Relevant bibliographies by topics / Approximation of solutions

Academic literature on the topic 'Approximation of solutions'

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Published: 14 March 2023

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Journal articles on the topic "Approximation of solutions"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Approximation of solutions"

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Morini, Massimiliano. "Free-discontinuity problems: calibration and approximation of solutions." Doctoral thesis, SISSA, 2001. http://hdl.handle.net/20.500.11767/3923.

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Tarkhanov, Nikolai. "Unitary solutions of partial differential equations." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2985/.

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Khan, Rahmat Ali. "Existence and approximation of solutions of nonlinear boundary value problems." Thesis, University of Glasgow, 2005. http://theses.gla.ac.uk/4037/.

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In chapter two, we establish new results for periodic solutions of some second order non-linear boundary value problems. We develop the upper and lower solutions method to show existence of solutions in the closed set defined by the well ordered lower and upper solutions. We develop the method of quasilinearization to approximate our problem by a sequence of solutions of linear problems that converges to the solution of the original problem quadratically. Finally, to show the applicability of our technique, we apply the theoretical results to a medical problem namely, a biomathematical model of blood flow in an intracranial aneurysm. In chapter three we study some nonlinear boundary value problems with nonlinear nonlocal three-point boundary conditions. We develop the method of upper and lower solutions to establish existence results. We show that our results hold for a wide range of nonlinear problems. We develop the method of quasilinearization and show that there exist monotone sequences of solutions of linear problems that converges to the unique solution of the nonlinear problems. We show that the sequences converge quadratically to the solutions of the problem in the C1 norm. We generalize the technique by introducing an auxiliary function to allow weaker hypotheses on the nonlinearity involved in the differential equations. In chapter four, we extend the results of chapter three to nonlinear problems with linear four point boundary conditions. We generalize previously existence results studied with constant lower and upper solutions. We show by an example that our results are more general. We develop the method of quasilinearization and its generalization for the four point problems which to the best of our knowledge is the first time the method has been applied to such problems. In chapter five, we extend the results to second order problems with nonlinear integral boundary conditions in two separate cases. In the first case we study the upper and lower solutions method and the generalized method of quasilinearization for the Integral boundary value problem with the nonlinearity independent of the derivative. While in the second case we show the nonlinearity to depend also on the first derivative. Finally, in chapter six, we study multiplicity results for three point nonlinear boundary value problems. We use the method of upper and lower solutions and degree arguments to show the existence of at least two solutions for certain range of a parameter r and no solution for other range of the parameter. We show by an example that our results are more general than the results studied previously. We also study existence of at least three solutions in the pressure of two lower and two upper solutions for some three-point boundary value problems. In one problem, we employ a condition weaker than the well known Nagumo condition which allows the nonlinearity f(t, x, x’) to grow faster than quadratically with respect to x’ in some cases.
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Chidume, Chukwudi Soares de Souza Geraldo. "Iteration methods for approximation of solutions of nonlinear equations in Banach spaces." Auburn, Ala., 2008. http://repo.lib.auburn.edu/EtdRoot/2008/SUMMER/Mathematics_and_Statistics/Dissertation/Chidume_Chukwudi_33.pdf.

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Rouy, Elisabeth. "Approximation numérique des solutions de viscosité des équations d'Hamilton-Jacobi et exemple." Paris 9, 1992. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=1992PA090010.

