. "Approximation of Capacity in MIMO Systems." Doklady BGUIR 20, no. 2 (April 5, 2022): 53–61. http://dx.doi.org/10.35596/1729-7648-2022-20-2-53-61.

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This paper introduces functional approximations to the MIMO capacity over flat Rayleigh fading channels, which allow for analytical solutions to network resource optimization problems. This approximation allows to solve the problem of resource allocation optimization in radio networks and in other systems used to transfer information. The precision of the suggested approximations is assessed and is shown to provide a very close match to the exact capacity expression.

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25

Kaliszewski, Ignacy, and Janusz Miroforidis. "Primal–Dual Type Evolutionary Multiobjective Optimization." Foundations of Computing and Decision Sciences 38, no. 4 (December 1, 2013): 267–75. http://dx.doi.org/10.2478/fcds-2013-0013.

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Abstract A new, primal-dual type approach for derivation of Pareto front approximations with evolutionary computations is proposed. At present, evolutionary multiobjective optimization algorithms derive a discrete approximation of the Pareto front (the set of objective maps of efficient solutions) by selecting feasible solutions such that their objective maps are close to the Pareto front. As, except of test problems, Pareto fronts are not known, the accuracy of such approximations is known neither. Here we propose to exploit also elements outside feasible sets with the aim to derive pairs of Pareto front approximations such that for each approximation pair the corresponding Pareto front lies, in a certain sense, in-between. Accuracies of Pareto front approximations by such pairs can be measured and controlled with respect to distance between such approximations. A rudimentary algorithm to derive pairs of Pareto front approximations is presented and the viability of the idea is verified on a limited number of test problems.

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26

Ergaliev, E. R., M. N. Madiyarov, N. M. Oskorbin, and L. L. Smolyakova. "Ellipsoidal Approximation of the Set of Solutions of Interval Systems of Linear Algebraic Equations." Izvestiya of Altai State University, no. 4(120) (September 10, 2021): 97–101. http://dx.doi.org/10.14258/izvasu(2021)4-15.

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The article presents the results of the approximation of the set of solutions of interval systems of linear algebraic equations. These systems are used in the problems of modeling linear deterministic processes. It is assumed that the modeled process is described by an output variable and a set of input variables, the measurement errors of which are assumed to be set by known intervals symmetric with respect to the zero value. Traditionally, the sets of solutions of interval systems of linear algebraic equations in applied problems are approximated by a hyper-rectangular whose sides are parallel to the axes of the selected coordinate system. In this paper, we propose to use an ellipsoidal approximation of these sets, which is more efficient. The main results of the work include the substantiation of assumptions about the properties of the modeled process, the choice of a mathematical method for constructing an approximating ellipsoid, the proposed method for forming boundary points, and a numerical method for solving the problem. A computer simulation of the problem of estimating the parameters of a linear process is performed in Excel, which is used for a comparative study of approximations of solutions of interval systems of linear algebraic equations by a hyper-rectangular and an ellipse.

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27

Pizio, O. A., and Z. B. Halytch. "Structural Properties of the Ion-Dipole Model of Electrolyte Solutions in the Bulk and Near a Charged Hard Wall.Application of the Truncated Optimized Cluster Series." Zeitschrift für Naturforschung A 46, no. 1-2 (February 1, 1991): 8–18. http://dx.doi.org/10.1515/zna-1991-1-203.

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AbstractAn ion-dipole model of electrolyte solutions in the bulk case and near a charged or uncharged hard wall is considered. A method to derive the terms of optimized cluster expansions for the distribution functions of ions and dipoles which provides a set of approximations beyond the mean spherical approximation is given. The third cluster coefficient approximation is investigated

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28

Rocha, Cesar B., William R. Young, and Ian Grooms. "On Galerkin Approximations of the Surface Active Quasigeostrophic Equations." Journal of Physical Oceanography 46, no. 1 (January 2016): 125–39. http://dx.doi.org/10.1175/jpo-d-15-0073.1.

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AbstractThis study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have superior accuracy than A at high wavenumbers. Because B lacks the conservation of energy, this study recommends approximation C for constructing solutions to the surface active QG equations using the Galerkin series with standard vertical modes.

