Showing papers on "Free boundary problem published in 1983"
TL;DR: In this article, a boundary element method for the analysis of free vibrations in solid mechanics is developed using a non-standard boundary integral approach, utilizing the statical fundamental solution and employing a special class of coordinate functions, the algebraic eigenvalue problem results.
668 citations
235 citations
TL;DR: In this paper, the authors considered boundary value problems in non-smooth domains with conic points and isolated non-regular points on the boundary and showed that the solutions of these problems are asymptotic in the neighbourhood of a conic boundary point.
Abstract: CONTENTS Introduction Chapter I. General elliptic boundary-value problems § 1. The solubility of general elliptic boundary-value problems in domains with conic points § 2. The asymptotic behaviour of the solutions of a general boundary-value problem in the neighbourhood of a conic boundary point § 3. General boundary-value problems in non-smooth domains Chapter II. Boundary-value problems for the equations of mathematical physics in non-smooth domains § 1. Boundary-value problems for the system of elasticity theory § 2. Problems of hydrodynamics in domains with a non-smooth boundary § 3. The biharmonic equation Chapter III. Second-order elliptic equations in domains with a non-smooth boundary § 1. Boundary-value problems for second-order elliptic equations in an arbitrary domain § 2. Boundary-value problems in domains with isolated non-regular points on the boundary § 3. Second-order elliptic equations in domains with edges § 4. Boundary-value problems in domains that are diffeomorphic to a polyhedron Chapter IV. Parabolic and hyperbolic equations and systems in non-smooth domains § 1. Parabolic equations and systems in non-smooth domains § 2. Hyperbolic equations and systems in domains with singular points on the boundary References
217 citations
TL;DR: In this article, the authors considered elliptic and parabolic equations of the form * and * where and are positive homogeneous functions of the first order homogeneity with respect to, convex upwards with respect and satisfying a uniform condition of strict ellipticity.
Abstract: This paper considers elliptic equations of the form (*)and parabolic equations of the form (**)where and are positive homogeneous functions of the first order of homogeneity with respect to , convex upwards with respect and satisfying a uniform condition of strict ellipticity. Under certain smoothness conditions on and boundedness from above of the second derivatives of with respect to , solvability is established for these equations of a problem over the whole space, of the Dirichlet problem in a domain with a sufficiently regular boundary (for the equation (*)), and of the Cauchy problem and the first boundary value problem (for equation (**)). Solutions are sought in the classes , and their existence is proved with the aid of internal a priori estimates in . Bibliography: 29 titles.
193 citations
TL;DR: In this article, it is shown that the use of the Brinkman equation to obtain this correction is not justified, because of the limitations of that equation, and an alternative procedure based on a model in which the porous medium is sandwiched between two fluid layers, and the Beavers-Joseph boundary condition is applied at the interfaces, is described, and expression for the correction is obtained.
Abstract: The no-slip condition on rigid boundaries necessitates a correction to the critical value of the Rayleigh–Darcy number for the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below. It is shown that the use of the Brinkman equation to obtain this correction is not justified, because of the limitations of that equation. These limitations are discussed in detail. An alternative procedure, based on a model in which the porous medium is sandwiched between two fluid layers, and the Beavers–Joseph boundary condition is applied at the interfaces, is described, and an expression for the correction is obtained. It is found that the correction can be of either sign, depending on the relative magnitudes of the parameters involved.
175 citations
TL;DR: In this article, a boundary integral approach for eigenvalue and transient dynamic analyses in solid mechanics is presented, where the free vibration case is reduced to an algebraic eigenva problem, while transient vibrations can be solved by using a direct time integration procedure.
150 citations
TL;DR: In this paper, the authors prove that the temperature and the Stefan problem are continuous in the two-phase Stefan problem, a model for ice-water melting which admits a unique weak solution.
Abstract: The classical two-phase Stefan problem, a model for ice-water melting, gives rise to a singular, nonlinear partial differential equation which admits a unique weak solution Here we prove that this solution, and therefore the temperature in the Stefan problem, are continuous
144 citations
TL;DR: In this paper, it was shown that the GKS stability criterion has a physical interpretation in terms of group velocity, and that if the finite difference model together with its boundary conditions can support a set of waves at the boundary with group velocities pointing into the field then it is unstable.
