bluuebunny/crossref_metadata_2025_split huggingface dataset:
Bifurcation results for a class of periodic quasi‐linear parabolic equations
Authors: de Mottoni P. , Schiaffino A. , Hadeler K. P. 
Published: March 2007

AbstractWe consider the problem where a and f are 1‐periodic in t, a is positive, f satisfies appropriate decreasing conditions; smoothness of a, f, ∂Ω is also assumed. Denote by λ0 the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non‐negative periodic solution exists but zero, and any solution with continuous non‐negative initial datum converges to zero uniformly as t → ∞; (b) if ε > 0, then a unique non trivial non‐negative 1‐periodic solution u* exists, and any solution with continuous, non‐negative not identically zero initial datum approaches uniformly u* as t → ∞.
Asymptotic relationship between telegraphic and diffusion equations
Authors: Mika J. , Stankiewicz R. , Neunzert H. 
Published: March 2007

AbstractIn the paper it is considered the generalized telegraphic equation represented by the system of four evolution equations, containing a small positive parameter multiplying some of the time derivatives. It is shown that if the small parameter tends to zero, the solution to the telegraphic equation tends asymptotically to the solution of the diffusion equation obtained from the former one by putting the small parameter equal to zero.
Free boundary problems for the laplace equation: A priori estimates, existence theorems, asymptotic behaviours
Authors: Dervieux A. , Kawarada H. , Fujita H. 
Published: March 2007

AbstractThis work is devoted to Free Boundary Problems with Laplace equation Δu = f in the domain and the two conditions on the Free Boundary u = 1 and |grad u| = λ = const. In the model problems which we study, three cases arise: 1) f = 0, λ is given, 2) f = 0, λ is unknown and the length of the Free Boundary is given, 3) f ≠ 0, λ = 0 (Obstacle Problem).
Sur une Interprétation Mathématique de l'Intégrale de Rice en Théorie de la Rupture Fragile
Authors: Destuynder P. , Djaoua Palaiseau Et M. , Chesnay Le , Nedelec J. C. 
Published: March 2007

AbstractWe formulate the assumptions under which Rice J integral and the critical energy release rate, which occur in fragile fracture mechanic, are equivalent.
A constructive approximation theorem for the oblique derivative problem in potential theory
Authors: Freeden W. , Kersten H. , Leis R. 
Published: March 2007

AbstractThe (exterior) oblique derivative problem of potential theory magnified image is considered where l is a C(1,λ)‐vector field on a regular boundary ∂Ge in Euclidean space R3 and the direction l at any point of ∂Ge forms with the outside normal n an angle ∢ (l, n) satisfying cos ∢ (l, n) ≥ c > 0. An approximation of the uniquely determined solution is given by use of trial functions {Φn}n=0,1,… harmonic in some region containing Ge and suitable for numerical purpose (for instance: solid spherical harmonics, certain sequences of fundamental solutions). The system {l▽Φn+hΦn}n=0,1,… defined on ∂Ge is shown to be closed and complete in the pre‐Hilbert space C(0,λ)(∂Ge) of λ‐Hölder continuous functions.
The decay of solutions of the Carleman model
Authors: Illner R. , Reed M. C. , Neunzert H. 
Published: March 2007

AbstractWe prove that for positive initial data u0, v0 ϵ C1 (R) ∩ L1 (R) vanishing at infinity, the solution u(x, t) v(x, t) of the Carleman model satisfies the estimate The constant C depends only on the initial mass m.
Scattering techniques for a one dimensional inverse problem in geophysics
Authors: Carroll R. , Santosa F. , Paynec L. 
Published: March 2007

AbstractA one dimensional problem for SH waves in an elastic medium is treated which can be written as vtt = A−1 (Avy)y, A = (ϱμ)1/2, ϱ = density, and μ = shear modulus. Assume A ϵ C1 and A′/A ϵ L1; from an input vy(t, 0) = ∂(t) let the response v(t, 0) = g(t) be measured (v(t, y) = 0 for t < 0). Inverse scattering techniques are generalized to recover A(y) for y > 0 in terms of the solution K of a Gelfand‐Levitan type equation, .
New asymptotic expansion algorithm for singularly perturbed evolution equations
Authors: Mika J. , Eckhaus W. 
Published: March 2007

AbstractA new asymptotic expansion algorithm related to the Chapman‐Enskog expansion in kinetic theory is applied to systems of linear evolution equations. The uniform convergence of the asymptotic solution to the exact one is shown. The algorithm is applied to the linearized Carleman model of the Boltzmann equation, to the neutron transport equation, and to the Fokker‐Planck equation.
A singular perturbation problem for linear operators with an application to electrostatic and magnetostatic boundary and transmission problems
Authors: Engl H. W. , Kress R. , Meister E. 
Published: March 2007