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Bazgan, Cristina, Stefan Ruzika, Clemens Thielen, and Daniel Vanderpooten. "The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems." Theory of Computing Systems 66, no. 1 (November 22, 2021): 395–415. http://dx.doi.org/10.1007/s00224-021-10066-5.

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AbstractWe determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

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30

Giorno, V., A. G. Nobile, and L. M. Ricciardi. "On some diffusion approximations to queueing systems." Advances in Applied Probability 18, no. 4 (December 1986): 991–1014. http://dx.doi.org/10.2307/1427259.

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For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

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Giorno, V., A. G. Nobile, and L. M. Ricciardi. "On some diffusion approximations to queueing systems." Advances in Applied Probability 18, no. 04 (December 1986): 991–1014. http://dx.doi.org/10.1017/s0001867800016244.

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For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

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32

Friedrich, Tobias, Jun He, Nils Hebbinghaus, Frank Neumann, and Carsten Witt. "Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models." Evolutionary Computation 18, no. 4 (December 2010): 617–33. http://dx.doi.org/10.1162/evco_a_00003.

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The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical explorations on this subject. We consider the approximation ability of randomized search heuristics for the class of covering problems and compare single-objective and multi-objective models for such problems. For the VertexCover problem, we point out situations where the multi-objective model leads to a fast construction of optimal solutions while in the single-objective case, no good approximation can be achieved within the expected polynomial time. Examining the more general SetCover problem, we show that optimal solutions can be approximated within a logarithmic factor of the size of the ground set, using the multi-objective approach, while the approximation quality obtainable by the single-objective approach in expected polynomial time may be arbitrarily bad.

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Honet, Antoine, Luc Henrard, and Vincent Meunier. "Exact and many-body perturbation solutions of the Hubbard model applied to linear chains." AIP Advances 12, no. 3 (March 1, 2022): 035238. http://dx.doi.org/10.1063/5.0082681.

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This study reports on the accuracy of the GW approximation for the treatment of the Hubbard model compared to exact diagonalization (ED) results. We consider not only global quantities, such as the total energy and the density of states, but also the spatial and spin symmetry of wavefunctions via the analysis of the local density of states. GW is part of the more general Green’s function approach used to develop many-body approximations. We show that, for small linear chains, the GW approximation corrects the mean-field (MF) approach by reducing the total energy and the magnetization obtained from the MF approximation. The GW energy gap is in better agreement with ED, especially in systems of an even number of atoms where, in contrast to the MF approximation, no plateau is observed below the artificial predicted phase transition. In terms of density of states, the GW approximation induces quasi-particles and side satellite peaks via a splitting process of MF peaks. At the same time, GW slightly modifies the localization (e.g., edges or center) of states. We also use the GW approximation results in the context of Löwdin’s symmetry dilemma and show that GW predicts an artificial paramagnetic–antiferromagnetic phase transition at a higher Hubbard parameter than MF does.

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34

Nguyen, Van-Luat. "Solutions for elastic moduli of three-phase composite with random distribution of coated-ellipse inclusions." Functional Composites and Structures 4, no. 4 (October 31, 2022): 045003. http://dx.doi.org/10.1088/2631-6331/ac9c42.

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Abstract Some solutions in this work are developed to estimate the elastic moduli of three-phase isotropic composite with random coated-ellipse inclusion in the matrix. Solutions to the macro-elastic moduli of materials in two-dimensional space using approximation and numerical methods including equivalent-inclusion (EI), polarization approximation (PA), differential approximations (DA), and fast Fourier transformation (FFT). In which, there is a combination of those methods to give approximations such as EI-PA, EI-DA, FFT-EI. The construction algebraic expressions can be directly applied to the random coated-ellipse model, in special cases it can be used for circular aggregate particles. The numerical solutions using FFT analysis will be compared with EI-PA, EI-DA, and Hashin–Shtrikman’s bounds. From this, it is possible to indicate the best solution that engineers can use to determine the elastic modulus of the coated-ellipse model.

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Zlatanovska, Biljana, and Donc̆o Dimovski. "A modified Lorenz system: Definition and solution." Asian-European Journal of Mathematics 13, no. 08 (May 20, 2020): 2050164. http://dx.doi.org/10.1142/s1793557120501648.

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Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII 1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33].

