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  • Food Science, Agricultural, Medicinal and Pharmaceutical Chemistry
  • Mathematics and Statistics
  • Wiley-Blackwell  (744)
  • Copernicus
  • 1985-1989  (703)
  • 1975-1979  (41)
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  • Wiley-Blackwell  (744)
  • Copernicus
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Year
  • 1
    Electronic Resource
    Electronic Resource
    On the low wave number asymtotics for the two-dimensional exterior dirichlet problem for the reduced wave equationDedicated to E. Martensen on the occasion of his 60th birthday (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 335-341 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new proof is given for the low wave number asymptotics for the solutions to the exterior Dirichlet problem for the reduced wave equation in two dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670090126
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  • 2
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    Diffraction d'une Onde Acoustique par une Paroi Absorbante: Nouvelles Equations Intégrales (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 431-454 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present here a new formulation of the exterior Robin problem of the reduced wave equation, using boundary integrals with the jumps of the function and its normal derivative as variables. Our system of integral equations present an important advantage for any numerical purpose: it can be transformed in a variational problem where the sesqui-linear form is coercive and is exempt of hyper singular integrals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670090131
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  • 3
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    An integral equation of the first kind for a free boundary value problem of the stationary Stokes' equations (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 550-575 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We investigate a free boundary value problem of the stationary Stokes' equations. In a previous paper adapted hydrodynamical potentials have been constructed and their jump relations have been discussed. Here we study a direct method to obtain an equivalent boundary integral equations' system of the first kind. Its solution properties are investigated in the framework of strongly elliptic pseudodifferential operators. For numerical purposes a suitable representation formula for the variational equation is given in terms of integro-differential operators which avoids the evaluation of hypersingular integrals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670090135
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  • 4
    Electronic Resource
    Electronic Resource
    Masthead (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988) 
    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.1670100101
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  • 5
    Electronic Resource
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    Explicity resolvable equations with functional non-linearities (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 477-485 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the non-linear evolution equation μdu/dt+Bu+n(u)L(u)=h, where n(u) is a functional, and introduce assumptions which allow its explicit solution. Several concrete applications of our procedure are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100411
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  • 6
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    Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 197-223 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper a thermomechanical model for the dynamics of structural phase transitions in the so-called ‘shape memory alloys’ is developed. These materials exhibit rather spectacular hysteresis phenomena. The resulting mathematical model consists of a coupled and highly non-linear system of partial differential equations reflecting the balance laws of linear momentum and energy. For an appropriate weak formulation the local-in-time existence of weak solutions is shown.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100302
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  • 7
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    Explicit formula for scalar non-linear conservation laws with boundary condition (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 265-287 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form \documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{\partial }{{\partial t}}u + \frac{\partial }{{\partial x}}f\left(u \right) = 0 $$\end{document}, with initial data and boundary condition. The scalar function u = u(x, t), x 〉 0, t 〉 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ∊ ∝ \documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{\partial }{{\partial t}}\left({x^\alpha u} \right) + \frac{\partial }{{\partial x}}\left({x^\alpha \frac{{u^2 }}{2}} \right) = 0 $$\end{document}.We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100305
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  • 8
    Electronic Resource
    Electronic Resource
    Masthead (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988) 
    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.1670100401
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  • 9
    Electronic Resource
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    Unilateral contact with adherence (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 367-381 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The dissipative adhesive properties of a viscoelastic solid are investigated. An approximate problem is considered and an existence theorem is proved. Numerical computations are presented.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100403
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  • 10
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    A dual method to the collocation method (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 439-445 
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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 consider methods which are related to the collocation method by interchanging the test and the trial spaces. Error estimates are derived. As a by-product we obtain some extensions to the known convergence results for the collocation method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100408
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  • 11
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    A stability result for distributed parameter identification in bilinear systems (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 447-456 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we give a stability result for the problem of identifying distributed parameters (for instance, a conductivity matrix) in partial differential equations. The result is used to establish an error estimate for an algorithm to identify the parameter.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100409
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  • 12
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    Time-periodic solutions of wave equations on ∝1 and ∝3 (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 457-475 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The classical Paiey-Wiener theorem and Hilbert space methods are used to show the existence of time-periodic solutions of the wave equation wtt-wrr+λw=h, 0 〈 r 〈 + ∞, which are radially symmetric and have exponential decay as r → + ∞. This problem is obtained when considering a one-dimensional or three-dimensional problem, and should be thought of as a linearization of a semilinear problem in which the associated linear operator has point spectrum (- ∞, λ). When λ ≤ 0 there is uniqueness, otherwise there is a non-trivial finite-dimensional null space. Estimates on w, wt, wr are obtained, which show that in either case there is a continuous correspondence h → w, where w is a uniquely characterized solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100410
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  • 13
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    Unique continuation for Schrödinger operators and for higher powers of the Laplacian (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 531-542 
