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Dissertations / Theses on the topic 'Dirichlet solution'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Unsurangsie, Sumalee. "Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem." Thesis, North Texas State University, 1988. https://digital.library.unt.edu/ark:/67531/metadc331780/.

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In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.

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2

Neuberger, John M. (John Michael). "Existence of a Sign-Changing Solution to a Superlinear Dirichlet Problem." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc278179/.

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We study the existence, multiplicity, and nodal structure of solutions to a superlinear elliptic boundary value problem. Under specific hypotheses on the superlinearity, we show that there exist at least three nontrivial solutions. A pair of solutions are of one sign (positive and negative respectively), and the third solution changes sign exactly once. Our technique is variational, i.e., we study the critical points of the associated action functional to find solutions. First, we define a codimension 1 submanifold of a Sobolev space . This submanifold contains all weak solutions to our proble

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3

Samad, Mostafa. "Approximation de la solution du problème de Dirichlet relatif... dans un domaine extérieur (Nantes, 1986)." Grenoble 2 : ANRT, 1986. http://catalogue.bnf.fr/ark:/12148/cb37601030j.

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4

Aibeche, Aïssa. "Quelques problèmes non linéaires dans des domaines à frontière polygonale, comportement singulier de la solution." Nice, 1985. http://www.theses.fr/1985NICE4052.

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5

Melin, Jaron Patric. "Examples of discontinuity for the variational solution of the minimal surface equation with Dirichlet data on a domain with a nonconvex corner and locally negative mean curvature." Thesis, Wichita State University, 2013. http://hdl.handle.net/10057/10639.

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The purpose of this thesis is to investigate the role of smoothness, specifically the smoothness of the boundary ∂Ω, in the behavior of the variational solution f on a domain Ω to the Dirichlet problem for the Minimal Surface Equation at a point O ∈ ∂Ω when the (generalized) curvature of ∂Ω has a negative upper bound in a neighborhood of O. We give examples which show that the assumption of boundary-regularity which Simon made in [12] or at least some weaker boundary-regularity assumption which excludes nonconvex corners in the boundary of the domain is necessary in order to guarantee that the

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6

Eschke, Andy. "Analytical solution of a linear, elliptic, inhomogeneous partial differential equation with inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions for a special rotationally symmetric problem of linear elasticity." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-149965.

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The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary condition

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7

Sargent, Ethan. "Radial Solutions to Semipositone Dirichlet Problems." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/hmc_theses/229.

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We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

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8

Kis, Ludovic. "Perturbations du problème de Dirichlet." Metz, 1998. http://docnum.univ-lorraine.fr/public/UPV-M/Theses/1998/Kis.Ludovic.SMZ9832.pdf.

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Dans toute la suite (a) sera un opérateur elliptique du second ordre, (oméga) un domaine de dimension (n), (lp) l'espace des fonctions ayant la p-ième puissance intégrable, (h) l'espace des fonctions ayant des dérivées jusqu'au premier ordre au carré intégrable, (ho) l'espace des fonctions de (h) s'annulant sur le bord, (wp) l'espace des fonctions ayant des dérivées jusqu'au deuxième ordre avec la p-ième puissance intégrable. On met entre parenthèses un symbole mathématique. Le contenu de ce travail est divisé en trois parties. Dans la première partie on étudie des équations semilinéaires de l

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9

Hassanpour, Mehran. "Sufficient Conditions for Uniqueness of Positive Solutions and Non Existence of Sign Changing Solutions for Elliptic Dirichlet Problems." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc279227/.