104 citations
01 Jan 1983
TL;DR: In this paper, the boundary surface is decomposed into a finite number of segments and the boundary functions are approximated by corresponding finite elements, the boundary elements, and appropriately discretized version of the boundary integral equation then provides a finite system of linear approximate equations whose coefficient matrix, the influence matrix is fully distributed.
Abstract: Nowadays the most popular numerical methods for solving elliptic boundary value problems are finite differences, finite elements and, more recently, boundary element methods. The latter are numerical methods for solving integral equations (or their generalizations) on the boundary Γ of the given domain. The reduction of interior or exterior stationary boundary value problems as well as transmission problems to equivalent boundary integral equations is by no means unique, the two most popular reductions are the “direct method” and the “method of potentials”. In all these cases one needs a fundamental solution of the differential equations explicitly since it will be used in numerical computations. This restricts the boundary integral methods to cases of simple computability of a fundamental solution, i.e. essentially to differential equations with constant coefficients. The formulation on the boundary surface F reduces the dimensions of the original problem by one. For the computational treatment the boundary surface is decomposed into a finite number of segments and the boundary functions are approximated by corresponding finite elements, the boundary elements. The appropriately discretized version of the boundary integral equation then provides a finite system of linear approximate equations whose coefficient matrix, the influence matrix is fully distributed.
98 citations
Book•
31 Dec 1983
TL;DR: The boundary-value problems of Riemann and Hilbert as mentioned in this paper have been applied to the theory of analytic functions and their application to hydrodynamics, as well as to the study of boundary properties of analytical functions.
Abstract: Part I Boundary-value problems in the theory of analytic functions and their application to hydrodynamics: I. The boundary-value problems of Riemann and Hilbert II. Singular operators in spaces of summable functions. Application to boundary-value problems and to the study of boundary properties of analytic functions III. The mixed boundary-value problem with free boundary IV. Flows of an incompressible fluid with free boundaries Part II Generalized solutions of quasilinear systems of equations in the mechanics of continuous media: V. Quasiconformal mappings and generalized solutions of elliptic systems of equations on the plane VI. Boundary-value problems VII. Boundary-value problems in hydrodynamics and subsonic gas dynamics VIII. Problems in filtration theory for a fluid with free boundaries IX. Some planar problems with an unknown boundary in elasticity theory.
96 citations
01 Jan 1983
TL;DR: In this paper, a flow with a free boundary is represented by a jet of fluid travelling through a region of constant pressure, and two typical situations are shown in Figure 1 : a jet impinging on a fixed wall and a jet emerging from a hole in the wall of a large reservoir.
Abstract: One example of a flow with a free boundary is that of a jet of fluid travelling through a region of constant pressure. There are two typical situations which are shown in Figure. The first is a jet impinging on a fixed wall and the second is a jet emerging from a hole in the wall of a large reservoir. These situations may either be two or three-dimensional, but we can make more analytical progress in the two-dimensional case.
TL;DR: In this article, the initial and boundary value problem for a class of degenerate parabolic equations is studied, where the boundary value is a function of the initial value of the parabolic equation.
Abstract: (1983). The initial and boundary value problem for a class of degenerate parabolic equations. Communications in Partial Differential Equations: Vol. 8, No. 7, pp. 693-733.
TL;DR: In this article, boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third-order boundary value problems which govern the motion of thin liquid films subject to viscous, capillary and gravitational forces are derived.
Abstract: This paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third-order boundary value problems which govern the motion of thin liquid films subject to viscous, capillary and gravitational forces. Precise conditions specifying where and when the third-order derivative terms in the differential equations can be neglected are derived, and improved estimates for the actual solutions in terms of solutions of the lower-order models are constructed. The paper also contains a technique for replacing a third-order problem with an asymptotically equivalent second-order one that may have wider applicability.