AbstractThe treatment of certain electro‐ and magnetostatic boundary and transmission problems by boundary integral equations leads to parameter‐dependent integral equations of the second kind. The integral operators involved have the property that the dimension of their nullspaces changes between two nonzero values (depending on the geometry of the problem) as the parameter tends to zero. We investigate the continuous dependence of solutions to these equations on the parameter. To this end, we treat the problem of continuous dependence of solutions to parameter‐dependent linear operator equations of the second kind in a Banach space in the framework of generalized inverses.
Study of motions of a vibrating string subject to solid friction
Authors: Cabannes H. , Neunzert H. 
Published: March 2007

AbstractIn this paper we consider a string moving in its plane and subject to solid friction (Coulomb's law). It is known that when the time increases indefinitely the string reaches an equilibrium position and by analogy with the case of a mass point we ask if the equilibrium is reached after a finite time. We prove that this is the case when the string is initially at rest and 1. when the initial shape possesses a second derivative bounded by certain limits, or 2. when the initial shape is formed by two straight line segments. In the last section we obtain some partial results when the string is initially at rest in the shape of a polygonal line. The case of an arbitrary initial position is still an open problem.
Bifurcation of periodic orbits in coupled chemical reactors I
Authors: Kirchgraber U. , Hagedorn P. 
Published: March 2007

AbstractThis paper provides a rigorous approach to the problem of bifurcation of periodic orbits of a chemical reactor system recently studied by Marek and Stuckl, and by Neu. Using transformation techniques and invariant manifold theory the problem is reduced to a two‐parameter bifurcation equation in R.
Mathematische Behandlung ernes Modells der Haemodialyse
Authors: Klingelhöfer D. , Koch G. F. , Krabs W. , Hadeler K. P. 
Published: March 2007

AbstractIn this paper a simple mathematical model for the process of hemodialysis is presented. This model is based on a system of two linear differential equations of first order with partly discontinuous coefficients that describe the time‐development of the concentrations of a certain toxin (like urea) in the intra‐ and extracellular part of the human body. The main result is the existence of periodic positive solutions of this system under the natural assumption that the generation of the toxin and its removal by hemodialysis are periodic processes. These periodic positive solutions are also computed numerically for a realistic choice of the coefficients of the modelling differential equations.
On the Vlasov hierarchy
Authors: Spohn H. , Neunzert H. 
Published: March 2007

AbstractMotivated by a recent paper of H. Narnhofer and G. Sewell, we investigate the problem of existence and uniqueness of solutions of the Vlasov hierarchy. It is shown that the unique solution of the Vlasow hierarchy is induced by the flow on the space of probability measures on R6 which is obtained from the solution of the Vlasov equation.
The impedance boundary value problem for the time harmonic Maxwell equations
Authors: Colton D. , Kress R. , Hsiao G. C. 
Published: March 2007

AbstractThe existence of a unique solution to Maxwell's equations defined in an exterior domain with impedance boundary condition is established for all frequencies. This is accomplished by reducing this problem to that of solving a system of singular integral equations and then regularizing this system such that the Riesz theory is applicable. We also consider the inverse problem in which it is desired to determine the impedance from a knowledge of the far field pattern. By restricting the impedance to lie a priori in a compact set results are obtained on the existence, uniqueness, and stability of the solution to this inverse scattering problem.
Regularity theorems for Maxwell's equations
Authors: Weber C. , Werner P. 
Published: March 2007

AbstractMethods using the theory of distributions and Hilbert space operators have been very powerful in the past to achieve uniqueness and existence results for Maxwell's equations. In this paper conditions are given when such abstract “Hilbert space”‐solutions represent differentiable “regular” functions which satisfy Maxwell's equations, boundary conditions, and transmission conditions in the classical sense.
On spherical spline interpolation and approximation
Authors: Freeden W. , Törnig W. 
Published: March 2007

AbstractSpherical spline functions are introduced by use of Green's surface functions with respect to the (Laplace‐)Beltrami operator of the (unit) sphere. Natural (spherical) spline functions are used to interpolate data discretely given on the sphere. A method is presented that allows the smoothing of irregularities in measured values or experimental data. Extensions of Peano's theorem and Sard's theory of best approximation to the spherical case are given by integral formulas. Schoenberg's theorem is transcribed into spherical nomenclature.
Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3
Authors: Zajaczkowski W. M. , Piskorek A. 
Published: March 2007

