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In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow-up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd.

A new defect-correction method for the stationary Navier–Stokes equations based on local Gauss integration is considered in this paper. In both defect step and correction step, a locally stabilized technique based on the Gaussian quadrature rule is used. Moreover, stability and convergence of the presented method are deduced. Finally, we provide some numerical experiments to show good stability and effectiveness properties of the presented method. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper we consider the non-characteristic Cauchy problem ut−a(x)uxx−b(x)ux−c(x)u = 0, x ∈ (0, l), t ∈ I, u(0, t) = φ(t), ux(0, t) = 0, t ∈ I, where I = ℝ or I = ℝ+ and u(x, 0) = 0, x ∈ [0, l], in the case I = ℝ+. Assuming an a priori bound for ‖u(l,.)‖, we derive the exact Holder type dependence of on ‖u(x,.)‖ on ‖φ‖.

We describe in this paper a finite element method for the solution of viscous incompressible flow problems which incorporates an approximate form of the incompressibility condition automatically into the finite element basis. Several examples of such finite elements are presented and applied to a simple test problem.

In this paper we consider a free boundary problem of general type for the heat equation in one space dimension. We formulate this problem as an optimal control problem and derive necessary conditions for a solution of it. In order to compute a solution of the control problem, we apply the projection-gradient-method. Simple numerical examples illustrate the results.

Mixed boundary value problems for the wave-equation in the x-y-plane, with boundaries parallel to the x-axis, are usually treated by Laplacetransformation. To use this method numerically we present it here by means of the Mikusinski-calculus. In this paper our main interest lies in the treatment of mixed initial-boundary value problems arising in unsteady subsonic cascade flow.

In connection with the free boundary value problem of determining the earth's surface from measurements of gravitational potential and force-field (“the geodetic boundary problem”), an oblique derivative problem arises, where D0 is some bounded domain, star shaped with respect to the origin. In order to prove a uniquencess theorem for the geodetic boundary problem, it is essential to give estimates for (weighted) L2-norms of the second derivatives of the solutions so that their bounds can be estimated numerically if bounds for the function describing the boundary are known. In this paper a Fredholm inverse for the above problem is constructed and the second derivatives of the solutions are estimated in the desired form.

This paper deals with so-called general linear stochastic processes (GLSP), defined by T. Kawata in 1972 in generalization of work of R. Lugannani and J. B. Thomas of 1967/71. These second order processes (which are not necessarily stationary nor have independent increments) are described by rather weak requirements, so that several processes such as some random noise and pulse train processes are specific models of these GLSP. Part I is concerned with two general theorems giving asymptotic expansions (including those for the density function) in the central limit theorem for such GLSP, together with error rates. The assumptions for the corresponding θ– and o–error estimates seem rather natural: in the former, apart from assumptions on the inherent structure of such GLSP, the existence of certain moments of higher order as well as a Cramer-type condition are assumed, in the latter in addition a Lindeberg-type condition of higher order. Fourier analytic machinery is used for the proofs.

In this paper boundary controllability of one-dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off. This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null-controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported beam.

For any differential operator L, separable in some coordinate system, we construct a set of commuting operators Si such that 1) each Si maps solutions of Lu = 0 into solutions, and 2) the separated solutions of Lu = 0 are simultaneous eigenfunctions of Si. Moreover, a certain description of the separable coordinate system by Si is possible. The paper generalizes results obtained by W. Miller and others on many equations of mathematical physics.

Ordinary bifurcation problems of the form (1) typically have at most one nontrivial, nonnegative solution for λ > 0. The paper shows that this is in general not true for discrete analogues to (1) no matter how small the step width h > 0 is chosen.

The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {n} be a sequence of vector-valued functions on G where the L2-norms of {n}, {curl n}, and {div(β n)} are bounded, and where all n either satisfy x n = 0 or (β Fn) = 0 at the boundary ∂G of G ( = normal to ∂G): then {n} has a L2-convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ∂G if β is interpreted as electric dielectricity ϵ or as magnetic permeability μ, respectively. These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations.

In this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k2U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L2(∂G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L2(∂G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G. The well-known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.

It is well known that the free boundary of the one-phase Stefan problem (1.1–5) depends continuously on the boundary data [1]. In this paper we prove that, in addition, the solution operator S which, to each g. assigns the corresponding free boundary, is continuously Frechet differentiable and we give the defining formulas of the Frechet derivative.

The change of variable for the temperature Θ in the one-phase Stefan problem leads to the evolution inequality, (ut – Δu – f)(v – u) ⩾ 0 for all regular v ⩾ 0, where u ⩾ 0 is required. This inequality is to hold over a space-time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ∂ Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {um}. The sequence {um} may be interpolated linearly in t to obtain an approximation UΔt of u. The following results are obtained in this paper: (i) a two-sided weak maximum principle for um – um-1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of UΔt and ∇UΔt, to u and ∇u, respectively, on D; (iii) the uniform convergence to the Holder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence xΔt, where xΔt interpolates xΔt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.