TL;DR: In this article, a general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation, and the latter is pursued and applied in detail.
Abstract: Boundary integral equation methods are presented for the solution of some two-dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two-dimensional soil freezing with a time-dependent boundary condition. Agreement between simulated and measured histories is good.
TL;DR: In this article, an artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one, and the solution in the exterior domain is represented by an integral equation over the artificial boundary, incorporated into a velocity pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem.
Abstract: In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.
TL;DR: In this article, some typical free boundary problems, such as the obstacle problem, the seepage surface problem, Stefan problem, and the Elenbaas equation, are all picked up into the scheme.
TL;DR: In this paper, the boundary value problems for the differential equations if A = 0, considered in [7,8] and the references therein, for the integro-differential equations of the Volterra type if A is 0, were considered.
Abstract: where R(t) = (x(t), x’(t), . . . , X@ -‘J(t)) and A is a continuous operator map: C(“-l)[al, a,] + C[ar, a,]. The function f is continuous in all of its arguments. The problem (l.l), (1.2) includes several particular cases, for example the boundary value problems for the differential equations if A = 0, considered in [7,8] and the references therein, for the integro-differential equations of the Volterra type if
TL;DR: In this paper, a nonlinear elliptic eigenvalue problem with neumann boundary conditions, with an application to population genetics, is studied. But the authors focus on population genetics only.
Abstract: (1983). On a nonlinear elliptic eigenvalue problem with neumann boundary conditions, with an application to population genetics. Communications in Partial Differential Equations: Vol. 8, No. 11, pp. 1199-1228.
TL;DR: In this article, a direct biharmonic boundary integral equation (BBIE) method was used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.
Abstract: Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.
TL;DR: In this paper, an attempt was made to generalize results for arbitrary multidimensional difference schemes for dissipative difference schemes, but the problem may not be resolved for all such schemes.
Abstract: An attempt is made to generalize results for arbitrary multidimensional difference schemes. Severe obstacles are encountered, however, for dissipative difference schemes the problem may be resolved.
TL;DR: In this paper, a numerical scheme to treat the open lateral boundary of a limited-area primitive equation model was formulated, which was tested in the numerical integrations of prognostic equations for a Haurwitz-type wave.
Abstract: A numerical scheme to treat the open lateral boundary of a limited-area primitive equation model was formulated. Although overspecification of the boundary condition is inevitable in the pointwise boundary setting, the scheme was designed to keep the overspecification to a minimum degree. To impose the boundary conditions, a damping technique was used. Special care was taken to deal with the boundary layer winds at the lateral boundary. The above scheme is most suitable when gravity waves do not prevail in the vicinity of the open boundary. The scheme was tested in the numerical integrations of prognostic equations for a Haurwitz-type wave. Experimental results are presented which indicate the utility of the proposed method.
TL;DR: In this article, a procedure for the solution of the vibration problem of a Bernoulli-Euler beam with time-dependent boundary conditions is presented, where the dependent variable in the original partial differential equation can be changed to produce homogeneous boundary conditions and at the same time maintain a homogeneous differential equation.
TL;DR: In this paper, the Hausdoff measure estimates of a free boundary for a minimum problem is used to estimate the free boundary of the minimum problem in the context of partial differential equations.
Abstract: (1983). Hausdoff measure estimates of a free boundary for a minimum problem. Communications in Partial Differential Equations: Vol. 8, No. 13, pp. 1409-1454.
TL;DR: In this article, the analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered and optimal error estimates and numerical examples for problems posed on polyhedral domains are provided.
Abstract: The analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered. The results are given for linear second order elliptic partial differential equations and for the nonlinear stationary Navier-Stokes equations. For certain easily implemented boundary treatments, optimal error estimates and numerical examples are provided for problems posed on polyhedral domains.
TL;DR: In this article, the boundary value problem for differential equations with regular singularities was introduced, which was solved in [4] and then it was completely solved by Wang et al. in [5].