AbstractIn this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners

where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary,
(2) the existence of solutions of the following evolution problem for given velocity vector

(3) the method of successive approximations, using solvability of problems (1) and (2).
On finite element methods for convection dominated phenomena
Authors: Kikuchi Fumio , Ushijima Teruo , Fujita H. 
Published: March 2007

AbstractThe aim of this work is to give theoretical justification of several types of finite element approximations to the initial‐boundary value problems of first order linear hyperbolic equations. Our approximate scheme is obtained by the piecewise linear continuous finite element method for space variable, x, and the Euler type step by step integration method for time variable, t. An artificial viscosity technique, up‐stream type methods are considered within the frame work of L2‐theory. The convergence and the error estimate of the approximate solutions to the true one are discussed.
Boundary integral equations for the Helmholtz equation: The third boundary value problem
Authors: Angell T. S. , Kleinman R. E. , Hsiao G. C. 
Published: March 2007

AbstractIn this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L∞ (∂D), ∂D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.
Nontrivial periodic solutions of a nonlinear beam equation
Authors: Chang K. C. , Sanchez Luis , Rabinowitz P. 
Published: March 2007

AbstractWe consider the existence of a nontrivial solution of the following equation:wheregis a nondecreasing function defined on R1, satisfiesg(O) = O, and some other additional conditions.Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]‐[8]:.
A generalized Lane‐Emden‐Fowler equation
Authors: Van Broek W. J. Den , Verhulst F. , Eckhaus W. 
Published: March 2007

AbstractThe nonlinear potential problem Δ ϕ = –r2m 0, n = m is studied with r ⩾ 0, n ε (½, ∞), m ε (−1, ∞) and n m > 0.With various additional conditions we have regular and singular solutions which can be studied explicitly by homology transformation. It is proved that if m − n + 1 = 0 the solution with ϕ(0) = 1, ≠ (0) = 0 is radiusstable. Numerical analysis of a number of cases with m − n + 1 = 0 yields the same conclusion.
Monotone schemes for semilinear elliptic systems related to ecology
Authors: Leung Anthony , Rabinowitz P. H. 
Published: March 2007

AbstractSemilinear elliptic systems of partial differential equations related to ecology are studied, with Dirichlet boundary conditions. Monotone sequences of functions which satisfy scalar equations are constructed so that they will converge to upper and lower bounds for the solutions of the systems. In case a related system has a unique positive solution, then these sequences will converge to the solution of the original system. Applications of the monotone sequences to uniqueness and stability are also given.
Galerkin approximations for the equation Lu + Nu = f
Authors: Göthel R. , Jones D. S. 
Published: March 2007

AbstractIntroducing the concept of a supercontinuous operator, we obtain a general convergence theorem for Galerkin approximations. Under the stronger assumption that N is a monotone operator with N(0) = 0, we show norm convergence of the unique Galerkin approximations to the unique solution.
Existence and uniqueness for an equation of mixed type in a rectangle
Authors: Aziz A. K. , Bell J. B. , Schneider M. , Gilbert R. P. 
Published: March 2007

AbstractA boundary value problem is considered for an equation of mixed type in a rectangle. An energy method is used to prove uniqueness for appropriate boundary conditions. Similar considerations for the adjoint problem allow one to prove the existence of a generalized solution to the problem. It is also shown that the energy method leads naturally to a finite element method. The method is described for a special case and convergence is shown for the resulting numerical procedure.
Biorthogonal sequences of solutions of the generalized feller equation
Authors: Lehnigk Siegfried H. , Roach G. F. 
Published: March 2007

AbstractThe generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.
Eigenvalue bounds for elliptic operators
Authors: Lortz Dietrich , Mayer‐Spasche Rita , Törnig W. 
Published: March 2007

AbstractLower bounds for the real parts of the points in the spectrum of elliptic equations

are derived. These bounds, depending only on the diameter L of the domain G and on the maximum norm M of the coefficients a, b, are optimal. They are always positive and thus the spectrum is bounded away from the imaginary axis. This result is then used to prove an “anti‐dynamo theorem” for magnetic fields with plane symmetry in the case of a compressible steady flow surrounded by a perfect conductor.
Zur theorie rotierender und schwingender schaufelkränze in einer unterschallstromung durch einen ringkanal
Authors: Buggle G. , Meister E. 
Published: March 2007