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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 unique continuation property for Schrödinger operators and its application for proving the non-existence of positive eigenvalues (embedded in the continuous spectrum). We also use the estimate given by Jerison and Kenig9 to prove unique continuation for higher powers of the Laplace operator.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100504
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  • 14
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    The optimal stability estimate for some ill-posed Cauchy problems for a parabolic equation (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 575-583 
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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 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 = ∝ or I = ∝+ and u(x, 0) = 0, x ∊ [0, l], in the case I = ∝+. Assuming an a priori bound for ‖u(l,.)‖L2, we derive the exact Hölder type dependence of on ‖u(x,.)‖L2 on ‖ϕ‖L2.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100507
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  • 15
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    Inequalities related to stationary flows of granulated media (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 27-41 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We investigate some variational inequalities associated with the equations of the stationary motion of granulated media with a constant density. These inequalities replace the usual equations of motion in the case when some additional constraints are imposed on the flow.We prove the existence of solutions of the inequalities, study their regularity, uniqueness, dependence on data and relations to solutions of the equations of motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110102
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  • 16
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    Some remarks on Saint-Venant's principle (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 71-77 
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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 a three-dimensional hyperelastic cylinder in Ω = D × [0, ∞]. We study the asymptotic behaviour of the deformations of the cross-sections in an equilibrium state. In this case we show that the solutions either have exponential decay or exponential growth. We give some initial conditions such that the latter case occurs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110105
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  • 17
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    On the continuation for variational inequalities depending on an eigenvalue parameter (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 95-104 
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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 generalize recent theoretical results on the local continuation of parameter-dependent non-linear variational inequalities. The variational inequalities are rather general and describe, for example, the buckling of beams, plates or shells subject to obstacles. Under a technical hypothesis that is satisfied by the simply supported beam, we obtain the existence of a continuation of both the solution and the eigenvalue with respect to a local parameter. A numerical continuation method is presented that easily overcomes turning points. Numerical results are presented for the non-linear beam.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110107
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  • 18
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    Existence result and discontinuous finite element discretization for a plane stresses Hencky problem (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 169-184 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We hereafter propose and analyse a discontinuous finite element method for a plane stress Hencky problem. For that purpose we begin by proving an existence result for the continuous problem. A kind of Green's formula between \documentclass{article}\pagestyle{empty}\begin{document}$$ BD\left(\Omega \right) = \left\{{u \in {\rm{L}}^1 \left(\Omega \right),\varepsilon _{ij} (u) \in M_1 \left(\Omega \right)} \right\}{\rm{and}}H\left(\Omega \right) = \left\{{\sigma \in L^\infty \left(\Omega \right),div\sigma \in {\rm{L}}^2 \left(\Omega \right)} \right\} $$\end{document} and other intermediate results that may be of independent interest are presented and established separately. Then we formulate the discretized problem, give an existence result for it and prove a result of weak convergence of a subsequence of discrete solutions to a solution of the continuous problem.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110202
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  • 19
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    Quasilinear hyperbolic-hyperbolic singular perturbation problem: Study of shock layer (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 237-252 
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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 the asymptotic analysis of a quasilinear hyperbolic-hyperbolic singular perturbation problem in one dimension. The leading part of the analysis concerns the construction of some shock layers associated with discontinuities of a hyperbolic problem. This study is a generalization of the case of viscous perturbation for a hyperbolic problem.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110205
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  • 20
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    Anisotropic Hele-Shaw flows (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 417-429 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The procedure of Elliott and Janovsky is abstracted to a fairly general variational problem. As an application of the methodology a non-isotropic Hele-Shaw flow is treated.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110401
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  • 21
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    A penalty approximation for a unilateral contact problem in non-linear elasticity (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 447-458 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Using Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear-elastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials.Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110403
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  • 22
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    Stationary solutions of the non-linear Boltzmann equation in a bounded spatial domain (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 471-481 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A contraction mapping (or, alternatively, an implicit function theory) argument is applied in combination with the Fredholm alternative to prove the existence of a unique stationary solution of the non-linear Boltzmann equation on a bounded spatial domain under a rather general reflection law at the piecewise C1 boundary. The boundary data are to be small in a weighted L∞-norm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110405
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  • 23
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    On subsolutions of a non-linear diffusion problem (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 409-416 
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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 paper a subsolution of a non-linear diffusion problem in the radial case is constructed. Some integral equation methods are used.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110308
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  • 24
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    Three-dimensional stochastic vortex flows (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 431-445 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We discuss the stochastic structure of the Navier-Stokes flow in ∝3 and prove that it can be approximated by means of a finite-dimensional stochastic process. Such a process reduces to an algorithm already discussed in Reference 4 for the Euler case, when the viscosity coefficient vanishes.