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36

Rozmej, Piotr, and Anna Karczewska. "New Exact Superposition Solutions to KdV2 Equation." Advances in Mathematical Physics 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/5095482.

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New exact solutions to the KdV2 equation (also known as the extended KdV equation) are constructed. The KdV2 equation is a second-order approximation of the set of Boussinesq’s equations for shallow water waves which in first-order approximation yields KdV. The exact solutionsA/2dn2[B(x-vt),m]±m cn[B(x-vt),m]dn[B(x-vt),m]+Din the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, that is, the solitonic ones and periodic ones given by singlecn2ordn2Jacobi elliptic functions.

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37

Meftah, S., and B. Ghanmi. "APPROXIMATE SOLUTIONS OF WEAKLY NONLINEAR DIFFERENTIAL EQUATIONS." Advances in Mathematics: Scientific Journal 11, no. 7 (July 5, 2022): 591–600. http://dx.doi.org/10.37418/amsj.11.7.3.

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In this work, we study the most useful approximation methods for proving solutions to analytic approximation for solving a weak second-order nonlinear differential equation in a power series with small parameters. We prove the second-order periodic approximation solution and also the best third-order approximation of the weak nonlinear differential equation.

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38

Nguyen, Van-Luat. "FFT, DA, and Mori-Tanaka approximation to determine the elastic moduli of three-phase composites with the random inclusions." EPJ Applied Metamaterials 9 (2022): 9. http://dx.doi.org/10.1051/epjam/2022007.

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In this work, some solutions such as Mori-Tanaka approximation (MTA), Differential approximations (DA), and Fast Fourier transformation method (FFT) were applied to estimate the elastic bulk and shear modulus of three-phase composites in 2D. In which two different sizes of circular inclusions are arranged randomly non-overlapping in a continuous matrix. The numerical solutions using FFT analysis were compared with DA, MTA, and Hashin-Strikman's bounds. The MTA and DA reasonably agreeable solution with the FFT solution shows the effectiveness of the approximation methods, which makes MTA, DA useful with simplicity and ease of application.

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39

Segalman, Daniel J., Michael J. Starr, and Martin W. Heinstein. "New Approximations for Elastic Spheres Under an Oscillating Torsional Couple." Journal of Applied Mechanics 72, no. 5 (December 2, 2004): 705–10. http://dx.doi.org/10.1115/1.1985430.

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The Lubkin solution for two spheres pressed together and then subjected to a monotonically increasing axial couple is examined numerically. The Deresiewicz asymptotic solution is compared to the full solution and its utility is evaluated. Alternative approximations for the Lubkin solution are suggested and compared. One approximation is a Padé rational function which matches the analytic solution over all rotations. The other is an exponential approximation that reproduces the asymptotic values of the analytic solution at infinitesimal and infinite rotations. Finally, finite element solutions for the Lubkin problem are compared with the exact and approximate solutions.

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40

Sabatier, Jocelyn. "Solutions to the Sub-Optimality and Stability Issues of Recursive Pole and Zero Distribution Algorithms for the Approximation of Fractional Order Models." Algorithms 11, no. 7 (July 12, 2018): 103. http://dx.doi.org/10.3390/a11070103.

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This paper analyses algorithms currently found in the literature for the approximation of fractional order models and based on recursive pole and zero distributions. The analysis focuses on the sub-optimality of the approximations obtained and stability issues that may appear after approximation depending on the pole location of the initial fractional order model. Solutions are proposed to reduce this sub-optimality and to avoid stability issues.

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41

Al-Nuaimy, Adawiyah, and Tariq Abdul-Razaq. "Approximation Solutions For Multicriteria Scheduling Problems." Journal of Al-Rafidain University College For Sciences ( Print ISSN: 1681-6870 ,Online ISSN: 2790-2293 ), no. 2 (October 15, 2021): 161–79. http://dx.doi.org/10.55562/jrucs.v34i2.288.

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This paper presents local search algorithms for finding approximation solutions of the multicriteria scheduling problems within the single machine context, where the first problem is the sum of maximum tardiness and maximum late work and the second problem is the sum of total late work and maximum late work. Late work criterion estimates the quality of a schedule based on durations of late parts of jobs. Local search algorithms (descent method (DM), simulated annealing (SA) and genetic algorithm (GA))are implemented. Based on results of computational experiments, conclusions are formulated on the efficiency of the local search algorithms.