AbstractIn this paper the three‐dimensional perturbation flow induced by a rotating and oscillating blade row which operates in a subsonic flow in axial direction of an annular channel is studied. The velocity potential is reduced to the infinite Hilbert space vector of Fourier coefficients of an eigen‐function expansion with respect to vanishing normal derivatives on both cylinder walls. These coefficients satisfy an infinite set of ordinary differential equations of second order after an application of a one‐dimensional Fourier transform in axial direction.Several canonical two‐part mixed boundary value problems are then investigated by reduction to “infinite two‐by‐two‐Wiener‐Hopf functional systems”. In case of strong factorizability of certain matrix‐operator‐valued functions on the line these systems may be solved explicitely. Criteria for the factorization are not given here.
Rounding errors in numerical solutions of two linear equations in two unknowns
Authors: Stummel F. , Wendland W. 
Published: March 2007

AbstractNew condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating‐point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.
Asymptotic behavior of a nonhomogeneous predator‐prey system with one diffusing and one sedentary species
Authors: Rothe F. , Hadeler K.P. 
Published: March 2007

AbstractIn general the predator‐prey system studied will have many equilibrium states with different “patches”, i.e. regions where the sedentary species vanishes. If one equilibrium is given, a Ljapunov functional can be constructed for a set of initial data with patches satisfying a certain compatibility condition. By the methods of Alikakos [1] and Rothe [28] this implies uniform bounds for the diffusing species in a Hölder space and the sedentary species in a Lp‐space. Considering pieces of the trajectory for time in some bounded interval and using the weak*‐topology for the sedentary species, one defines an ω‐limit set which is compact and nonvoid. Using the Ljapunov functional an asymptotic version of the differential equations is derived. It turns out that they admit only constant solutions except for a very special case. Thus convergence to the equilibrium follows.
A critical case for the solvability of stefan‐like problems
Authors: Fasano A. , Primicerio M. , Hadeler K.P. 
Published: March 2007

AbstractWe consider a one‐phase one‐dimensional Stefan problem with general data with the aim to investigate some open questions on existence of classical solutions. We show how existence and nonexistence are discriminated by the behavior of the initial datum in the neighborhood of the starting point of the free boundary.
A trace theorem for solutions of the wave equation, and the remote determination of acoustic sources
Authors: Symes W. W. , Payne L. 
Published: March 2007

AbstractThe determination of sources of acoustic wave motion in several dimensions from remote measurements is of considerable interest in many applications, and the underlying mathematical problem is quite ill‐posed. We separate the source determination problem into a control problem for the wave equation and an inverse mixed initial‐boundary value problem, and concentrate on the latter, in which the initial data for a solution of the wave equation are to be determined from its trace on a time‐like hyperplane. Though the geometry of this problem is simple, it exhibits some of the central analytic difficulties of more complex problems. We prove a uniqueness theorem, give examples of instability, establish regularity properties of the trace, and locate noncompact classes of stable functionals. The existence of these noncompact classes shows that the problem is “partially well‐posed”, i.e. that smoothing in all directions is not required to regularize the problem, and distinguishes it from most other ill‐posed problems, such as backwards diffusion and analytic continuation.
Periodic solutions of u'(t) = −f(u(t − 1)): Multiplicity results and small period solutions
Authors: Angelstorf N. , Kirchgassner K. 
Published: March 2007

AbstractIn this paper we are concerned with the differential delay equationmagnified imageFor odd and continuous nonlinearities f: R → R. We want to prove existence and multiplicity criteria for so called “slowly oscillating” periodic solutions of (A).The oddness condition is of course a rather restrictive one, but due to results of Nussbaum [9], [10] and Peters [13],[14] and numerical studies of Jürgens, Peitgen and Saupe [7] and Hadeler [6] we know that even for odd nonlinearities f (A) may display a complicated dynamical behaviour. On the other hand the oddness condition allows a classification of symmetry properties of periodic solutions of (A).
Asymptotic equipartition of energy for equations of elasticity
Authors: Sandefur J. T. , Payne L. 
Published: March 2007

AbstractIn this paper we give a review of the equipartition of energy results of Goldstein and Sandefur [3], [4], [5] as well as proving a new result in the case of a particular fourth order equation. These results are then applied to the equations of elasticity to give a weak asymptotic orthogonality for the shear and pressure waves. In the case of boundary value problems in the interior of a bounded domain we get weak asymptotic orthogonality in the average.
Error analysis of a numerical scheme for 3D Maxwell–Landau–Lifshitz system
Authors: Cimrák Ivan 
Published: March 2007

AbstractWe suggest a linear numerical scheme solving strongly non‐linear coupled Maxwell–Landau–Lifshitz (Maxwell–LL) system describing ferromagnetic phenomena. Using recent results on the regularity of the solutions to the Maxwell–LL system we are able to prove convergence and to derive the error estimates for this numerical scheme. We provide numerical examples which confirm the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.