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    URL: http://dx.doi.org/10.1002/mma.1670110402
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  • 25
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    Boundary control of a vibrating plate with internal damping (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 573-586 
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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 finite energy states of a vibrating viscoelastic plate of the Kelvin-Voigt type are, in general, not exactly controllable by L2-boundary controls. Accordingly, we present a result on approximative controllability. The method is general.
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    URL: http://dx.doi.org/10.1002/mma.1670110502
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  • 26
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    Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 697-723 
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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 a domain Ω in ∝n of the form Ω = ∝l × Ω′ with bounded Ω′ ⊂ ∝n-l. In Ω we study the Dirichlet initial and boundary value problem for the equation ∂t2 u + [(- ∂12 -… - ∂l2)m + (- ∂l+12 -… - ∂n2)m]u = fe-iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like tα (α rational, 0〈α〈1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.
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    URL: http://dx.doi.org/10.1002/mma.1670110601
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  • 27
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    Surface gradients and continuity properties for some integral operators in classical scattering theory (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 789-804 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The surface gradients of some of the most important boundary integral operators for the time-harmonic Helmholtz and Maxwell equations are computed. These results are used to give new and elementary proofs of the continuity properties of these boundary operators in Sobolev and Hölder spaces of arbitrary order.
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    URL: http://dx.doi.org/10.1002/mma.1670110605
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  • 28
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    Duality theorems in some overdetermined boundary value problems (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 805-819 
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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 several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110606
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  • 29
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    Optimal bounds for bilinear forms associated with linear equations (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 518-531 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The construction of upper and lower bounds to the bilinear quantity g0, f, where f is the solution of an operator equation Tf = f0, requires either an approximation for f or one for T-1. In this paper the question of “best” approximation of T-1 by an operator of the form B = βI, where β is a real constant, is investigated for linear operators that are either self-adjoint or can be related by suitable manipulations to others that are. Particular attention is paid to a special operator, previously studied by Robinson, of importance in predicting the dynamic polarisabilities of quantum-mechanical systems.
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    Wall effects on the slow steady motion of a particle in a viscous incompressible fluid (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 23-40 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section.In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.
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    Local existence of solutions of the vlasov-maxwell equations and convergence to the vlasov-poisson equations for infinite light velocity (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 533-558 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We prove the local existence of smooth solutions for the Vlasov-Maxwell equations in three space variables. The existence time for such solutions is independent of the light velocity c. Then we derive regularity results for both the Vlasov-Poisson and the Vlasov-Maxwell equations. The last part of the paper is devoted to a proof of weak and strong convergence of the Vlasov-Maxwell equations towards the Vlasov-Poisson equations, when the light velocity c goes to infinity.
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    Uniqueness for structural phase transitions in shape memory alloys (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 145-151 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The non-linear coupled equations arising from alloy mechanism \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {u_{tt} - a\left({u_x,\theta } \right)u_{xx} - \mu u_{xxt} - b\left({u_x,\theta } \right)\theta _x = f\left({x,t} \right),} \\ {c\left({u_x,\theta } \right)\theta _t - k\theta _{xx} - \alpha k\theta _{xxt} - \mu u_{xt}^2 - d\left({u_x,\theta } \right)u_{xt} = \lambda \left({x,t} \right),} \\\end{array} $$\end{document} have two important features: a may take negative values and c may be degenerate.The local existence has been proved in Reference 1, but the uniqueness was open. In this paper the uniqueness is proved. For a discussion of the physical model and for the justifications of the detailed technical assumptions to be made, we refer to Reference 1.
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    The capillary contact angle, II: The inclined plane (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 165-196 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In an earlier paper an hypothesis on the nature of the forces resisting motion of a liquid drop on a support surface II was introduced and tested in the (rotationally symmetric) case for which ll is a horizontal plane. In the present work II is chosen as an inclined plane, so that rotational symmetry fails. The hypothesis leads to a formal series development; in a restricted case the result is verified by a perturbation analysis, which yields the identical series.The series predictions are compared with results of computer calculations. A consequence of the analysis is an estimate for a Bohd number B0, such that a further increase in B would result in the fluid penetrating the support surface; thus an absolute upper bound B0, for stability of the envisaged configuration is obtained.