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Cette thèse concerne l'approximation numérique des solutions de viscosité, telles qu'elles ont été définies par Michael Grain Crandall et Pierre-Louis Lions, des équations Hamilton-Jacobi du premier ordre qui sont des équations aux dérivées partielles non linéaires, ainsi que l'étude d'un exemple issu du traitement d'images, le shape-from-shading, qui consiste en la reconstruction d'un relief à trois dimensions à partir de la donnée d'une image en deux dimensions, d'une photographie par exemple. Le premier chapitre est une présentation succincte des solutions de viscosité des équations d’Hamilton-Jacobi et de quelques résultats d'existence et d'unicité. Le second chapitre décrit les différentes méthodes développées pour approcher ces solutions, et relève de l'analyse numérique. Le troisième chapitre, plus appliqué, a pour but d'expliquer comment, concrètement, on peut écrire un schéma d'approximation des solutions de viscosité. Enfin, l'exemple est étudié de façon précise (en reprenant les différents développements des premiers chapitres de la thèse): on montre comment le relief peut être interprété comme la solution de viscosité d'une équation d’Hamilton-Jacobi; on étudie les différentes formalisations possibles pour les bords de l'image afin de parvenir à des résultats d'existence et d'unicité satisfaisants. Puis un schéma est construit et applique à la reconstruction numérique de différentes images
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Badra, Mehdi. "Stabilisation par feedback et approximation des équations de Navier-Stokes." Toulouse 3, 2006. http://www.theses.fr/2006TOU30242.

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Cette thèse est consacrée à l'étude de problèmes de stabilisation par retour d'état ou "feedback" des équations de Navier-Stokes autour d'une solution stationnaire instable. Les cas d'un contrôle correspondant à une force exercée dans une partie du domaine et celui d'un contrôle localisé sur la frontière du domaine sont considérés. Le contrôle s'exprime en fonction du champ de vitesse à l'aide d'une loi de feedback linéaire. Celle-ci est fournie par la solution d'une équation de Riccati algébrique, dont la dérivation fait appel à la théorie du contrôle optimal. La question de l'approximation de ces problèmes de contrôle est aussi considérée. Nous nous intéressons d'abord à l'approximation générale du système de Navier-Stokes linéarisé autour d'un état stationnaire (système de Oseen). Nous donnons des estimations d'erreur dans le cas d'une condition de Dirichlet peu régulière et dans le cas d'une condition de divergence peu régulière. Le cas particulier d'un approximation de Galerkin est alors traité. Puis nous montrons un théorème général pour l'approximation non conforme des systèmes linéaires contrôlés obtenus à l'aide de l'opérateur de Riccati. Ce théorème est ensuite appliqué dans le cas du système de Oseen soumis à un contrôle feedback distribué et dans le cas du système de Oseen soumis à un contrôle feedback frontière
This thesis deals with some feedback stabilization problems for the Navier-Stokes equations around an unstable stationary solution. The case of a distributed control localized in a part of the geomatrical domain and the case of a boundary control are considered. The control is expressed in function of the velocity field by a linear feedback law. The feedback law is provided by an algebraic Riccati equation which is obtained with the tools of the optimal control theory. The question of approximating such controlled systems is also considered. We first study the approximation of the linearized Navier-Stokes equations (the so-called Oseen equations) for rough boundary and divergence data. General error estimates are given and Galerkin methods are investigated. We also prove a general nonconform approximation theorem for closed-loop systems obtained from the Riccati theory. We apply this theorem to study the approximation of the Oseen closed-loop system
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Hugot, Hadrien. "Approximation et énumération des solutions efficaces dans les problèmes d'optimisation combinatoire multi-objectif." Paris 9, 2007. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=2007PA090028.

