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  • Applied Mathematics  (85)
  • Wiley-Blackwell  (85)
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  • 1990-1994  (85)
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  • 1990  (85)
  • 1
    Electronic Resource
    Electronic Resource
    Some decomposition theorems and their application to non-linear potential theory and Hodge theory (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 35-52 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The paper considers Dirichlet (or Neumann type) boundary value problems of generalized potential theory \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm d}\alpha = f,\;\delta \varepsilon \left(\alpha \right) = g\,{\rm in}\,M, $$\end{document} \documentclass{article}\pagestyle{empty}\begin{document}$$ \alpha = 0\,{\rm on}\,\partial M $$\end{document} on Lipschitz manifolds with boundary. Here ∊ denotes a permissible non-linearity. The existence theory is developed in the framework of monotone operators. The approach covers a variety of applications including fluid dynamics and electro- and magneto-statics. Only fairly weak regularity assumptions are required (e.g. Lipschitz boundary, L∞-coefficients). As a by-product we obtain a non-linear Hodge theorem generalizing a result by L. M. Sibner and R. J. Sibner (‘A non-linear Hodge-DeRham theorem’, Acta Math., 125, 57-73 (1970)).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670120103
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  • 2
    Electronic Resource
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    Time-invariant uniqueness for reacting viscous fluids near chemical equilibrium (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 137-141 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A system of equations is investigated which models a reacting mixture of viscous fluids acted upon by an external body force. Through logarithmic convexity arguments the solution is shown to be unique when either the initial or the final data are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130204
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  • 3
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    Remarks on the wave front of polarization: Polarized solutions for the evolution three-dimensional Lame system (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 181-188 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Some remarks complementary to the work of N. Dencker in [1] concerning the wave front of polarization are given. The theorem due to N. Dencker about the propagation of a wave front of polarization is applied to solutions for the evolution of the three-dimensional Lame system. The propagation of polarized solutions is established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130208
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  • 4
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    On a contact problem for a system of parabolic equations in the thermal conductivity of electric machines: II (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 249-261 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The non-linear contact problem for the parabolic system of second order in the sense of Pietrovski, which is the generalization of the problem considered in Part I (preceding paper), is formulated. The matrix of fundamental solutions for parabolic systems of second order with coefficients containing unknown functions and their first-order derivatives is constructed and used to reduce the problem to the equivalent system of integral equations which is then reduced to a system of Volterra type of the second kind. The existence of the solution of the system obtained is proved by using the Schauder fixed-point theorem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130306
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  • 5
    Electronic Resource
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    The scattering of electromagnetic waves by a perfectly conducting infinite cylinder (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 503-518 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the scattering of a plane time-harmonic electromagnetic wave by a perfectly conducting infinite cylinder with axis in the direction k, where k is the unit vector along the z axis. Suppose the incident wave propagates in a direction perpendicular to the cylinder. For a given observation angle θ, let FD(θ, α)k be the far-field pattern of the electric field corresponding to an incident wave with direction angle α polarized perpendicular to the axis and let FN(θ; α)k be the far-field pattern of the magnetic field corresponding to an incident wave with direction angle α polarized parallel to the z axis. Let {αn}n=1∞ be a distinct set of angles in [ - π, π] and μ a complex number. Then, necessary and sufficient conditions are given for the set {(1 - μ)FD(θ;αn) + μFN(θ;αn)}n = 1∞ to be complete in L2[ - π, π]. Applications, together with numerical examples, are given to the inverse scattering problem of determining the shape of the cylinder from a knowledge of the far-field data.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670120605
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  • 6
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    Some solutions for the spin glass system and for systems originating from it (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 31-41 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: For the spin glass system the solutions, u = νΦ and u = νx/|x| are considered, where ν is a scalar function and Φε∝3 is constant. For a system originating from the spin glass one, a solution u = tan[q(|x|/t)]x/|x| is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130104
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  • 7
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130201
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  • 8
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    A study of a non-linear kinetic model for an ionized gas (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 95-109 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A non-linear one-dimensional kinetic model for an ionized gas is considered. In this model the collisions between charged and neutral particies and the field generated by the charges are taken into account. The Cauchy problem for the resulting system of equations is examined. An existence and uniqueness theorem of a global in time solution is proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130202
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  • 9
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    A two-species plasma in the limit of large ion mass (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 159-167 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Classical solutions of the relativistic Vlasov-Maxwell system are considered, describing a collisionless plasma with two species of particles. ions and electrons. It is shown that as the ion mass m tends to infinity, the corresponding solution of the relativistic Vlasov-Maxwell system tends to the solution of a system, in which the ions are given by a fixed ion background and only the electrons move. The convergence is uniform on compact time intervals, with an asymptotic convergence rate of m-1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130206