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    A free boundary problem of biological interest (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 79-93 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: During fertilization of certain echinoderms, a long actin-filled tube is extended by the sperm towards the interior of the egg. This yields a parabolic free boundary problem, which differs from the classical one-phase Stefan problem by the presence of a convective term in the partial differential equation, and because the equilibrium interface condition θ(s(t),t) = 0 is here replaced by a kinetic law s′(t) = vθ(s(t),t). This problem is set in variational form and the existence of a solution is proved by means of a Faedo-Galerkin approximation procedure.
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    Multiple scales expansions for eddy current computation in layered media (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 125-138 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: An asymptotic expansion is derived by the multiple scales method for the electromagnetic potential in a one-dimensional model problem for eddy current computations in periodic non-homogeneous layered media in presence of an integral constraint for each period.Error estimates for approximations with partial sums of any order are established. Numerical computations of correctors up to the fourth order are performed and the results are compared with the exact solution in some significant cases.
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    General conical singularities in three-dimensional Poisson problems (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 215-235 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Additional Material: 14 Ill.
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    Stability of linear systems of differential equations with variable structure and impulse effect (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 271-278 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In the present paper sufficient conditions for stability of the solutions of linear systems of differential equations with variable structure and impulse effect are found.
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    Continuous dependence on the time and spatial geometry for the equations of thermoelasticity (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 317-329 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we establish the stability of the solution of the standard initial-boundary value problem of linear anisotropic thermoelasticity under perturbations of the initial time geometry and of the spatial geometry. This is done by deriving appropriate explicit a priori inequalities which permit us to bound in particular the L2 integral of the perturbation in terms of some well defined measure of the perturbation in the geometry.
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    Analysis of Generalised Galerkin methods in the numerical solution of elliptic equations (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 3-15 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The Petrov-Galerkin projection method is outlined for the solution of the linear elliptic equation Lu = f with homogeneous boundary conditions. By choosing appropriate finite dimensional trial and test spaces, the methods of weighted residuals, collocation, and H1 Galerkin can be interpreted within the Petrov-Galerkin projection method framework.The important question of how best to choose the trial and test functions to suit a particular type of problem is then discussed. Objective criteria associated with the matrix which governs the Petrov-Galerkin numerical process are proposed.
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    The construction of solutions to the heat equation backward in timeDedicated to the memory of our “Doctor-Father” Professor Arthur Erdélyi. (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 32-39 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Solutions to the backwards heat equation are approximated by solutions of a pseudo-heat equation. Solutions to this modified equation are constructed by means of a fundamental solution and potential theory, and it is shown that the fundamental solution can be approximated by various expansions in special functions.
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    On the treatment of free boundary problems with the heat equation via optimal control (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 40-61 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: 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.
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    The convergence of the horizontal line method for Maxwell's equations (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 101-113 
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    Notes: The interior initial-boundary value problem for Maxwell's equations is considered for certain homogeneous boundary conditions and currents which satisfy corresponding compatibility conditions. Under the assumption that a solution exists, it is shown by means of suitable normed spaces that the sequence of discrete solutions obtained from the horizontal line method converges discretely to this solution. A priori estimations are given both for the discrete solutions and the solution of the given problem.
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    Masthead (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Error estimates for sampling approximation of non-bandlimited functionsThis research was carried out under the DFG „Schwerpunktprogramm Digitale Signalverarbeitung“ (Bu 166/29, 31). (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 127-137 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: A new proof of the sampling theorem for non-bandlimited signals is presented. It leads to better rates of convergence for the envolved approximation process. Furthermore, the order of convergence is given for an extended class of functions.
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    Hopf-Bifurkation bei Lasern (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 194-213 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The semiclassical equations describing a ring laser show two successive bifurcations, one stationary and one Hopf bifurcation. This phenomenon is analyzed mathematically. The initial value problem for the laser equations and the stability of the stationary solutions are discussed in detail. The transition to ultrashort laser pulses is shown to be a Hopf bifurcation. The direction of the bifurcation is determined for a numerical example. It turns out that it depends on the parameters of the system.
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    Masthead (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    On the integral equation method for the plane mixed boundary value problem of the Laplacian (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 265-321 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).
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    On a class of elliptic singular perturbations with applications in population genetics (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 432-441 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: With the maximum principle for differential equations asymptotic estimates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to the stationary solution of the Kolmogorov backward equation in population genetics.
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    Zum 70. Geburtstag von Henry Görtler (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 443-444 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    On the time-dependent Nernst-Planck equationsProf. Henry Görtler zum 70. Geburtstag (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 480-490 
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    Notes: For the one-dimensional Nernst-Planck equations of the theory of ion-transport in thin layers or membranes which mathematically represent a nonlinear coupled parabolic integro-differential system some global existence and uniqueness theorems for inner and outer problems are given using a priori estimates of the maximum principle type.
