THE EXISTENCE OF LIMIT CYCLES FOR THE SYSTEM X=Q(x,y),Y=P(x)

ＴＨＥＥＸＩＳＴＥＮＣＥＯＦＬＩＭＩＴＣＹＣＬＥＳＦＯＲＴＨＥＳＹＳＴＥＭX＝Ｑ（ｘ，ｙ），Y＝Ｐ（ｘ）ＸｕＲｏｎｇ－ｌｉａｎｇ（徐荣良）；ＺｈｏｕＧｕｏ－ｃａｉ（周国才）；ＳｕｎＺｈａｏ（孙昭）（ＴａｉｙｕａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏ...     

The existence of limit cycles for system\dot x=Q(x,y), Y=P(x)

In [1], by a transformation on the Lienard equation system

Existence and Asymptotic Behavior of Solutions of the Cauchy Problem x (x′)γ = f(t,x, x′), x(0) = 0

We consider a singular Cauchy problem and prove the existence of continuously differentiable solutions with required asymptotic properties.     

On the singular equation β(u)t=Δu

Communicated by H. Brezis     

On the (u, p) problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements u and loads p simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors u and p are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 100–103, May–June, 2006.     

On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations

The paper deals with the use of the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of viscous compressible flows. We start with a scalar convection–diffusion equation and present a discretization with the aid of the non‐symmetric variant of DGFEM with interior and boundary penalty terms. We also mention some theoretical results. Then we extend the scheme to the system of the Navier–Stokes equations and discuss the treatment of stabilization terms. Several numerical examples are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

On the numerical solution of the equations of propagation by the method of characteristics

Summary A new application of the method of characteristics for the numerical solution of the problems of one-dimensional propagation is stated. The main concept is to consider a Massau grid and to compel it into a net of rectangles, having the sides parallel to the axes of the space-time plane.     

On (g, f)-factorizations of graphs

I.IntroductionLetGbeafinitegraph(withoutloops),V(G)andE(G)bethevenicessetandedgesset,respectively.Foreachxos(G),wedenotebydo(x)thedegreeofxinG.Letg,fbetwononnegativeinteger-valuedfunctionsdefinedonV(G)andg相似文献   

A Petrov-Galerkin method for the numerical solution of the Bradshaw-Ferriss-Atwell turbulence model

The Bradshaw-Ferriss-Atwell model for 2D constant property turbulent boundary layers is shown to be ill-posed with respect to numerical solution. It is shown that a simple modification to the model equations results in a well-posed system which is hyperbolic in nature. For this modified system a numerical algorithm is constructed by discretizing in space using the Petrov-Galerkin technique (of which the standard Galerkin method is a special case) and stepping in the timelike direction with the trapezoidal (Crank-Nicolson) rule. The algorithm is applied to a selection of test problems. It is found that the solutions produced by the standard Galerkin method exhibit oscillations. It is further shown that these oscillations may be eliminated by employing the Petrov-Galerkin method with the free parameters set to simple functions of the eigenvalues of the modified system.     

The semi-infinite strip x≥0, −1≤y≤1; completeness of the Papkovich-Fadle eigenfunctions when Φxx(0,y), Φyy(0,y) are prescribed