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  • 10
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    Asymptotic analysis of the quantum Liouville equation (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 143-157 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present an asymptotic analysis of the quantum Liouville equation with respect to the Planck's constant, which models the temporal evolution of the (quasi)distribution of an ensemble of electrons under the action of a potential. We consider two cases: firstly a smooth potential, and secondly a potential modelled by a δ-distribution. In both cases the zeroth-order term behaves classically. In the smooth case the classical Liouville equation is satisfied and in the case for the δ-potential an interface condition is derived, so that everything is reflected at the potential barrier.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130205
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  • 11
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130101
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  • 12
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    An existence theorem for Poiseuille flow with surface tension in an open channel (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 23-30 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Poiseuille-type flow in an open channel is studied taking surface tension and wall adhesion into account. A proof is presented of the existence of a solution under certain conditions.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670130103
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  • 13
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    Low-frequency asymptotics for dissipative Maxwell's equations in bounded domains (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 81-93 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Consider a bounded domain Ω surrounded by a perfect conductor and containing a conducting cavity D. The behaviour of the solutions of the time harmonic Maxwell problem as frequency tends to 0 is analysed in this situation. Necessary and sufficient conditions on the excitations are given which guarantee the existence of a limit. This limit turns out to be the solution of some static problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130107
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  • 14
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    Zero resonances in local perturbations of parallel-plane waveguidesHerrn Professor Rolf Leis zum 60. Geburtstag am 22·7·1991 gewidment (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 111-136 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In a joint paper with K. Morgenröther [9] we have studied the propagation of scalar waves in domains of the type Ω = Ωo-B̄ with Ωo:=∝2 × (0, 1), where B is a smooth bounded domain with B̄ ⊏ Ωo. In particular, we have shown that the solution of Neumann's initial and boundary value problem for the wave equation with time-independent right-hand side ƒ increases with a logarithmic rate as t → ∞ if the integral over ƒ does not vanish. The main purpose of the present paper is to extend this result to arbitrary smooth local perturbations of Ωo.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130203
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  • 15
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    On the ‘one and one-half dimensional’ relativistic Vlasov-Maxwell system (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 169-179 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The time evolution of a collisionless plasma is studied in the case when the Viasov density ƒ is a function of the time, one space variable and two velocity variables. The electromagnetic fields E, B also have a special structure, and the magnetic field B is non-trivial. It is shown that smooth, consistent initial values generate a uniquc smooth global solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130207
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  • 16
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    On the uniqueness of motion of viscous gaseous stars (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 391-404 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of solutions of the evolutionary equations of motion of a star regarded as a compressible viscous fluid with self-gravitation, bounded by a free surface, has recently been considered [3, 4]. In this paper I study the uniqueness of the above solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130504
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  • 17
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    An optimal control problem for the Helmholtz equation in a stratified Shallow oceanThis research was supported in part by the U.S. AFOSR86-0269. (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 405-419 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we shall consider a kind of optimal control of the Robin problem. First, we discuss the solvability of the Robin problem, which is not uniquely solvable in general. Then we find the representation of the far-field pattern in terms of the minimal solution to an integral equation. Based on this representation, an optimization problem is discussed and a Galerkin scheme for the solution is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130505
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  • 18
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    Higher moments for kinetic equations: The Vlasov-Poisson and Fokker-Planck cases (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 441-452 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let us consider a solution f(x,v,t)∊L1(∝2N × [0,T]) of the kinetic equation where |v|α+1 fo,|v|α ∊L1 (∝2N × [0, T]) for some α〈 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set Kx, we haveWe use this result to improve the regularity of the local density ρ(x,t) = ∫ƒdν for the Vlasov-Poisson equation, which corresponds to g = Eƒ, where E is the force field created by the repartition ƒ itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L2 belongs to L2([0, T]; H1/2(∝x,v2N)).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130508
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  • 19
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    On the mathematical foundations of hyperthermia therapy (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 467-479 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Hyperthermia is an aid to standard cancer therapy. The aim is to heat up a tumour region inside the patient's body with the help of microwave radiation. The microwayes are generated by some suitable antenna array. Here we investigate the control properties of such antenna configurations and determine their resolution capabilities. We will define the hyperthermia operator, which can produce every field of any antenna array up to any degree of accuracy. We shall see that this operator is compact and we are able to compute a singular system under general assumptions explicitly. This, together with a detailed knowledge of the decay of the singular values, constitutes the tools with which we treat the previously stated problems. We examine the propagation of microwave radiation in homogeneous media and then carry over the results obtained to the case of inhomogeneous media.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130603