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    The direction of the secondary wave motion in the Ekman boundary layerDedicated to Professor Henry Görtler on the occasion of his 70th birthday (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 491-497 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The secondary wave motion in the free Ekman layer, on which a constant shear stress acts, is examined by an analytic method. The results show, that the waves move away from the center of rotation.
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    An existence theorem for the unmodified Vlasov Equation (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 530-544 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The initial value problem for the modified Vlasov equation with a mollification parameter δ 〉 0, as introduced by Batt, has a unique global solution in the weak sense whenever f0 ε L1 and f0 ≧ 0 λ-a.e. Assuming boundedness of f0 and boundedness of the kinetic energy, it is shown that, as δ → 0, there are subsequences δn → 0 such that the corresponding solutions converge weakly in the measure-theoretical sense. The limits are shown to be global weak solutions of the initial value problem for Vlasov's equation, and these solutions are seen to be weakly continuous with respect to t. For the plasma physical case, boundedness of the kinetic energy is a consequence of energy conservation.
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    A finite element method for the solution of a potential theory integral equation (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 581-587 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: This paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H1/2(Γ), frd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.
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    Finite elements for the eigenvalue problem of differential operators in unbounded intervals (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 1-39 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The eigenvalue problem for one-dimensional differential operators with a possible essential spectrum is discretized with finite elements defined on a bounded interval together with a fundamental system of the differential equation outside of the interval. A non-pollution property of the discrete spectra is proved and the error in the approximation of isolated eigenvalues and corresponding eigenvectors is estimated. The convergence of some numerical algorithms for the solution of the subsequent discrete nonlinear eigenvalue problem is proved. The method is tested in some numerical examples.
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    On spline collocation methods for boundary integral equations in the plane (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 74-89 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods.In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8].The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.
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    A method of iterative data refinement and its applications (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 108-123 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: Iterative data refinement (IDR) is a general procedure for producing a sequence of estimates of the data that would be collected by a measuring device which is idealized to a certain extent, starting from the data that are collected by an actual measuring device. Following a discussion of the fundamentals of IDR, we present a number of previously published procedures which are special cases of it. We concentrate on examples from medical imaging. In particular, we discuss beam hardening correction in x-ray computerized tomography, attenuation correction in emission computerized tomography, and compensation for missing data in reconstruction from projections. We also show that a standard method of numerical mathematics (the parallel chord method) as well as a whole family of constrained iterative restoration algorithms are special cases of IDR. Thus IDR provides a common framework within which a number of originally different looking procedures are presented and discussed. We also present a result of theoretical nature concerning the initial behavior of IDR.
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    Zur Asymptotik der wellengleichung und der wärmeleitungsgleichung in zweidimensionalen außenräumenHerrn Professor Claus Müller zum 65. Geburtstag am 20. 2. 1985 gewidmet. (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 170-201 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We study initial and boundary value problems for the wave equation and the heat equation with a time-independent right-hand term f in two space dimensions in the exterior of a closed curve C. In the case of Neumann's boundary condition ∂u/∂n = 0 on C, the solutions increase with a logarithmic rate as t → ∞ if ∫ fdx ≠ 0. In contrast to this, the solutions of the corresponding Dirichlet problems converge to the solution of the related static problem as t → ∞. In the case of the wave equation, these results have already been obtained by L: A. Muravei under the additional assumption that the curvature of C is positive, by using high frequency estimates for the reduced wave equation Δ U + ϰ2 U = 0. The analysis presented here is based on different methods, which can be applied to arbitrary smooth curves.
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    Asymptotic analysis of differential-delay equations and nonuniqueness of periodic solutions (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 223-237 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We study the differential-delay equation x′(t) = -αf(x(t-1)), where α is a positive parameter and f is an odd function which decays like x-r at infinity. In particular, we consider the case r ≤ 2, and prove the existence of periodic solutions with special symmetries which are different from previously known periodic solutions of minimal period 4. For r = 2 we prove sharp asymptotic estimates for the minimal periods of these solutions.Our results disprove a conjecture of R. D. Nussbaum.
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    An integral equation of elasticity (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 261-268 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In this work a certain mixed problem in R+3 for the Lamé equations in the theory of elasticity is reduced to the integral equationAn explicit formula for the solutions is given for either Ω = {x:x2 〉 0} or Ω = {x:∥x∥〈r}. The question of smoothness of the solutions is also discused. Another formulae on Ω = {x:∥x∥〈r} are found in [5], [4]. It seems that the techniques used below allows a deeper investigation of properties of the solutions to problem (*).