Abstract: In a symposium on hyperfunctions and partial differential equations held at Research Institute for Mathematical Science, Okamoto introduced the following Helgason's conjecture: Any simultaneous eigenfunction of the invariant differential operators of a Riemannian symmetric space of non-compact type has a Poisson integral representation of a hyperfunction on its maximal boundary. There had been several affirmative results in some cases. But the method used there was hard to apply to general cases. On the other hand, taking this introduction by Okamoto, we constructed the concept of boundary value problems for differential equations with regular singularities in [5] to solve this conjecture and then it was completely solved in [4]. Recently the conjecture, which is now solved, reveals many important applications in the theory of unitary representations. But the method in [5] is not always easy to be understood by every person. In this note we introduce another definition of the boundary values in an elementary way. Also we give several results concerning the definition, which are sufficient to solve \"the conjecture\" and moreover Corollary 5.5 in [6] which determines the image of the Poisson transformation of Schwartz's distributions on the boundary. Such applications of this note will appear in another paper.
TL;DR: In this paper, the Landau-Lifschitz equation for ferromagnets is written as a compatibility condition of two linear differential equations and the inverse scattering problem for this system is ruduced to the matrix Riemann boundary problem on a torus and then to a certain Fredholm integral equation.
Abstract: The Landau-Lifschitz equation for ferromagnets is written as a compatibility condition of two linear differential equations. The inverse scattering problem for this system is ruduced to the matrix Riemann boundary problem on a torus and then to a certain Fredholm integral equation. The N-soliton solitoons are also obtaine.
TL;DR: In this article, the boundary element method as applied to two-dimensional elastostatic problems is implemented with quadratic variation in boundary variables and geometry and additional parameters are associated with nodes at segment intersections to allow for discontinuous tractions.
Abstract: The boundary element method as it applies to two-dimensional elastostatic problems is implemented with quadratic variation in boundary variables and geometry. Additional parameters are associated with nodes at segment intersections to allow for discontinuous tractions. Supplementary equations consistent with linear elasticity are used to augment the regular boundary integral equations for certain mixed and displacement boundary conditions. Zoning or subregionalization capabilities are included to provide solutions to piecewise nonhomogeneous problems or problems on regions of irregular shape. Example problems are given to demonstrate the accuracy of the technique.
TL;DR: In this paper, it was shown that the two-dimensional free boundary problem for incompressible irrotational water waves without surface tension has exactly eight nontrivial conservation laws.
Abstract: The two-dimensional free boundary problem for incompressible irrotational water waves without surface tension is proved to have exactly eight nontrivial conservation laws. Included is a discussion of what constitutes a conservation law for a general free boundary problem, and a characterization of conservation laws for two-dimensional free boundary problems involving a harmonic potential proved using elementary methods from complex analysis. Introduction. The main purpose of this paper is to prove that the free boundary problem describing the motion of gravity waves over a two-dimensional irrotational, incompressible ideal fluid in the absence of surface tension (\"water waves\") has exactly eight independent conservation laws. Extensions to three-dimensional waves, with or without surface tension are indicated, but not explicitly proven. This result carries a number of implications for the interpretation of the qualitative and quantitative properties of real water waves by soliton models such as the KortewegdeVries equation, which we discuss at length in §2. The proof of such a result must incorporate a precise definition of the concept of a conservation law for a free boundary problem, which, to my knowledge, has not appeared in the literature to date. §3 elaborates on the physical and mathematical motivations for the definition proposed here, which is more general than what one might, by analogy with the corresponding concept for systems of partial differential equations, be tempted to use. The present definition of a conservation law is formulated so as to be applicable to a wide class of free boundary problems. A second result of more general applicability is an interesting characterization of conservation laws for two-dimensional free boundary problems in which the field variables consist of a single harmonic potential. In essence, the time derivative of the conserved density must equal the sum of a divergence and an analytic contribution, the latter being the unusual feature of this result; see §5. I would like to thank T. Brooke Benjamin for the vital encouragement needed to complete this work. Received by the editors November 20, 1981 and, in revised form, May 3, 1982. 1980 Mathematics Subject Classification. Primary 35R35, 35Q20, 76B15.