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  • 20
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    Non-homogeneous non-linear damped wave equations in unbounded domainsSupported by the SFB 256 of the Deutsche Forschungsgemeinschaft at the University of Bonn, F.R.G. (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 481-491 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present a global existence theorem for solutions of utt - ∂iaik (x)∂ku + ut = ƒ(t, x, u, ut, ∇u, ∇ut, ∇2u), u(t = 0) = u0, ut(=0)=u1, u(t, x), t ⋝ 0, x∊Ω.Ω equals ∝3 or Ω is an exterior domain in ∝3 with smoothly bounded star-shaped complement. In the latter case the boundary condition u|∂Ω = 0 will be studied. The main theorem is obtained for small data (u0, u1) under certain conditions on the coefficients aik.The Lp - Lq decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130604
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  • 21
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    Global solutions of Yang-Mills-Higgs systems on four-dimensional Minkowski space: I. Existence (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 153-165 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of global solutions to the Cauchy problem for Yang-Mills-Higgs systems in the temporal gauge is examined. The results apply to the self-coupling of the Higgs field. Results from other works are strengthened.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670120206
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  • 22
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    Global solutions of Yang-Mills-Higgs systems on four-dimensional Minkowski space: II. Some properties of solutions (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 167-176 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The paper concerns the properties of global solutions to the Cauchy problem of Yang-Mills-Higgs systems in the temporal gauge. The following properties are considered: global analyticity of solutions, an approximation property, conservation of the topological class of solutions, scattering of a part of the energy of solutions through a light cone.
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    URL: http://dx.doi.org/10.1002/mma.1670120207
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  • 23
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    Continuous dependence on data for solutions of a conducting dusty gas model backwards in time (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 183-197 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A conducting dusty gas flow in the presence of a given applied magnetic field is modelled by an initial boundary value problem. The continuous dependence of solutions backward in time to this model is studied. The electric and induced magnetic fields are assumed to be negligible and the dust particles are non-conducting. A uniqueness result is also derived.
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    URL: http://dx.doi.org/10.1002/mma.1670120302
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  • 24
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    The propagation problem and far-field patterns in a stratified finite-depth ocean (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 199-208 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we investigate the direct problem associated with the scattering of ‘plane waves’ from an object submerged in an ocean of finite depth. An integral representation for the Dirichlet problem is found, from which a formula for the far-field pattern evolves.A density theorem is established concerning the set of all far-field patterns. This theorem is essential for the reconstruction of the submerged object, the ‘inverse’ problem [2], [4], [5].
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    URL: http://dx.doi.org/10.1002/mma.1670120303
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  • 25
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    Blow-up in non-linear three-dimensional thermoelasticity (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 267-273 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: It is shown that solutions of the equations of thermoelasticity in general will develop singularities in finite time.
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    URL: http://dx.doi.org/10.1002/mma.1670120308
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  • 26
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    On some quasilinear hyperbolic-parabolic initial boundary value problems (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 315-339 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove a local existence result for a coupled hyperbolic-parabolic initial boundary value problem.
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    URL: http://dx.doi.org/10.1002/mma.1670120404
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  • 27
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670120501
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  • 28
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    On a boundary value problem in the theory of linear water waves (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 393-404 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A body Ω floating in a fluid is subjected to small periodic displacement. Under idealized conditions the resulting wave pattern can be described by a linear boundary value problem for the Laplacian in an unbounded domain with a non-coercive boundary condition on part of the boundary. Nevertheless uniqueness can be shown if Ω is confined to certain subsets of the fluid which can be described explicitly. This extends a result of V. G. Maz'ja saying that uniqueness holds provided that the exterior normal for ∂Ω avoids certain directions.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670120504
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  • 29
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    A uniqueness condition for the polyharmonic equation in free space (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 275-291 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Consider the polyharmonic wave equation ∂t2u + (- Δ)mu = f in ∝n × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u (x, t) as t → ∞ and show that u(x, t) either converges or increases with order tα or In t as t → ∞. In the first case we study the limit \documentclass{article}\pagestyle{empty}\begin{document}$ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $\end{document} and give a uniqueness condition that characterizes u0 among the solutions of the polyharmonic equation ( - Δ)mu = f in ∝n. Furthermore we prove in the case 2m ≥ n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.