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    Addendum (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 330-331 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Relative influence of two small parameters in non uniformly elliptic homogenization problems (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 367-381 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We study three elliptic problems depending on two small parameters (∊ = homogenization parameter and δ = perturbation parameter which causes non uniform ellipticity). In each case, the homogenized operator corresponding to the second order operator is independent of the way (∊, δ) → (0, 0), but the convergence results and the limit solution do depend on the relative size of ∊ and δ; the relevant parameter is δ∊-2.
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    Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 385-415 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques.By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.
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    On the absolutely continuous spectrum of MHD plasma confined in the flat torus (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 432-442 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The spectral properties of the linearized MHD operator are investigated applying the theory of trace class perturbation. Especially, it is shown that the essential spectrum in the positive (stable) part of the spectrum contains the absolutely continuous one and the corresponding part of the spectrum is unitarily equivalent to some multiplication operator.
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    The exterior dirichlet problem for the Laplace equation (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 461-469 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.
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    Masthead (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986) 
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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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    On a free boundary value problem of physical geodesy, II (existence)This paper is part of the author's “Habilitationsschrift”, accepted by the “Fachbereich 6 der Universität - Gesamthochschule - Essen”. (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 1-22 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The determination of the figure of the earth is considered as a local free boundary value problem of potential theory. In the C1,ε-topology local existence is proved by means of Legendre transforms. This method also provides an elementary argument for the regularity of the solutions.
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    An optimal design problem for submerged bodies (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 50-76 
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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 finding the shape of a smooth body submerged in a fluid of finite depth which minimizes added mass or damping is considered. The optimal configuration is sought in a suitably constrained class so as to be physically meaningful and for which the mathematical problem of a submerged body with linearized free surface condition is uniquely solvable. The problem is formulated as a constrained optimization problem whose cost functional (e.g. added mass) is a domain functional. Continuity of the solution of the boundary value problem with respect to variations of the boundary is established in an appropriate function space setting and this is used to establish existence of an optimal solution. A variational inequality is derived for the optimal shape and it is shown how finite dimensional approximate solutions may be found.
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    The two-dimensional, linear, orthotropic plate: The traction problem (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 102-116 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In [13] we extended the analysis of Ciarlet and Destuynder [5] to the clamped orthotropic plate. For the present paper we shall apply these methods to the orthotropic plate under traction. In particular, we shall be considering the type of problem posed in Friedrichs and Dressler [10] for the isotropic plate and make use of the fact that the variational problem will split just as was the case for the partial differential equation formulation. With the present approach we shall be able to produce a proper convergence analysis for the formal asymptotics used in Friedrichs and Dressler.
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    On optimal shape design of systems governed by mixed Dirichlet-Signorini boundary value problems (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 157-181 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: A problem for finding optimal shape for systems governed by the mixed unilateral boundary value problem of Dirichlet-Signorini-type is considered. Conditions for the solvability of the problem are stated when a variational inequality formulation and when a penalty method is used for solving the state problem in question. The asymptotic relation of design problems based on these two formulations is presented. The optimal shape design problem is discretized by means of finite element method. The convergence results for the approximation are proved. The discretized versions are then formulated as a non-linear programming problem. Results of practical computations of the problem in question are reported.
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    Functional spaces for norton-hoff materials (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 206-222 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Some specific functional spaces are required for the study of Norton-Hoff materials. For this purpose we establish here a Korn's inequality and miscellaneous results of duality, traces, density and orthogonality.
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    A reaction-diffusion system modelling the spread of bacterial infections (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 234-246 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We investigate a system of reaction-diffusion equations which model the spread of a bacterial infection in a human population. A decoupling technique together with global bifurcation theory is used to study the steady-state solutions of the system. The asymptotic behaviour of solutions is discussed by using sub and subersolutions and the quasimonotonicity of the system.
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    Periodic solutions of quasilinear non-autonomous systems with impulses (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 247-255 
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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 considers a system of differential equations with impulse perturbations at fixed moments in time of the formwhere x ∊ Rn, ε is a small parameter,Sufficient conditions have been found for existence of the periodic solution of the given system in the critical and non-critical cases.
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    A note on the nonlinear stability of a spatially symmetric vlasov-possion flow (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 284-288 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We give sufficient conditions for the nonlinear stability of a, possibly non-smooth, spatially homogeneous Vlasov-Poisson flow.Research partially supported by Italian CNR and Ministero della Pubblica Istruzione.
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    A note on the solvability of Boltzmann's equation via comparison methods (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 332-339 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We consider the Cauchyproblem for the space-inhomogeneous Boltzmann equation. For the so-called soft-interaction case we prove unique local solvability, analogous to a result of Palczewski [1]. However, since in our proof a comparison argument is used rather than the familiar contraction mapping principle, we are able to extend the previously known time-interval of existence. In a final section the two main difficulties in generalizing the result are discussed in some detail.