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In this paper we study the uniqueness of positive solutions as well as the non existence of sign changing solutions for Dirichlet problems of the form $$\eqalign{\Delta u + g(\lambda,\ u) &= 0\quad\rm in\ \Omega,\cr u &= 0\quad\rm on\ \partial\Omega,}$$where $\Delta$ is the Laplace operator, $\Omega$ is a region in $\IR\sp{N}$, and $\lambda>0$ is a real parameter. For the particular function $g(\lambda,\ u)=\vert u\vert\sp{p}u+\lambda$, where $p={4\over N-2}$, and $\Omega$ is the unit ball in $\IR\sp{N}$ for $N\ge3$, we show that there are no sign changing solutions for small $\lambda$ and als

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10

Ali, Ismail 1961. "Uniqueness of Positive Solutions for Elliptic Dirichlet Problems." Thesis, University of North Texas, 1990. https://digital.library.unt.edu/ark:/67531/metadc330654/.

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In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB, where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order. We also p

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11

Meng, Tan Chee. "Infinitely Many Radial Solutions to a Superlinear Dirichlet Problem." Scholarship @ Claremont, 2007. https://scholarship.claremont.edu/hmc_theses/202.

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My thesis work started in the summer of 2005 as a three way joint project by Professor Castro and Mr. John Kwon and myself. A paper from this joint project was written and the content now forms my thesis.

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12

Kurepa, Alexandra. "Radially Symmetric Solutions to a Superlinear Dirichlet Problem in a Ball." Thesis, North Texas State University, 1987. https://digital.library.unt.edu/ark:/67531/metadc330725/.

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13

SILVA, CLAUSON CARVALHO DA. "STOCHASTIC REPRESENTATION FOR SOLUTIONS OF THE DIRICHLET PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2016. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=27261@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>FUNDAÇÃO DE APOIO À PESQUISA DO ESTADO DO RIO DE JANEIRO<br>PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO<br>BOLSA NOTA 10<br>Como motivação, apresentaremos alguns problemas que ilustram a conexão entre a teoria da probabilidade e algumas equações diferenciais parciais. Suas soluções mesclam os dois assuntos e provocam a suspeita de que alguns processos estocásticos e operadores diferenciais caminham juntos. Em seguida, exibiremos a teoria das difusões de Itô. Mostrar

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14

Srinivasan, Venkatraman. "Analytical solutions for sequentially coupled multi-species reactive transport problems." Auburn, Ala., 2007. http://repo.lib.auburn.edu/07M%20Theses/SRINIVASAN_VENKATRAMAN_13.pdf.

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Thesis (M.S.)--Auburn University, 2007.<br>Abstract. Vita. "This thesis has produced the following three journal publications: 1) V. Srinivasan, T.P. Clement, and K.K. Lee. "Domenico solution -- Is it valid?", Ground Water, 25(2): 136-146, May 2007 ; 2) V. Srinivasan and T.P. Clement. "Analytical solutions for sequentially coupled reactive transport problems. Part I: Mathematical derivations", submitted May 2007, Advances in Water Resources ; 3) V. Srinivasan and T.P. Clement. "Analytical solutions for sequentially coupled reactive transport problems. Part II: Special cases, implementation and

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15

Paneah, Boris, and Bert-Wolfgang Schulze. "On the existence of smooth solutions of the dirichlet problem for hyperbolic : differential equations." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2517/.

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16

Antoniouk, Alexandra, Oleg Kiselev, Vitaly Stepanenko, and Nikolai Tarkhanov. "Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point." Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/6198/.

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The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a for

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17

Schreffler, Morgan F. "Approximation of Solutions to the Mixed Dirichlet-Neumann Boundary Value Problem on Lipschitz Domains." UKnowledge, 2017. http://uknowledge.uky.edu/math_etds/47.

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We show that solutions to the mixed problem on a Lipschitz domain Ω can be approximated in the Sobolev space H1(Ω) by solutions to a family of related mixed Dirichlet-Robin boundary value problems which converge in H1(Ω), and we give a rate of convergence. Further, we propose a method of solving the related problem using layer potentials.

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18

Tabet, Tchamba Thierry Wilfried. "Comportement asymptotique des solutions du problème de Cauchy-Dirichlet généralisé pour des équations de Hamilton-Jacobi visqueuses." Thesis, Tours, 2010. http://www.theses.fr/2010TOUR4007/document.