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    A solution to the equations of shell theory (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 341-363 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper the Vekua-Dikmen formulation of the equations of shell theory as a heirarchy are presented as a hypercomplex system. Using the function theory associated with this algebra the torsion problem is solved as an integral representation.
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    Control of continuous sedimentation of ideal suspensions as an initial and boundary value problem (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 533-548 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We present the control of continuous sedimentation in an ideal thickener as an initial and boundary value problem and construct the entropy solution.
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    Global weak solutions to the problem of continuous sedimentation of an ideal suspension (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 1-22 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We present the sedimentation of an ideal suspension in a continuous thickener as an initial value problem and study the transient states by constructing the corresponding global weak solutions. Finally, we obtain a way to control the operation of the thickener.
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    A one-dimensional dynamic contact problem in linear viscoelasticity (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 55-79 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The initial-boundary value problem associated with a linear viscoelastic bar which moves along its axis and against a stationary obstacle perpendicular to the axis is discussed. The existence of a solution is established by the penalty method and a multiplier technique. The uniqueness of the solution is also proved in a special case.
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    An effective direct solution method for certain boundary element equations in 3D (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 43-53 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The boundary element method for the Dirichlet problem in a three-dimensional rotational domain leads to a system of linear equations with a full dense matrix having a special block structure. A direct solution method for such systems is presented, which requires O(N3/2 ln N) arithmetical operations only, using a Fast Fourier Transformation (FFT), where N denotes the number of unknowns on the boundary surface.
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    Long-time behaviour of absorbing boundary conditions (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 189-203 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: A new class of computational far-field boundary conditions for hyperbolic partial differential equations was recently introduced by the authors. These boundary conditions combine properties of absorbing conditions for transient solutions and properties of far-field conditions for steady states. This paper analyses the properties of the wave equation coupled with these new boundary conditions: well-posedness, dissipativity and convergence in time.
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990) 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Abstract Riccati equations in an L1 space of finite measure and applications to transport theory (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 53-67 
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    Notes: We consider operator-valued Riccati initial-value problems of the form R′(t) + TR(t) + R(t)T = TA(t) + TB(t)R(t) + R(t)TC(t) + R(t)TD(t)R(t), R(0) = R0. Here A to D and R0 have values as non-negative bounded linear operators in L1 (μ), where μ is a finite measure, and T is a closed non-negative operator in L1 (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.
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    Dense sets and the projection theorem for acoustic harmonic waves in a homogeneous finite depth ocean (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 69-76 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The scattering of a ‘plane wave’ off a submerged body situated in an ocean of finite depth is investigated. The index of refraction is considered to be depth-independent. It is shown that the far field is not unique; hence, the problem of determining the shape of an object from its far field is not well-posed. If solutions are sought among a restricted class of problems the ‘dense set’ property implies that the problem can be made well-posed.
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    Partition of energy in strongly damped generalized wave equations (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 95-103 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Dissipative perturbations of hyperbolic equations such as utt + But + A2u = 0 with positive operators A, B are considered. The rates of decay and partition of energy theorems are established for solutions of these equations.
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    Linear stability condition for explicit Runge-Kutta methods to solve the compressible Navier-Stokes equations (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 139-151 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: A linear stability condition is dervied for explicit Runge-Kutta methods to solve the compressible Navier-Stokes equations by central second-order finite-difference and finite-volume methods. The equations in non-conservative form are simplified to quasilinear form, and the eigenvalues of the resulting coefficient matrices are determined for general co-ordinates. Assuming a well-posed Cauchy problem with constant coefficients, the von Neumann stability analysis yields sufficient stability conditions for viscous-inviscid operator-splitting schemes. They have been applied in computational aerodynamics to solve the compressible Navier-Stokes equations by an unsplit explicit Runge-Kutta finite-volume method.
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    Examples of embedded eigenvalues for the Dirichlet-Laplacian in domains with infinite boundaries (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 177-182 
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    Notes: For certain unbounded domains the Laplace operator with Dirichlet condition is shown to have an unbounded sequence of eigenvalues which are embedded into the essential spectrum. A typical example of such a domain is a locally perturbed cylinder with circular cross-section whose diameter in some bounded subset is greater than at infinity.