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Cette thèse porte sur la résolution de problèmes d'optimisation combinatoire multi-objectif. La résolution de ces problèmes passe par la détermination de l'ensemble des solutions efficaces. Cependant, il peut s'avérer que le nombre de solutions efficaces soit très grand. Approcher l'ensemble des solutions efficaces d'un tel problème constitue, dès lors, un sujet de recherche central dans ce domaine. Les approches existantes sont souvent basées sur des méthodes approchées, de type (méta-)heuristiques, donc sans garantie sur la qualité des solutions trouvées. Des algorithmes d'approximation (à garantie de performance) ont aussi été développés pour certains problèmes, sans toutefois avoir été conçus en vue d'une mise en œuvre pratique. Dans cette thèse, nous nous sommes attachés à concevoir des approches visant à concilier à la fois les qualités des méthodes approchées et celles des méthodes d'approximation. Pour ce faire, nous proposons, dans un contexte général où les solutions sont comparées à l'aide d'une relation de préférence pouvant être non-transitive, un cadre de Programmation Dynamique Généralisée (PDG). Ce cadre est basé sur une extension du concept de relations de dominance utilisées dans la PD. Il permet, notamment, de concevoir des méthodes exactes et d'approximation qui se sont avérées particulièrement efficaces en pratique pour résoudre le problème du sac-à-dos multi-objectif 0-1. Enfin, une dernière partie de notre travail a porté sur l'apport d'une modélisation multicritère pour résoudre, dans un contexte réel, le problème d'association de données. Ceci nous a conduits à nous intéresser au problème d'affectation multi-objectif et à sa résolution au sein de notre cadre de PDG
This thesis deals with the resolution of multi-objective combinatorial optimization problems. A first step in the resolution of these problems consists in determining the set of efficient solutions. Nevertheless, the number of efficient solutions can be very huge. Approximating the set of efficient solutions for these problems constitutes, then, a major challenge. Existing methods are usually based on approximate methods, such as heuristics or meta-heuristics, that give no guarantee on the quality of the computed solutions. Alternatively, approximation algorithms (with provable guarantee) have been also designed. However, practical implementations of approximation algorithms are cruelly lacking and most of the approximation algorithms proposed in the literature are not efficient in practice. This thesis aims at designing approaches that conciliate on the one hand the qualities of the approximate approaches and on the other hand those of the approximation approaches. We propose, in a general context, where the preference relation used to compare solutions is not necessarily transitive, a Generalized Dynamic Programming (GDP) framework. GDP relies on an extension of the concept of dominance relations. It provides us, in particular, with exact and approximation methods that have been proved to be particularly efficient in practice to solve the 0-1 multi-objective knapsack problem. Finally, a last part of our work deals with the contributions of a multi-criteria modelling for solving, in real context, the data association problem. This led us to study the multi-objective assignment problem and, in particular, the resolution of this problem by the means of our GDP framework
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Milišić, Vuk. "Approximation cinétique discrète de problèmes de lois de conservation avec bord." Bordeaux 1, 2001. http://www.theses.fr/2001BOR12449.

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Nous étudions l'approximation cinétique discrète de lois de conservation scalaires quasi-linéaires dans le quart d'espace positif. Cette approximation est obtenue par l'introduction de systèmes de type BGK relaxant la loi scalaire. Nous démontrons la convergence des systèmes semi-linéaires vers la loi scalaire. Nous discrétisons ces modèles pour obtenir une gamme de schémas numériques adaptés au problème avec bord. Dans une troisième partie, nous appliquons ces schémas à un certain nombre de cas test numériques.
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Bouhar, Mustapha. "Comportement limite de solutions d'équations quasi-linéaires dans des cylindres infinis." Tours, 1991. http://www.theses.fr/1991TOUR4002.

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On étudie le comportement asymptotique des solutions d'équations elliptiques quasi-linéaires avec un potentiel singulier dans un domaine au voisinage de zéro, au voisinage de l'infini du plan. L'étude de cette équation nous permet d'obtenir un certain nombre de configurations semblables à celles obtenues pour les solutions positives de la même équation sans potentiel dans un domaine borné et ouvert en dimension supérieure par P. L. Lions et Richard-Veron pour le cas sous-critique, P. Avilés pour le cas critique et Gidas-Spruck pour le cas sur-critique et dans le cas radial par M. F. Bidaut-Veron. A l'aide d'un changement de variable de type logarithmique, notre équation se transforme en une équation autonome. Notre travail est le suivant: a) étude de solutions radiales; étude complète de l'ensemble des solutions du problème stationnaire dans un cercle avec une quantification de leurs énergies et obtention des représentations exactes de ces solutions; c) obtention d'une borne à priori pour les solutions positives; Résultats de convergence quand t tend vers l'infini; e) étude des solutions globales; f) étude du problème dans des cylindres avec bord
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Yevik, Andrei. "Numerical approximations to the stationary solutions of stochastic differential equations." Thesis, Loughborough University, 2011. https://dspace.lboro.ac.uk/2134/7777.