For the problem of bending of a semi-infinite strip x0, –1y1, with the sides y=±1 clamped, we give a proof that the end-data% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaarmWu51MyVXgaiuGacqWFgpGzdaWgaaWcbaGaaeiEaiaabIha% aeqaaGqbaOGae4hiaaIaaiikaiaaicdacaGGSaGae4hiaaIaamyEai% aacMcacqGFGaaicqGH9aqpcqGFGaaicaWGMbGaaiikaiaadMhacaGG% PaGaaiilaaqaaiab-z8aMnaaBaaaleaacaqG5bGaaeyEaaqabaGccq% GFGaaicaGGOaGaaGimaiaacYcacqGFGaaicaWG5bGaaiykaiab+bca% Giabg2da9iab+bcaGiaadAgacaGGOaGaamyEaiaacMcacaGGSaaaaa% a!5D6D!\[\begin{array}{l} \phi _{{\rm{xx}}} (0, y) = f(y), \\ \phi _{{\rm{yy}}} (0, y) = f(y), \\ \end{array}\] where f(y), g(y) are arbitrary independent functions prescribed on (–1,1), may be expanded as a series of the bi-orthogonal Papkovich-Fadle eigenfunctions for the strip. This represents an advance on the standard work of R. T. C. Smith [6], who proved such an expansion, but under conditions which are often not satisfied in practice. In particular we are able to solve this bi-harmonic boundary value problem when f, g do not satisfy the side conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaacaWGMbGaaiikaiabgglaXkaaigdacaGGPaqedmvETj2BSbac% faGae8hiaaIaeyypa0Jae8hiaaIaamOzamaaCaaaleqabaGaai4jaa% aakiab-bcaGiaacIcacqGHXcqScaaIXaGaaiykaiab-bcaGiabg2da% 9iab-bcaGiaaicdacaGGSaaabaGaam4zaiaacIcacqGHXcqScaaIXa% Gaaiykaiab-bcaGiabg2da9iab-bcaGiaadEgadaahaaWcbeqaaiaa% cEcaaaGccqWFGaaicaGGOaGaeyySaeRaaGymaiaacMcacqWFGaaicq% GH9aqpcqWFGaaicaaIWaGaaiilaaaaaa!6222!\[\begin{array}{l} f( \pm 1) = f^' ( \pm 1) = 0, \\ g( \pm 1) = g^' ( \pm 1) = 0, \\ \end{array}\]and when the conditions of consistency% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qmaeaacaWGNbGaaiikaiaadMhacaGGPaqedmvETj2BSbacfaGa% e8hiaaIaamizaiaadMhacqWFGaaicqWF9aqpcqWFGaaidaWdXaqaai% aadMhacaWGNbGaaiikaiaadMhacaGGPaGae8hiaaIaamizaiaadMha% cqWFGaaicqGH9aqpcqWFGaaicaaIWaaaleaacqWFsislcqWFXaqmae% aacqWFXaqma0Gaey4kIipaaSqaaiabgkHiTiaaigdaaeaacaaIXaaa% niabgUIiYdaaaa!5A1B!\[\int_{ - 1}^1 {g(y) dy = \int_{ - 1}^1 {yg(y) dy = 0} } \]are not satisfied.The present completeness proof thus answers questions raised recently (in the mathematically equivalent context of Stokes flow) by Joseph [3], and Joseph and Sturges [5], who showed that if the side conditions (A), (B) are relaxed then the corresponding eigenfunction series may still converge; but they left open the more difficult question of whether these series still converge to the data.The method of proof used here also succeeds in proving a corresponding completeness theorem for the Williams eigenfunctions for the wedge with the data.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaadaabciqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaa% daqadiqaamaalaaabaGaaGymaaqaaiaadkhaaaqedmvETj2BSbacfi% Gae8NXdygacaGLOaGaayzkaaaacaGLiWoadaWgaaWcbaGaamOCaiab% g2da9iaaigdaaeqaaGqbaOGae4hiaaIaeyypa0Jae4hiaaIaamOzai% aacIcacqaH4oqCcaGGPaGaaiilaaqaamaaeiGabaWaaSaaaeaacqGH% ciITdaahaaWcbeqaaiaaikdaaaGccqaHgpGzaeaacqGHciITcqaH4o% qCdaahaaWcbeqaaiaaikdaaaaaaOWaaeWaceaadaWcaaqaaiaaigda% aeaacaWGYbaaaiab-z8aMbGaayjkaiaawMcaaaGaayjcSdWaaSbaaS% qaaiaadkhacqGH9aqpcaaIXaaabeaakiab+bcaGiabg2da9iab+bca% GiaadEgacaGGOaGaeqiUdeNaaiykaiaacYcaaaaa!6B9C!\[\begin{array}{l} \left. {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = f(\theta ), \\ \left. {\frac{{\partial ^2 \phi }}{{\partial \theta ^2 }}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = g(\theta ), \\ \end{array}\]prescribed on –<<, (where 2 is the wedge angle).Department of Mathematics, University of ManchesterOn leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A 9259 and A9117.