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990) 
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    Steady states in plasma physics - the Vlasov-Fokker-Planck equation (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 471-487 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we investigate the non-linear Vlasov-Fokker-Planck (VFP) equation, a both physically and mathematically interesting modification of Vlasov's equation, which describes a plasma in a thermal bath. We prove existence, uniqueness and representation results for steady states of the VFP equation both in the case of a mollified interaction potential and for the VFP-Poisson system. The uniqueness and representation results are of special interest since they distinguish special solutions of the Vlasov equation.
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    The Sommerfeld half-plane problem revisited, V: The bifurcated guide with mixed boundary conditions on the septumFrom a paper presented at the conference on ‘Wiener-Hopf Problems and Applications’ held in Oberwolfach, West Germany, 20-26, July 1986. (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 369-386 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We discuss the solution of the boundary value problem in a duct with a centered septum [9]. On the lower wall of the duct a Neumann condition is applied while on the upper wall a Dirichlet condition is applied. On the septum we apply a Dirichlet condition on the lower side and a Neumann condition on the upper one. This problem is formulated as a pair of integral equations of the Wiener-Hopf type for which we supply solutions for two modes of excitation as well as real and complex wave number. A critical examination is made of the construction, which reduces the problem to one in complex analysis. For real wave number, the physical parameters are provided in very simple forms.
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    On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singlular initial data (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 463-470 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We study the solutions of the Navier-Stokes equations when the initial vorticity is concentrated in small disjoint regions of diameter ∊. We prove that they converge, uniformily in ∊. for vanishing viscosity to the corresponding solutions of the Euler equations and they are connected to the vortex model.
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    On a problem in condensed two-phase combustion (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 519-531 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: An exothermic binary chemical reaction of first order is studied, assuming that one of the reactants can undergo a change of phase. In the limit of low activation energy, a one-dimensional problem is considered and the well-posedness of the corresponding system of two parabolic equations with a free boundary is proved in a classical sense.
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    The kinetics equations in a multiplying medium with free boundaries (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 263-272 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We study a mathematical model of neutron multiplication in a slab S, by taking into account temperature feedback effects and considering one group of delayed neutrons. The thickness 2a of S is time dependent because of temperature variations due to the energy released by fissions.Starting from a quite detailed picture of the physical phenomena occurring in S, we derive a system of three coupled ordinary differential equations for the total number of neutrons F̂ = F̂(t), for the total number of precursors Ĉ = Ĉ(t), and for the half-thickness of S, a = a(t).We finally examine some stability properties of such a system of ordinary differential equations.
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    On non-stationary flows of incompressible asymmetric fluids (1990)
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    Mathematical Methods in the Applied Sciences 13 (1990), S. 219-232 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We consider an initial boundary value problem for the system of equations describing non-stationary flows of incompressible asymmetric fluids. We prove the existence of a local in time, weak solution of the problem in the case when the initial density is not separated from zero by a positive constant.
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    Inversion of the exponential X-ray transform. II: Numerics (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 205-218 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The exponential X-ray transform arises in single photon emission computed tomography and is defined on functions on the plane by Pμf(ϕ,x) = ∫- ∞∞f (x + tϕ)eμt where μ is a constant. In [MMAS(10), 561-574, 1988], we derived analytical formulae for filters K corresponding to a general point spread function E that can be used to invert the exponential X-ray transform via a filtered backprojection algorithm. Here, we use those formulae to derive expressions suitable for numerical computation of the filters corresponding to a specific family of bandlimited point spread functions and give the results of reconstructions of a mathematical phantom using these filters. Also included is an analogue of the Shepp-Logan ellipse theorem, [IEEE Trans. Nucl. Sci. (21), 21-43, 1974], for the exponential X-ray transform.
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    Masthead (1990)
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    Mathematical Methods in the Applied Sciences 13 (1990) 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    A boundary element Galerkin method for a hypersingular integral equation on open surfaces (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 281-289 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Topics: Mathematics
    Notes: A hypersingular boundary integral equation of the first kind on an open surface piece Γ is solved approximately using the Galerkin method. As boundary elements on rectangles we use continuous, piecewise bilinear functions which vanish on the boundary of Γ. We show how to compensate for the effect of the edge and corner singularities of the true solution of the integral equation by using an appropriately graded mesh and obtain the same convergence rate as for the case of a smooth solution. We also derive asymptotic error estimates in lower-order Sobolev norms via the Aubin-Nitsche trick. Numerical experiments for the Galerkin method with piecewise linear functions on triangles demonstrate the effect of graded meshes and show experimental rates of convergence which underline the theoretical results.