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Cette thèse, constituée de trois grandes parties, a pour objet l’étude générale ducomportement, en temps grands, de l’unique solution du problème de Cauchy-Dirichlet pour deséquations de Hamilton-Jacobi visqueuses de type sur et sous quadratiques. Après un bref rappeldes notions de base de la théorie sur les solutions de viscosité qui constitue le cadre de ce travail, lapremière partie établit des résultats sur l’existence globale en temps et l’unicité de la solution deviscosité dudit problème de Cauchy-Dirichlet. La deuxième partie s’intéresse au comportement decette solution pour des Hamilto

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19

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 d

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20

Collins, Michael [Verfasser], Jens [Akademischer Betreuer] Habermann, and Jens [Gutachter] Habermann. "Existence for Variational Solutions to Cauchy-Dirichlet Problems on Metric Measure Spaces / Michael Collins ; Gutachter: Jens Habermann ; Betreuer: Jens Habermann." Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2020. http://d-nb.info/121973683X/34.

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21

Schnieders, Inka [Verfasser], Guido [Gutachter] Sweers, and Dirk [Gutachter] Horstmann. "Positivity and regularity of solutions to higher order Dirichlet problems on smooth domains / Inka Schnieders ; Gutachter: Guido Sweers, Dirk Horstmann." Köln : Universitäts- und Stadtbibliothek Köln, 2021. http://d-nb.info/1226933378/34.

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22

Brada, Alain. "Comportement asymptotique de solutions d'équations elliptiques semi-linéaires dans un cylindre." Tours, 1987. http://www.theses.fr/1987TOUR4010.

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On étudie le comportement asymptotique pour T tendant vers l'infini de solutions d'équations d'évolution semi-linéaires du second ordre : l'étude se fait d'abord dans un ouvert borné de l'espace successivement avec une condition de Neumann, puis de Dirichlet au bord et ensuite dans l'espace

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23

Abatangelo, Nicola. "Large solutions for fractional Laplacian operators." Thesis, Amiens, 2015. http://www.theses.fr/2015AMIE0019/document.

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La thèse étudie les problèmes de Dirichlet linéaires et semilinéaires pour différents opérateurs du type Laplacien fractionnaire. Les données peuvent être des fonctions régularières [régulières] ou plus généralement des mesures de Radon. Le but est de classifier les solutions qui présentent une singularité au bord du domaine prescrit. Nous remarquons d'abord l'existence de toute une gamme de fonctions harmoniques explosant au bord et nous les caractérisons selon une nouvelle notion de trace au bord. A l'aide d'une nouvelle formule d'intégration par parties, nous élaborons ensuite une théorie f

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24

Micaux, Bertrand. "Flots stochastiques d’opérateurs dirigés par des bruits gaussiens et poissonniens." Paris 11, 2007. http://www.theses.fr/2007PA112218.

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Dans l’article Integration of Brownian vector fields (2002), Le Jan et Raimond étudient des ÉDS dirigées par des champs de vecteurs browniens non nécessairement réguliers. Sous certaines hypothèses de symétrie, ils montrent qu’elles admettent des solutions fortes sous forme de flots d’opérateurs aléatoires qu’ils appellent solutions statistiques fortes de l’équation. L’objet de cette thèse est de généraliser ces résultats dans le cas où l’on ajoute un processus ponctuel de Poisson aux ÉDS étudiées. Au moyen de la théorie des chaos de Wiener-Poisson, nous établissons un théorème d’existence et

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25

Andersson, Tomas. "An iterative solution method for p-harmonic functions on finite graphs with an implementation." Thesis, Linköping University, Department of Mathematics, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-18162.