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This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.
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Books on the topic "Approximation of solutions"

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Bustamante, Jorge. Algebraic Approximation: A Guide to Past and Current Solutions. Basel: Springer Basel AG, 2012.

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Funaro, Daniele. Polynomial approximation of differential equations. Berlin: Springer-Verlag, 1992.

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Bent, Fuglede, North Atlantic Treaty Organization. Scientific Affairs Division., and NATO Advanced Research Workshop on Approximation by Solutions of Partial Differential Equations, Quadrature Formulae, and Related Topics (1991 : Hanstholm, Denmark), eds. Approximation by solutions of partial differential equations. Dordrecht: Kluwer Academic Publishers, 1992.

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Fuglede, B., M. Goldstein, W. Haussmann, W. K. Hayman, and L. Rogge, eds. Approximation by Solutions of Partial Differential Equations. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2436-2.

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Polynomial approximation of differential equations. Berlin: Springer-Verlag, 1992.

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Quarteroni, Alfio. Numerical approximation of partial differential equations. 2nd ed. Berlin: Springer, 1997.

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1953-, Valli A., ed. Numerical approximation of partial differential equations. Berlin: Springer-Verlag, 1994.

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Chen, Mingjun. Approximate solutions of operator equations. Singapore: World Scientific, 1997.

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Burstein, Joseph. Approximation by exponentials, their extensions & differential equations. Boston: Metrics Press, 1997.

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Křížek, M. Finite element approximation of variational problems and applications. Harlow, Essex: Longman Scientific & Technical, 1990.

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Book chapters on the topic "Approximation of solutions"

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Gauthier, P. M., J. Heinonen, and D. Zwick. "Axiomatic Approximation." In Approximation by Solutions of Partial Differential Equations, 79–85. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2436-2_8.

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Tarkhanov, Nikolai N. "Uniform Approximation." In The Analysis of Solutions of Elliptic Equations, 191–270. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8804-1_5.

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Tarkhanov, Nikolai N. "Mean Approximation." In The Analysis of Solutions of Elliptic Equations, 271–318. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8804-1_6.

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Tarkhanov, Nikolai N. "BMO Approximation." In The Analysis of Solutions of Elliptic Equations, 319–44. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8804-1_7.

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Shakarchi, Rami. "Approximation with Convolutions." In Problems and Solutions for Undergraduate Analysis, 183–87. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1738-1_12.

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Deutsch, Frank. "Generalized Solutions of Linear Equations." In Best Approximation in Inner Product Spaces, 155–92. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4684-9298-9_8.

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Sun, Shu-Ming, Ning Zhong, and Martin Ziegler. "Computability of the Solutions to Navier-Stokes Equations via Effective Approximation." In Complexity and Approximation, 80–112. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-41672-0_7.

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Bardi, Martino, and Italo Capuzzo-Dolcetta. "Approximation and perturbation problems." In Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, 359–96. Boston, MA: Birkhäuser Boston, 1997. http://dx.doi.org/10.1007/978-0-8176-4755-1_6.

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Frontini, M., G. Rodriguez, and S. Seatzu. "An algorithm for computing minimum norm solutions of finite moment problem." In Algorithms for Approximation II, 361–68. Boston, MA: Springer US, 1990. http://dx.doi.org/10.1007/978-1-4899-3442-0_31.

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Bagby, T., and P. M. Gauthier. "Uniform Approximation by Global Harmonic Functions." In Approximation by Solutions of Partial Differential Equations, 15–26. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2436-2_3.

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Conference papers on the topic "Approximation of solutions"

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van der Herten, Joachim, Dirk Deschrijver, and Tom Dhaene. "Fuzzy local linear approximation-based sequential design." In 2014 IEEE Symposium on Computational Intelligence for Engineering Solutions (CIES). IEEE, 2014. http://dx.doi.org/10.1109/cies.2014.7011825.