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    Masthead (1990)
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    Mathematical Methods in the Applied Sciences 13 (1990) 
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    The initial boundary value problem for the Schröudinger equation (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 385-390 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We study the stability properties of the one-dimensional Schrödinger equation with boundary conditions that involve the derivative in the direction of propagation (or time). We show that this type of boundary condition might cause a strong growth of the amplitude of the solution. Such a model is not useful for numerical computations. One example is the parabolic wave equation in underwater acoustics for wave propagation in a downsloping duct with the normal derivative condition ∂u/∂n =0 at the bottom.
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    An asymptotic-numerical approach to the coronal loop problem (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 431-439 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: The coronal loop problem is characterized by mixed boundary conditions and the loop length condition, which is global. Using singular perturbation methods one can identify and construct two boundary layers at the base of the loop.Extending this to a combined asymptotic-numerical treatment it is possible to construct two static solutions satisfying the same conditions; this unusual feature arises from the presence of the first boundary layer, which corresponds with a steep temperature gradient and an energy balance dominated by conduction and radiation losses.
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    Advective estimates (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 453-465 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Consider the partial differential equation: wt + bw + a· ∇w = g in ∝n × [0, T] (x1 〈0) and if the function a1 is positive, then, in the left half-space, the solution w is an O (ν) for small Viscosity ν.
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    Asymptotic expansions for hyperbolic-parabolic systems with a small parameter (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 493-514 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we consider initial-boundary value problems for systems with a small parameter ∊. The problems are mixed hyperbolic-parabolic when ∊ 〉 0 and hyperbolic when ∊ = 0. Often the solution can be expanded asymptotically in ∊ and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers.
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990) 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    On a contact problem for a system of parabolic equations in the thermal conductivity of electric machines: I (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 233-248 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Thermal phenomena arising in some electric motors are described (physical model) and an adequate mathematical model is constructed. The model, in the form of a contact problem, is then approximated in a numerical way. The stability of the difference scheme obtained is investigated.
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    URL: http://dx.doi.org/10.1002/mma.1670130305
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    Remarks on non-trivial solutions of a Volterra integral equation (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 273-279 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper non-linear integral equations describing shock wave phenomena are presented. Some necessary and sufficient conditions for the existence of non-trivial solutions to equations of this type are given.
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    URL: http://dx.doi.org/10.1002/mma.1670130308
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    Dissipative Maxwell's equations at low frequencies (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 305-322 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A study of the low-frequency behaviour of solutions to dissipative Maxwell's equations with discontinuous coefficients is given.
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    URL: http://dx.doi.org/10.1002/mma.1670130404
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    Minimal-energy splines: I. Plane curves with angle constraints (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 351-372 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This is the first in a series of papers on minimal-energy splines. The paper is devoted to plane minimal-energy splines with angle constraints. We first consider minimal-energy spline segments, then general minimal-energy spline curves. We formulate problems for minimal-energy spline segments and curves, prove the existence of solutions, justify the Lagrange multiplier rules, and obtain some nice properties (e.g., the infinite smoothness). Finally, we report our computational experience on minimal-energy splines.
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    URL: http://dx.doi.org/10.1002/mma.1670130407
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670120201
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    A hyperbolic singular perturbation of Burgers' equation (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 77-90 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper is concerned with the effect of perturbing Burgers' equation by a small term ∊2 Utt. It is shown by means of an energy estimate that the solution of Burgers' equation provides a uniform O (∊) approximation of the solution of the full hyperbolic problem. Existence and uniqueness of classical solutions for both problems is proved. A related linear problem is first addressed using the Faedo-Galerkin method to obtain key estimates. Important for the hyperbolic problem is the introduction of an ∊-dependent energy in order to track the order-∊ behaviour of various higher-order derivatives. Subsequent use of Schauder technique and Banach contraction mapping principle yields solutions of the semilinear problems.
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    A constructive method for solving the displacement boundary-value problem of elastostatics by use of global basis systemsDedicated to Claus Müller on the occasion of his 70th birthday. (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 105-128 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we are concerned with the three-dimensional homogeneous isotropic elastostatic boundary-value problem. Global basis vector fields are shown to be constructive structures for approximating the solution. The theoretical tool is a regularity condition developed from the classical integral equation approach. The paper ends with some reflections on the practical computability and numerical applicability.