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    Optimal controllability in viscoelasticity of rate type (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 368-386 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Recently, the control engineers interest turns upon material with structural damping, such as hysteretic effects in elastoplastic systems [27] and flexible space structures [30] and in particular with damping due to the viscoelastic nature of the material, in the sense of Kelvin, [7], [12]. The reason for this renewed interest is the possibility of constructing finite rank compensators with small spillover in the observation [1], [5], [11], [25].However, a control theory for the infinite system does not seem to exist in the literature. It is shown that the trajectories, initially in a certain subspace, can be steered to rest in any time T 〉 0 using distributed load controlsThe reesults are partially contained in [19].. Furthermore, and perhaps more interesting, the L2 (0, T, H)-norm of the control decreases as T increases. This gives rise to the problem of time-optimal-controllability and the correlation to the problem of minimum norm controllability.It is shown that the time-optimal control is characterized by a weak “bang-bang” principle.
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    An algorithm for tracing the boundary curves of mushy regions in some degenerate diffusion problems (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 458-477 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We present an algorithm for approximating the solution of the degenerate diffusion problem ut = (φ(u))xx in (0,1) × R+ (with zero Dirichlet boundary conditions, and nonnegative initial datum u0), where φ(u) = min {ku1} for some ϰ 〉 0. The algorithm also provides an approximation for the interface curves which represent the boundary of the Mushy Region M = {(x, t): φ (u(x, t)) = 1}. The convergence of the algorithm is proved.
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    Finite elements for incompressible flow (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 16-31 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: 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.
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    On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 89-100 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions φλ to the equation φλ - K λ φ λ = fλ as λ → 0 where {Kλ: X → X, λ ∊ (0,α)}, α 〉 0, denotes a family of compact linear operators such that I - Kλ (I identity) is bijective for λ∊(0,α) whilst I - K0 is not and ‖ Kλ - K0‖ →, 0, ‖fλ - f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.
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    Zur Theorie der Integraltransformationen (1979)
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    Mathematical Methods in the Applied Sciences 1 (1979), S. 114-125 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Let A be a selfadjoint uniformly elliptic differential operator. Let the underlying domain be bounded. Eigenvalue problems can be solved then, and an arbitrary square integrable function may be developed in a Fourier series relative to the eigenfunctions. In general elliptic differential operators have a continuous spectrum, if the underlying domain is unbounded. In this case the spectral theorem provides a representation of a given function by an integral transformation. The spectral projector can be calculated, if the outgoing and incoming solutions are known (radiation condition). Thus integral transformations may be derived very easily. Four examples will be given: the Fourier sine transform, the Lebedev transform, the transformation belonging to the Dirichlet problem of the plate equation and, finally, the Fourier transformation.
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    Ein algebraisches Verfahren zur Lösung von gemischten Anfangsrandwertproblemen aus der Theorie instationärer GitterströmungenHerrn Prof. Dr. H. Söhngen zu seinem 70. Geburtstage am 30. November 1978 gewidmet. (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 158-186 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: 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 Mikusiński-calculus. In this paper our main interest lies in the treatment of mixed initial-boundary value problems arising in unsteady subsonic cascade flow.
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    Asymptotic expansions for central limit theorems for general linear stochastic processes. I: General theorems on rates of convergence (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 241-264 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: 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.
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    Asymptotic expansions for central limit theorems for general linear stochastic processes. II: Models of the general random noise and pulse train processes (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 346-353 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: This part is concerned with the applications of the theorems on error estimates for asymptotic expansions in the central limit theorem for general linear stochastic processes of Part I to two important typical models. These are the general random noise process and a class of pulse train processes that occur in stochastic signal theory and in a variety of digital communication schemes. The assumptions on the inherent structure of the general theorems turn out to be simple here.
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    On boundary controllability of one-dimensional vibrating systems (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 322-345 
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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 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.