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Elizalde-Blancas, Francisco, and Ismail B. Celik. "On the Representation of Numerical Solutions Using Taylor Series Approximation." In ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-31247.

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In this study, the representation of discretization error using Taylor series in finite difference solutions is investigated as well as the behavior of the exact solutions to the finite difference equations as a function of the grid size and grid refinement factor. The results are compared to the classical Richardson Extrapolation method whereby the numerical solution (or the error) is explicitly expressed as a Taylor series expansion. The exact finite difference solutions are used to demonstrate that oscillatory convergence is a common occurrence. The expansion of the numerical solutions in Taylor series is based on the exact finite difference solutions that are obtained using different discretization schemes. It is shown that in some cases the numerical solution exhibited a singular behavior which can not be remedied easily. Some exact finite difference solutions also exhibited oscillatory behavior which was not due to the use of mixed order terms as is usually believed by the Computational Fluid Dynamics community. Moreover, representation of the numerical solution using Taylor series is not always satisfactory even in case of relatively simple one-dimensional problems.
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Peng, Ya-Xin, Xi-Yan Hu, and Lei Zhang. "An Iterative Method for Bisymmetric Solutions and Optimal Approximation Solution of AXB=C." In Third International Conference on Natural Computation (ICNC 2007). IEEE, 2007. http://dx.doi.org/10.1109/icnc.2007.231.

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El-Shafei, A. "Modeling Finite Squeeze Film Dampers." In ASME Turbo Expo 2002: Power for Land, Sea, and Air. ASMEDC, 2002. http://dx.doi.org/10.1115/gt2002-30637.

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Most closed form solutions of Reynolds’ equation assume either a short bearing approximation or a long bearing approximation. These closed form approximations are used in rotordynamic simulation applications, otherwise a Finite Difference solution of Reynolds’ equation would be prohibitively time consuming. Recently, there have been proposed series solutions for Reynolds’ equation for special cases. In this paper, a perturbation solution to the governing equations is proposed to obtain a closed form solution of Reynolds’ equation for a finite squeeze film damper executing a circular centered orbit. The pressure field and velocity profiles are obtained. It is shown that in the limit the finite damper solution approaches either the appropriate short or long damper. This perturbation solution can be used with appropriate boundary conditions, for various damper sealing configurations, and provides insight into the damper performance.
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Jank, Gerhard, and Gábor Kun. "Solutions of generalized Riccati differential equations and their approximation." In Third CMFT Conference. WORLD SCIENTIFIC, 1999. http://dx.doi.org/10.1142/9789812833044_0022.

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Dobkevich, Mariya, Felix Sadyrbaev, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "Types of solutions and approximation of solutions of second order nonlinear boundary value problems." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS: International Conference on Numerical Analysis and Applied Mathematics 2009: Volume 1 and Volume 2. AIP, 2009. http://dx.doi.org/10.1063/1.3241443.

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Allphin, Devin, and Joshua Hamel. "A Parallel Offline CFD and Closed-Form Approximation Strategy for Computationally Efficient Analysis of Complex Fluid Flows." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-38691.