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    Singular integral equations applied to free surface seepage from non-linear channels (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 209-219 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The problem of two-dimensional seepage from a non-linear channel through a homogeneous medium underlain at a finite depth by a drain will be considered. A new approach is given, transforming the seepage problem to a non-linear singular integral equation for which the unique existence of the solution is proved. The free boundaries of the seepage region can be described explicitly.
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    URL: http://dx.doi.org/10.1002/mma.1670120304
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    On the exponential convergence of the h-p version for boundary element Galerkin methods on polygons (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 413-427 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper applies the technique of the h-p version to the boundary element method for boundary value problems on non-smooth, plane domains with piecewise analytic boundary and data. The exponential rate of convergence of the boundary element Galerkin solution is proved when a geometric mesh refinement towards the vertices is used.
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    On a free boundary problem in electrostatics (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 387-392 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let Ωi ⊂ ∝N, i = 0, 1, be two bounded separately star-shaped domains such that \documentclass{article}\pagestyle{empty}\begin{document}$ \Omega _0 \supset \bar \Omega _1 $\end{document}. We consider the electrostatic potential u defined in \documentclass{article}\pagestyle{empty}\begin{document}$ \Omega : = \Omega _0 \backslash \bar \Omega _1 $\end{document}: \documentclass{article}\pagestyle{empty}\begin{document}$$ \Delta u = 0\,{\rm in}\,\Omega,\,u = 0\,{\rm on}\,\Gamma _{\rm 0} : = \partial \Omega _0,\,u = 1\,\Gamma _1 : = \partial \Omega _1. $$\end{document} The geometry of the two boundary components Γ0 and Γ1 is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.
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    On critical exponents for some quasilinear parabolic equations (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 429-438 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study the Cauchy problem for the quasilinear parabolic equation where p 〉 1 is a parameter and ψ is a smooth, bounded function on (1, ∞) with - ≤ sψ′(s)/ψ(s) ≤ θ for some θ 〉 0. If 1 〈 p 〈 1 + 2/N, there are no global positive solutions, whereas if p 〉 1 + 2/N, there are global, positive solutions for small initial data.
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    An improved boundary element method for the charge density of a thin electrified plate in ∝3 (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 291-303 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A weakly singular integral equation of the first kind on a plane surface piece Γ is solved approximately via the Galerkin method. The determination of the solution of this integral equation (with the single-layer potential) is a classical problem in physics, since its solution represents the charge density of a thin, electrified plate Γ loaded with some given potential. Using piecewise constant or piecewise bilinear boundary elements we derive asymptotic estimates for the Galerkin error in the energy norm and analyse the effect of graded meshes. Estimates in lower order Sobolev norms are obtained via the Aubin-Nitsche trick. We describe in detail the numerical implementation of the Galerkin method with both piecewise-constant and piecewise-linear boundary elements. Numerical experiments show experimental rates of convergence that confirm our theoretical, asymptotic results.
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    Continuous dependence on the heat source and non-linear stability for convection with internal heat generation (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 373-383 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: It is shown that the solution to the boundary-initial value problem for a heat-conducting viscous fluid depends continuously on changes in the heat supply function, for the improperly posed backward in time problem. A non-linear convection threshold is also determined for the problem of a layer of fluid heated internally (non-uniformly), with zero neat flux on the lower boundary and constant-temperature upper surface.
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    Determination of an unknown function in a parabolic equation with an overspecified conditionThis work was funded by NSERC Canada. (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 421-430 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we consider the inverse problems of identifying some space-dependent unknown coefficients in parabolic equations subject to initial boundary value conditions along with an overspecified condition at the final time t = T. We use the overspecified information to transform the problems into non-linear parabolic equations involving a functional of the solution with respect to the time variable. This transformation allows us to establish existence theorems for these inverse problems by employing the Schauder fixed-point theorem.
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    On the instability of resonances in parallel-plane waveguides under local perturbations of the boundaryHerrn Professor Erhard Meister zum 60. Geburtstag am 12.2.1990 gewidmet (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 1-34 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study the large-time asymptotics for solutions u(x, t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω0 ≔ ∝2 × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω0. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω0 at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ2U = - f in Ω, U = 0 on ∂Ω is reduced to a compact operator equation.