We consider the equation of mixed type (k(y) ⪌ 0 whenever y ⪌ 0) in a region G which is bounded by the curves: A piecewise smooth curve Γ lying in the half-plane y > 0 which intersects the line y = 0 at the points A(-1, 0) and B(0, 0). For y < 0 by a piecewise smooth curve Γ through A which meets the characteristic of (1) issued from B at the point P and the curve Γ which consists of the portion PB of the characteristic through B. We obtain sufficient conditions for the uniqueness of the solution of the problem L[u] = f, dnu: = k(y)uxdy – uydx|γ0 = = Ψ(s) for a “general” function k(y), when r(x, y) is not necessarily zero and Γ1 is of a more general form then in the papers of V. P. Egorov [6], [7].

In this paper we study the elastic wave diffraction in R3 through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.

In this paper we derive a uniformly valid asymptotic approximation of the periodic solution of a self-excited system given by the differential equation and β1,β2, are positive constants. By uniformly valid asymptotic approximation we mean that no secular terms are present. Our procedure makes use of a nonlinear change of independent variable that transforms the problem from one in which the discontinuities are ϵ dependent to one in which the discontinuities are ϵ independent. We obtain an asymptotic approximation up to order ϵ of the periodic solution and an asymptotic approximation up to order ϵ2 of the period. Some comparisons between our asymptotic results and numerically derived results are given. Application of our technique to other examples of self-excited systems is discussed. The equation is investigated in detail.

The paper starts with a short survey of the treatment of initial-boundary-value problems in temperature-free linear elasticity with unisotropic media. The main part of the paper is concerned with exterior initial-boundary-value problems in thermoelasticity. In this case the underlying differential operator A is no longer selfadjoint. Thus the spectrum of A has to be discussed. In 2.1 it is shown that all λ with Re λ < 0 belong to the resolvent set. In 2.2 the case G = R3 with homogeneous isotropic media is considered. Let Λ be the essential spectrum in this case. In 2.3 Λ depending on the thermic coupling parameter is discussed. 2.4 treats the spectrum of A assuming the medium to be homogeneous and isotrop outside a large ball. In this case Λ is the essential spectrum for A too. Radiation conditions are formulated. Finally 2.5 presents a short treatment of the time dependent case with Laplace-transformation.

The purpose of this paper is the application of Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.

This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ⊂ R3. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .

In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].

In the following paper a methodological survey with respect to applications of the horizontal line method (Rothe's method) to a class of initial boundary value problems is given. By means of results from abstract perturbation theory, convergence results and error estimates are established for several special initial boundary value problems of mathematical physics.

In this paper we describe a class of Gaus-Seidel iterations for the solution of large sparse nonlinear systems of equations. For suitable starting vectors every method of this class is monotonically convergent with respect to both sides. The result will be applied to the numerical solution of nonlinear boundary-value problems and boundary layer problems.

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Karman model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H1an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.

In 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.

In this paper we consider the reflection of acoustic waves at an unbounded surface which coincides with a plane outside a sufficiently large sphere. We prove uniqueness and existence theorems for the corresponding boundary value problems for the reduced wave equation with Dirichlet and Neumann data by employing integral equation methods.

In this paper a finite element approximation scheme for the system curl is considered. The use of pointwise approximation of the boundary condition leads to a nonconforming method. The error estimate is proved and numerically tested.

In 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.

Motivated 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 purpose of this paper is to give a systematic analysis for the linear energy dependent diffusion system in reactor dynamics, taking into consideration of delayed neutrons. This analysis includes the existence of a unique positive solution for the time dependent system, the asymptotic behaviour of the solution as t ∞, the stability and instability of steady-state solutions, and the existence and uniqueness of a steady-state solution for the time-independent system. Using the notion of upper and lower solution, we establish some threshold conditions for insuring the asymptotic behaviour of the solution and the stability and the instability of any given unperturbed solution, including steady-state solution. In fact, these conditions characterize the stability and the instability property of the solution, and yield a bifurcation result in terms of either the size of the diffusion domain or the physical parameters of the diffusion medium. We also discuss the case without the effect of delayed neutrons.

In this paper the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.

Methods 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.

In 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).

In the paper we prove the existence and uniqueness of solutions of the overdetermined elliptic system where b, ω are given functions, in a domain Ω C R3 with corners π/n, n = 2, 3, … The proof is divided on two steps, we construct a solution for the Laplace equation in a dihedral angle π/n, using the method of reflection and we get an estimate in the norms of the Sobolev spaces in some neighbourhood of the edge. In the dihedral angle system (A) reduces to the Dirichlet and Neumann problems for the Laplace equation. In the next step we prove the existence of solutions in the Sobolev spaces Wpl(Ω) using the existence of generalized solutions of (A).

In 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.

The 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.

This paper is concerned with spline methods in a reproducing kernel Hilbert space consisting of functions defined and harmonic in the outer space of a regular surface (e.g. sphere, ellipsoid, telluroid, geoid, (regularized) earth's surface). Spline methods are used to solve interpolation and smoothing problems with respect to a (fundamental) system of linear functional giving information about earth's gravity field. Best approximations to linear functionals are discussed. The spline of interpolation is characterized as the spline of best approximation in the sense of an appropriate (energy) norm.