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Computational fluid dynamics (CFD) solution approximations for complex fluid flow problems have become a common and powerful engineering analysis technique. These tools, though qualitatively useful, remain limited in practice by their underlying inverse relationship between simulation accuracy and overall computational expense. While a great volume of research has focused on remedying these issues inherent to CFD, one traditionally overlooked area of resource reduction for engineering analysis concerns the basic definition and determination of functional relationships for the studied fluid flow variables. This artificial relationship-building technique, called meta-modeling or surrogate/offline approximation, uses design of experiments (DOE) theory to efficiently approximate non-physical coupling between the variables of interest in a fluid flow analysis problem. By mathematically approximating these variables, DOE methods can effectively reduce the required quantity of CFD simulations, freeing computational resources for other analytical focuses. An idealized interpretation of a fluid flow problem can also be employed to create suitably accurate approximations of fluid flow variables for the purposes of engineering analysis. When used in parallel with a meta-modeling approximation, a closed-form approximation can provide useful feedback concerning proper construction, suitability, or even necessity of an offline approximation tool. It also provides a short-circuit pathway for further reducing the overall computational demands of a fluid flow analysis, again freeing resources for otherwise unsuitable resource expenditures. To validate these inferences, a design optimization problem was presented requiring the inexpensive estimation of aerodynamic forces applied to a valve operating on a simulated piston-cylinder heat engine. The determination of these forces was to be found using parallel surrogate and exact approximation methods, thus evidencing the comparative benefits of this technique. For the offline approximation, latin hypercube sampling (LHS) was used for design space filling across four independent design variable degrees of freedom (DOF). Flow solutions at the mapped test sites were converged using STAR-CCM+ with aerodynamic forces from the CFD models then functionally approximated using Kriging interpolation. For the closed-form approximation, the problem was interpreted as an ideal 2-D converging-diverging (C-D) nozzle, where aerodynamic forces were directly mapped by application of the Euler equation solutions for isentropic compression/expansion. A cost-weighting procedure was finally established for creating model-selective discretionary logic, with a synthesized parallel simulation resource summary provided.
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Djeridane, Badis, and John Lygeros. "Neural approximation of PDE solutions: An application to reachability computations." In Proceedings of the 45th IEEE Conference on Decision and Control. IEEE, 2006. http://dx.doi.org/10.1109/cdc.2006.377184.

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Bergmann, Ronny, and Dennis Merkert. "Approximation of periodic PDE solutions with anisotropic translation invariant spaces." In 2017 International Conference on Sampling Theory and Applications (SampTA). IEEE, 2017. http://dx.doi.org/10.1109/sampta.2017.8024347.

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Dong, Liang. "Analytical solutions for nonlinear waveguide equation under Gaussian mode approximation." In Lasers and Applications in Science and Engineering, edited by Jes Broeng and Clifford Headley III. SPIE, 2008. http://dx.doi.org/10.1117/12.774052.

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Reports on the topic "Approximation of solutions"

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Herzog, K. J., M. D. Morris, and T. J. Mitchell. Bayesian approximation of solutions to linear ordinary differential equations. Office of Scientific and Technical Information (OSTI), November 1990. http://dx.doi.org/10.2172/6242347.

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Kamai, Tamir, Gerard Kluitenberg, and Alon Ben-Gal. Development of heat-pulse sensors for measuring fluxes of water and solutes under the root zone. United States Department of Agriculture, January 2016. http://dx.doi.org/10.32747/2016.7604288.bard.

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The objectives defined for this study were to: (1) develop a heat-pulse sensor and a heat-transfer model for leaching measurement, and (2) conduct laboratory study of the sensor and the methodology to estimate leaching flux. In this study we investigated the feasibility for estimating leachate fluxes with a newly designed heat-pulse (HP) sensor, combining water flux density (WFD) with electrical conductivity (EC) measurements in the same sensor. Whereas previous studies used the conventional heat pulse sensor for these measurements, the focus here was to estimate WFD with a robust sensor, appropriate for field settings, having thick-walled large-diameter probes that would minimize their flexing during and after installation and reduce associated errors. The HP method for measuring WFD in one dimension is based on a three-rod arrangement, aligned in the direction of the flow (vertical for leaching). A heat pulse is released from a center rod and the temperature response is monitored with upstream (US) and downstream (DS) rods. Water moving through the soil caries heat with it, causing differences in temperature response at the US and DS locations. Appropriate theory (e.g., Ren et al., 2000) is then used to determine WFD from the differences in temperature response. In this study, we have constructed sensors with large probes and developed numerical and analytical solutions for approximating the measurement. One-dimensional flow experiments were conducted with WFD ranging between 50 and 700 cm per day. A numerical model was developed to mimic the measurements, and also served for the evaluation of the analytical solution. For estimation WFD, and analytical model was developed to approximate heat transfer in this setting. The analytical solution was based on the work of Knight et al. (2012) and Knight et al. (2016), which suggests that the finite properties of the rods can be captured to a large extent by assuming them to be cylindrical perfect conductors. We found that: (1) the sensor is sensitive for measuring WFD in the investigated range, (2) the numerical model well-represents the sensor measurement, and (2) the analytical approximation could be improved by accounting for water and heat flow divergence by the large rods.
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Hart, Carl, and Gregory Lyons. A tutorial on the rapid distortion theory model for unidirectional, plane shearing of homogeneous turbulence. Engineer Research and Development Center (U.S.), July 2022. http://dx.doi.org/10.21079/11681/44766.