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<p>In this paper I give a description and derivation of Dirichlet's problem, a boundary value problem, for p-harmonic functions on graphs and study an iterative method for solving it.The method's convergence is proved and some preliminary results about its speed of convergence are presented.There is an implementation accompanying this thesis and a short description of the implementation is included. The implementation will be made available on the internet at http://www.mai.liu.se/~anbjo/pharmgraph/ for as long as possible.</p>

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26

Sande, Olow. "Boundary Estimates for Solutions to Parabolic Equations." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-281451.

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This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of tw

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27

Romani, Giulio. "Positivity and qualitative properties of solutions of fourth-order elliptic equations." Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0359/document.

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Cette thèse concerne l'étude de certains problèmes elliptiques d'ordre 4 et, notamment, des propriétés qualitatives des solutions. Ces problèmes apparaissent dans de nombreux domaines, par exemple dans la théorie des plaques et dans la géométrie conforme, et, comparés à leurs homologues du deuxième ordre, ils présentent des difficultés intrinsèques, surtout liées à l'absence de principe de maximum. Premièrement on étudie la positivité des solutions dans le cas des conditions au bord de Steklov, qui sont intermédiaires entre les conditions de Dirichlet et de Navier. Elles apparaissent naturelle

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28

Cissé, Mamadou. "Méthodes probabilistes pour les conditions au bord artificielles d'équations aux dérivées partielles non linéaires en finance : problème d'arrêt optimal pour une diffusion régulière." Nice, 2008. http://www.theses.fr/2008NICE4025.

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Dans cette thèse, nous donnons un contrôle de l’erreur de localisation sur le système d’inéquations aux dérivées partielles paraboliques avec des conditions au bord de Dirichlet. Ce contrôle d’erreur se fait via l’interprétation probabiliste des inéquations variationnelles sous formes d’équations différentielles stochastiques rétrogrades (EDSRs). Ainsi les solutions de viscosité des inéquations variationnelles localisées avec des conditions de Dirichlet au bord s’interprètent comme des solutions des EDSRs réfléchies à temps final aléatoire borné. Nous établissons un théorème d’existence et d’u

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29

COSTA, Jackson Jonas Silva. "Soluções radiais e não radiais para a Equação de Hénon na bola unitária." Universidade Federal de Campina Grande, 2010. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1232.

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Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-24T16:13:25Z No. of bitstreams: 1 JACKSON JONAS SILVA COSTA - DISSERTAÇÃO PPGMAT 2010..pdf: 1326944 bytes, checksum: 5f5bb0430a12c8bdeabdea73796d8578 (MD5)<br>Made available in DSpace on 2018-07-24T16:13:25Z (GMT). No. of bitstreams: 1 JACKSON JONAS SILVA COSTA - DISSERTAÇÃO PPGMAT 2010..pdf: 1326944 bytes, checksum: 5f5bb0430a12c8bdeabdea73796d8578 (MD5) Previous issue date: 2010-04<br>Capes<br>Para visualizar o resumo recomendamos do download do arquivo uma vez que o mesmo utiliza fórmulas ou equações matemáticas que nã

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30

Bourguignon-Mirebeau, Jennifer. "Préconditionnement de méthodes de décomposition de domaine pour les problèmes de diffraction d'ondes électromagnétiques impliquant une cavité profonde." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00651957.

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Cette thèse est dédiée à la résolution numérique tridimensionnelle des équations de Maxwell harmoniques, par des méthodes de décomposition de domaine couplant des résolutions par équations intégrales entre elles. Pour traiter les problèmes de diffraction d'ondes, la méthode des équations intégrales est un outil précieux. Elle consiste à paramétrer le champ électromagnétique solution par une source définie sur la surface de l'objet diffractant, solution d'une nouvelle équation linéaire (l'équation intégrale). Pour des applications à haute fréquence, le grand nombre d'inconnues (de l'ordre du mi

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31

Cheaytou, Rima. "Etude des méthodes de pénalité-projection vectorielle pour les équations de Navier-Stokes avec conditions aux limites ouvertes." Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4715.