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    Examples of embedded eigenvalues for the Dirichlet Laplacian in perturbed waveguides (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 91-93 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: It is shown that there exist domains Ω ⊂ ∝N, which outside of some ball coincide with the strip ∝N - 1 × (0, π) and for which the Dirichlet Laplacian - Δ has eigenvalues within the subinterval (1, 4) of the essential spectrum (1, ∞).
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    A free boundary problem and stability for the rectangular plate (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 129-138 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the recent paper [13] we have answered the question of stability for the linear circular plate which is being axially compressed by a force greater than the critical value and contacts a plane obstacle. In this case there are radially symmetric solutions and the contact region is a disk of a smaller radius. This simplified the determination of the critical parameter values for which the plane jumps to another state. For the rectangular plate continuation has to be applied to the variational inequality in order to determine the contact region and evalute the stability criterion. A numerical method is developed for a discretization of the problem and is used to compute the critical load both in the simply supported and the clamped case.
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    Masthead (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Solution of a class of partial differential equations. Some diffusion equations (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 221-228 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article we present the solution of linear partial differential equations of the form ∂tf = L̂f, for initial value problems. Also the solution of some diffusion equations will be discussed.
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    Regularity of mixed boundary value problems in ∝3 and boundary element methods on graded meshes (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 229-249 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The solution of the three-dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A ‘tensor-product’ decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence of the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence.
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    An adapted Galerkin method for the resolution of Dirichlet and Neumann problems in a polygonal domain (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 251-265 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Neumann problem for Laplace's equation in a polygonal domain is associated with the exterior Dirichlet problem obtained by requiring the continuity of the potential through the boundary. Then the solution is the simple layer potential of the charge q on the boundary. q is the solution of a Fredholm integral equation of the second kind that we solve by the Galerkin method. The charge q has a singular part due to the corners, so the optimal order of convergence is not reached with a uniform mesh. We restore this optimal order by grading the mesh adequately near the corners. The interior Dirichlet problem is solved analogously, by expressing the solution as a double layer potential.
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    Junction between a 3-D Stokes flow and a 2-D thin film with pressure boundary conditions (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 293-314 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A 3-D Stokes equation modelling the flow in the inlet region of a lubricated device is studied. Boundary conditions on both the pressure and the velocity are considered. The geometry induces that the flow region is divided into two parts: the first one is a thin gap with height ∊h, where a change of scale is performed, and the second one is a cavity whose dimensions are large with respect to ∊. Two equivalent variational formulations are proposed and lead to accurate a priori estimates. The resulting asymptotic equation for the pressure in the thin gap is the currently used Reynolds equation, whose boundary conditions are obtained via the junction with the 3-D cavity. Both hydrostatic and hydrodynamic cases are considered.
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    A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 365-368 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let u be a vector field on a bounded Lipschitz domain in ∝3, and let u together with its divergence and curl be square integrable. If either the normal or the tangential component of u is square integrable over the boundary, then u belongs to the Sobolev space H1/2 on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solutions of Maxwell's equations on Lipschitz domains into L2.
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    Radially symmetric Poisson-Boltzmann equation in a domain expanding to infinity (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 405-412 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The electric potential u in a solution of an electrolyte around a linear polyelectrolyte of the form of a cylinder satisfies \documentclass{article}\pagestyle{empty}\begin{document}$$ \left({ru'} \right)^\prime = rf\left(u \right),\;0 〈 a 〈 r 〈 R, $$\end{document} \documentclass{article}\pagestyle{empty}\begin{document}$$ u'\left(a \right) = - \sigma /a,\;u\left(R \right) = 0. $$\end{document} We study the problem when R → ∞.
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    Some new fundamental solutions (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 439-461 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The fundamental solutions of non-decomposable evolution operators are represented by multi-dimensional parameter integration formulae. The method is applied to operators occurring in the theories of elasticity, magnetohydrodynamics and heat conduction.
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    URL: http://dx.doi.org/10.1002/mma.1670120508
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    On the numerical treatment of an integral equation arising from a cruciform crack problem (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 489-502 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the following two integral equations the latter subject to the condition ∫-11v(y)dy = 0, arising from the same problem of determining the distribution of stress in a thin elastic plate in the vicinity of a cruciform crack. In particular, we prove existence and uniqueness of the solutions of the two equations above in C[0, 1]. We also analyse the behaviour of φ(x) near the end-point x = 0.The therotical result obtained in the first part are then used in the last section to derive optimal rates of convergence for numerical methods, such as Galerkin, collocation and Nyström, applied to equation (1).
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