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The theory of near-surface atmospheric wind noise is largely predicated on assuming turbulence is homogeneous and isotropic. For high turbulent wavenumbers, this is a fairly reasonable approximation, though it can introduce non-negligible errors in shear flows. Recent near-surface measurements of atmospheric turbulence suggest that anisotropic turbulence can be adequately modeled by rapid-distortion theory (RDT), which can serve as a natural extension of wind noise theory. Here, a solution for the RDT equations of unidirectional plane shearing of homogeneous turbulence is reproduced. It is assumed that the time-varying velocity spectral tensor can be made stationary by substituting an eddy-lifetime parameter in place of time. General and particular RDT evolution equations for stochastic increments are derived in detail. Analytical solutions for the RDT evolution equation, with and without an effective eddy viscosity, are given. An alternative expression for the eddy-lifetime parameter is shown. The turbulence kinetic energy budget is examined for RDT. Predictions by RDT are shown for velocity (co)variances, one-dimensional streamwise spectra, length scales, and the second invariant of the anisotropy tensor of the moments of velocity. The RDT prediction of the second invariant for the velocity anisotropy tensor is shown to agree better with direct numerical simulations than previously reported.
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Gilsinn, David E. Approximating periodic solutions of autonomous delay differential equations. Gaithersburg, MD: National Institute of Standards and Technology, 2006. http://dx.doi.org/10.6028/nist.ir.7375.

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Campbell, Stephen L. Distributional Convergence of BDF (Backward Differentiation Formulas) Approximations to Solutions of Descriptor Systems. Fort Belvoir, VA: Defense Technical Information Center, November 1987. http://dx.doi.org/10.21236/ada190819.

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Eggertsson, Gauti, and Sanjay Singh. Log-linear Approximation versus an Exact Solution at the ZLB in the New Keynesian Model. Cambridge, MA: National Bureau of Economic Research, October 2016. http://dx.doi.org/10.3386/w22784.

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Domich, P. D. A near-optimal starting solution for polynomial approximation of a continuous function in the L₁ norm. Gaithersburg, MD: National Bureau of Standards, 1986. http://dx.doi.org/10.6028/nbs.ir.86-3389.

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Rojas-Bernal, Alejandro, and Mauricio Villamizar-Villegas. Pricing the exotic: Path-dependent American options with stochastic barriers. Banco de la República de Colombia, March 2021. http://dx.doi.org/10.32468/be.1156.

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We develop a novel pricing strategy that approximates the value of an American option with exotic features through a portfolio of European options with different maturities. Among our findings, we show that: (i) our model is numerically robust in pricing plain vanilla American options; (ii) the model matches observed bids and premiums of multidimensional options that integrate Ratchet, Asian, and Barrier characteristics; and (iii) our closed-form approximation allows for an analytical solution of the option’s greeks, which characterize the sensitivity to various risk factors. Finally, we highlight that our estimation requires less than 1% of the computational time compared to other standard methods, such as Monte Carlo simulations.
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Tal-Ezer, Hillel. Polynominal Approximation of Functions of Matrices and Its Application the the Solution of a General System of Linear Equations. Fort Belvoir, VA: Defense Technical Information Center, August 1987. http://dx.doi.org/10.21236/ada211390.

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Trenchea, Catalin. Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, February 2012. http://dx.doi.org/10.21236/ada567709.

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