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L'objectif de cette thèse consiste à étudier la méthode de pénalité-projection vectorielle notée VPP (Vector Penalty-Projection method), qui est une méthode à pas fractionnaire pour la résolution des équations de Navier-Stokes incompressible avec conditions aux limites ouvertes. Nous présentons une revue bibliographique des méthodes de projection traitant le couplage de vitesse et de pression. Nous nous intéressons dans un premier temps aux conditions de Dirichlet sur toute la frontière. Les tests numériques montrent une convergence d'ordre deux en temps pour la vitesse et la pression et prouv

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32

Sitar, Scott. "Entire Solutions to Dirichlet Type Problems." Thesis, 2007. http://hdl.handle.net/10012/3147.

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In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always e

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Pošta, Petr. "Baireovské a harmonické funkce." Doctoral thesis, 2017. http://www.nusl.cz/ntk/nusl-368917.

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Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent res

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Yan, Yan. "Smooth and Robust Solutions for Dirichlet Boundary Control of Fluid-Solid Conjugate Heat Transfer Problems." Thesis, 2015. https://doi.org/10.7916/D8H70DNK.

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This work offers new computational methods for the optimal control of the conjugate heat transfer (CHT) problem in thermal science. Conjugate heat transfer has many important industrial applications, such as heat exchange processes in power plants and cooling in electronic packaging industry, and has been a staple of computational methods in thermal science for many years. This work considers the Dirichlet boundary control of fluid-solid CHT problems. The CHT system falls into the category of multi-physics problems. Its domain typically consists of two parts, namely, a solid region subject to

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35

GUAN, WEN-GI, and 管文麒. "Positive radially symmetric solutions for senilinear elliptic partial differential equation on annualar with Dirichlet-Neumann boundary condition." Thesis, 1989. http://ndltd.ncl.edu.tw/handle/46273846456590631863.

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36

"On singular solutions of the Gelfand problem." Chinese University of Hong Kong, 1994. http://library.cuhk.edu.hk/record=b5888160.

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by Chu Lap-foo.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 1994.<br>Includes bibliographical references (leaves 68-69).<br>Introduction --- p.iii<br>Chapter 1 --- Basic Properties of Singular Solutions --- p.1<br>Chapter 1.1 --- An Asymptotic Radial Result --- p.2<br>Chapter 1.2 --- Local Uniqueness of Radial Solutions --- p.8<br>Chapter 2 --- Dirichlet Problem : Existence Theory I --- p.11<br>Chapter 2.1 --- Formulation --- p.12<br>Chapter 2.2 --- Explicit Solutions on Balls --- p.14<br>Chapter 2.3 --- The Moser Inequality --- p.19<br>Chapter 2.4 --- Existence of Solutions

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37

Jogiat, Aasif. "Evaluation of applying Crum-based transformation in solving two point boundary value problems." Thesis, 2016. http://hdl.handle.net/10539/22660.

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A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in ful llment of the requirements for the degree of Master of Science in Engineering, Johannesburg, 2016<br>The aim of this research project is evaluating the application of the Crum-based transformation in solving engineering systems modelled as two-point boundary value problems. The boundary value problems were subjected to the various combinations of Dirichlet, Non-Dirichlet and Affine boundary conditions. The engineering systems that were modelled were in the elds of elec

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38

Fokam, Jean-Marcel. "Forced vibrations via Nash-Moser iterations." Thesis, 2006. http://hdl.handle.net/2152/23960.

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In this thesis, we prove the existence of large frequency periodic solutions for the nonlinear wave equations utt − uxx − v(x)u = u3 + [fnof]([Omega]t, x) (1) with Dirichlet boundary conditions. Here, [Omega] represents the frequency of the solution. The method we use to find the periodic solutions u([Omega]) for large [Omega] originates in the work of Craig and Wayne [10] where they constructed solutions for free vibrations, i.e., for [fnof] = 0. Here we construct smooth solutions for forced vibrations ([fnof] [not equal to] 0). Given an x-dependent analytic potential v(x) previous works on (

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