In 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.

Data processing is an important tool in airspace surveillance and air traffic control today. In this paper the problem is treated how to reconstruct a flown trajectory from its correlated radar plots subject to a certain knowledge of the aircraft manoeuverability and the radar measurement statistics. A variational approach leads to a generalized smoothing spline method. Simulation results are presented.

In the paper a boundary value problem is studied for the equation of mixed typek(y)uxx + uyy + r(x, y)u = f(x, y) in the rectangular domain {(x, y)| −1 0, k(y) = sign y|y|m, m > 0 (and more generally for a function k = k(y) with k(O) = 0, k(y)y > O for y ≠ O). Specific for the stated problem is that no data are prescribed on the line {(x, yc), −1 < x < +1}. It is proved that the formulated problem is well-posed in the sense that there is at most one quasi-regular solution and that a generalized solution exists. The energy-integral-(abc-)method is used to show uniqueness and to obtain an apriori estimate for the solution of the adjoint problem whence the existence statement follows.

A half-plane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary condition on the other. These boundary conditions contrast the ones used by A. Sommerfeld in his classical paper. The present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method suggested by A. E. Heins in 1950.

This paper presents finite element methods to approximate inviscid incompressible flow problems. First we emphasize the conservation properties of these problems, and we show that finite element methods appear as a very natural way to find conservative schemes such as Arakawa's scheme. We give convergence theorems and an error analysis of finite element discretization schemes. We turn then to the time differencing problem. We derive stability and convergence results for a second-order semi-implicit scheme and for the leap-frog scheme.

In this paper the question of determining the dimension of the space of harmonic Dirichlet and Neumann differential forms on a Riemannian manifold with non-smooth boundary is answered for a wide class of boundaries. The admissible boundaries can be characterized using a generalized “global segment property”. The well-known relation between the Betti numbers and the dimension of these spaces is established in this more general case, too. Bounded and non-bounded manifolds are treated (“exterior and interior domains”).

In 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.

As is well known the real Heisenberg nilpotent group A(R) constitutes the group-theoretic embodiment of the Heisenberg canonical commutation relations (CCR) of classical quantum mechanics. In this connection, quantum mechanics stands for the quantum-mechanical description, at a given instant of time, of a non-relativistic microparticle moving in the one-dimensional configuration space R and having the plane R2 as its (flat) phase space. In fact, the (nilpotent) Lie algebra n of A(R) reflects the Weyl equations which are global versions of the Heisenberg CCR. Unfortunately, the subject of Heisenberg nilpotent groups is outside quantum mechanics and all the more outside mathematical physics not as commonly known as it should be considering its wide range of applications in a variety of different fields. The present paper which has two parts aims to develop a central topic of nilpotent harmonic analysis, to wit, the microparticle model, the lattice model which will be realized on the Heisenberg compact nilmanifold, and the complex wave model (or Bargmann-Fock-Segal model) of the linear Schrodinger representation of A(R) in order to examine geometrically several applications which are governed by the real Heisenberg nilpotent group A(R). These applications are in Part I the classical Whittaker-Shannon sampling theorem which is of basic importance in signal processing, to wit, for the transmission of digital signals as well as analog signals, and in Part II the Subbotin-Schoenberg existence and uniqueness theorem of cardinal spline interpolation. Moreover, Part II indicates briefly some connections of the aforementioned models to the Wigner phase-space quasiprobability density function of quantum statistical mechanics via the Schwartz kernels theory on unimodular Lie groups, an approach to the cross- and autoambiguity functions of radar synthesis, and to the Zak transform of solid state physics. The second part also points out some relations of harmonic analysis of the finite nilpotent group A(Z/NZ) to periodic spline interpolants admitting N equidistant knots on the one-dimensional compact torus group T. These last examples should serve mainly as hints for some further lines of investigations in the field of applications of nilpotent harmonic analysis.

In this part of the paper we deal with computational aspects of the bifurcation problem introduced in Part I, cf. [1]. It follows from Section 3 that the bifurcation behaviour is essentially determined by the expression KP(Ω) – ∧β, where β is some constant which was assumed to be nonzero, while P(Ω) is some 2π-periodic function which was implicitly defined in Section 2. In Section 4 we first derive easily computable expressions for P(Ω) and β from the definitions of Section 2. Secondly we offer a direct, although formal, derivation for these quantities. In Section 5 we briefly show how P(Ω). β can actually be computed by means of a high speed computer and discuss a number of examples. In Section 6 finally we give some asymptotic results for systems depending on an additional parameter.

In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of Rn(n ⩾ 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R3 with vanishing Dirichlet or Neumann boundary condition.

We prove asymptotic estimates for the Green's function of N-irregular eigenvalue problems My = λNγ with splitting boundary conditions. In contrast to the N-regular case the Green's function G(x,ζ,λ) grows exponentially for |λ| ∞ if x > ζ. These estimates are fundamental for the expansion of functions into a series of eigenfunctions of N-irregular eigenvalue problems. In a subsequent paper it will be shown that this irregular behavior of G(x,ζ,λ) implies that only a very small class of functions can be expanded into a series of eigenfunctions of such problems.