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42

Bubák, Pavel, Cornelis V. M. van der Mee, and André C. M. Ran. "Approximation of Solutions of Riccati Equations." SIAM Journal on Control and Optimization 44, no. 4 (January 2005): 1419–35. http://dx.doi.org/10.1137/s0363012903436843.

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43

Hamann, Uwe, and Günther Wildenhain. "Approximation by Solutions of Elliptic Equations." Zeitschrift für Analysis und ihre Anwendungen 5, no. 1 (1986): 59–69. http://dx.doi.org/10.4171/zaa/180.

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Idrissi, H., P. Loridan, and C. Michelot. "Approximation of solutions for location problems." Journal of Optimization Theory and Applications 56, no. 1 (January 1988): 127–43. http://dx.doi.org/10.1007/bf00938529.

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Miletta, Peter D. "Approximation of solutions to evolution equations." Mathematical Methods in the Applied Sciences 17, no. 10 (August 1994): 753–63. http://dx.doi.org/10.1002/mma.1670171002.

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Broadbridge, Philip, Alexander Kolesnik, Nikolai Leonenko, Andriy Olenko, and Dareen Omari. "Spherically Restricted Random Hyperbolic Diffusion." Entropy 22, no. 2 (February 14, 2020): 217. http://dx.doi.org/10.3390/e22020217.

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This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short- or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.

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Hong, Bum Il. "Regularity for Hamilton-Jacobi equations via approximation." Bulletin of the Australian Mathematical Society 51, no. 2 (April 1995): 195–213. http://dx.doi.org/10.1017/s0004972700014052.

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We prove new regularity results for solutions of first-order partial differential equations of Hamilton-Jacobi type posed as initial value problems on the real line. We show that certain spaces determined by quasinorms related to the solution's approximation properties in C(ℝ) by continuous, piecewise quadratic polynomial functions are invariant under the action of the differential equation. As a result, we show that solutions of Hamilton-Jacobi equations have enough regularity to be approximated well in C(ℝ) by moving-grid finite element methods. The preceding results depend on a new stability theorem for Hamilton-Jacobi equations in any number of spatial dimensions.

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CHEN, ZHAO-YOU, MIN LI, and CHUN-SHENG JIA. "APPROXIMATE ANALYTICAL SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH THE MANNING–ROSEN POTENTIAL MODEL." Modern Physics Letters A 24, no. 23 (July 30, 2009): 1863–74. http://dx.doi.org/10.1142/s0217732309030345.

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By approximating the centrifugal term in terms of a new approximation scheme, we solve the Schrödinger equation with the arbitrary angular momentum for the Manning–Rosen potential. The bound state energy eigenvalues and the unnormalized radial wave functions have been approximately obtained by using the supersymmetric shape invariance approach and the function analysis method. The numerical experiments show that our approximate analytical results are in better agreement with those obtained by using the numerical integration procedure than the analytical results obtained by using the conventional approximation scheme to deal with the centrifugal term.

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49

Huang, Weijun. "Solving Coplanar Power-Limited Orbit Transfer Problem by Primer Vector Approximation Method." International Journal of Aerospace Engineering 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/480320.

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The coplanar orbit transfer problem has been an important topic in astrodynamics since the beginning of the space era. Though many approximate solutions for power-limited orbit transfer problem have been developed, most of them rely on simplifications of the dynamics of the problem. This paper proposes a new approximation method called primer vector approximation method to solve the classic power-limited orbit transfer problem. This method makes no simplification on the dynamics, but instead approximates the optimal primer-vector function. With this method, this paper derives two approximate solutions for the power-limited orbit transfer problem. Numerical tests show the robustness and accuracy of the approximations.

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Co’, Giampaolo, and Stefano De Leo. "Hartree–Fock and random phase approximation theories in a many-fermion solvable model." Modern Physics Letters A 30, no. 36 (November 3, 2015): 1550196. http://dx.doi.org/10.1142/s0217732315501965.

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We present an ideal system of interacting fermions where the solutions of the many-body Schrödinger equation can be obtained without making approximations. These exact solutions are used to test the validity of two many-body effective approaches, the Hartree–Fock and the random phase approximation theories. The description of the ground state done by the effective theories improves with increasing number of particles.

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