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    URL: http://dx.doi.org/10.1002/mma.1670010303
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    Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the gummel map (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 455-492 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The initial/boundary-value problem for isothermal, lattice, semiconductor device modeling is described and analyzed. This nonlinear elliptic/parabolic system of reaction/diffusion/convection type is determined by a Maxwell equation, relating space/charge and the electric field, and by two continuity equations for the free electron and hole carrier concentrations. The Einstein relations for Brownian motion are not assumed in this analysis, so that the electrostatic potential, u, and the carrier concentrations, n and p, are the fundamental dependent variables of the system. The boundary conditions are Dirichlet conditions for dependent variable values on the contact portions of the device, and homogeneous Neumann conditions, expressing insulation, on the complement. Complicating the analysis are the transition singularity points between the mixed boundary conditions, and the field dependence of the mobility and diffusion coefficients. By means of a physically motivated analysis of the convective current component, we are able to uncouple the system by a cyclic horizontal line analysis, without an unreasonable time step restriction. The corresponding linear equations are solved by a contractive inner iteration. The outer iteration is shown to converge to a unique solution of the system, under singularity classification at the transition points. The definition of this outer iteration follows the steady-state Gummel iteration at discrete time steps. An existence theory is a by-product of the analysis, and is separated from uniqueness theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670090132
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    Plane potential flow past a cylinder with porous surface (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 587-605 
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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 paper two models for the potential flow past a cylinder with porous or perforated surface are reduced to nonlinear Riemann-Hilbert problems for the complex velocity function and general existence and uniqueness theorems for the solution of the problems are derived. Further the problems are reduced to nonlinear singular integral equations of Hilbert type whose solvability is investigated by means of the Schauder and Banach fixed point theorem.
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    URL: http://dx.doi.org/10.1002/mma.1670090137
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    Transport equations with boundary conditions of reverse reflection type (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 15-35 
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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 is performed of transport equations on arbitrary three-dimensional domains with boundary conditions of reverse reflection type. The existence of the dominant eigenvalue of the criticality problem is proved and its independence of the functional setting and its continuous dependence on a variety of data are established. The corresponding time-dependent problem is shown to be well-posed, also for a conservative boundary. The relationship between the criticality and the time-dependent problem is given explicitly.
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    URL: http://dx.doi.org/10.1002/mma.1670100103
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    Masthead (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/mma.1670100201
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    On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 1: Time-independent theory (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 125-144 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let Ω be a local perturbation of the n-dimensional domain Ω0 = Ropf;n - 1 × (0, π). In a previous paper8 we have introduced the notion of an admissible standing wave. We shall prove that the principle of limiting absorption holds for the Dirichlet problem of the reduced wave equation in Ω at ω ≥ 0 if Ω does not allow admissible standing waves with frequency ω. From Reference 8, this condition is satisfied for every ω ≥ 0 if Ω ≠ Ω0, and v·x′ ≤ 0 on δΩ, where x′ = (x1,…, xn - 1, 0) and v is the normal unit vector on δΩ pointing into the complement of Ω. In contrast to this, the principle of limiting absorption is violated in the case of the unperturbed domain Ω0 at the frequencies ω = 1,2,… if n ≤ 3. The second part of our investigation, which will appear in a subsequent paper, is devoted to the principle of limit amplitude.
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    URL: http://dx.doi.org/10.1002/mma.1670100203
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    On a class of probability distributions (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 210-219 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The application of a three parameter class of one-sided probability distributions is being discussed. For specific parameter values, this class contains as special cases a number of well-known distributions of statistics and statistical physics, namely, Gauss, Weibull, exponential, Rayleigh, Gamma, chi-square, Maxwell, and Wien (limiting case of Planck's distribution). One of the three parameters represents scale; the other two represent initial and terminal shape of the associated probability density function. A fourth parameter, shift, may be introduced. The distribution class discussed in this paper was introduced by L. Amoroso [2] in 1924. It is closely connected with a family of linear Fokker-Planck equations (generalized Feller equation). In fact, the class of probability density functions associated with the distribution class considered here is a special case of the set of all delta function initial condition solutions of the generalized Feller equation for a fixed value of the time variable. It will be shown that, as a function of the logarithm of the independent variable, the logarithm of the cumulative distribution function is asymptotically linear as the independent variable approaches zero from above. This fact leads to a general criterion for the applicability of the presented distribution family relative to given empirical data. The applicability criterion can be used to determine approximate values for the two shape parameters. They can subsequently be used as initial values in any of the established parameter estimation techniques.
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    URL: http://dx.doi.org/10.1002/mma.1670090116
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    On the construction of global weak solutions in the Kynch theory of sedimentation (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 245-264 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We construct global weak solutions to the different modes of sedimentation appearing in the theory of Kynch and show that, with constant initial concentration, only five modes of sedimentation exist. We also generalize the method of construction to the case of a monotonically decreasing initial concentration.
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    URL: http://dx.doi.org/10.1002/mma.1670100304
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    A mathematical study of a continuum-state cross-bridge model of muscle contraction (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 289-302 
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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 framework of the sliding-filament theory of muscle contraction, we develop a model which describes physioiogical cross-bridge activity with a continuum of states, from those which are detached and refractory to those which are attached. The latter are kinetically more important.An analytical study of the non-linear partial differential equations of the mathematical model shows existence and uniqueness of the solution.
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    URL: http://dx.doi.org/10.1002/mma.1670100306
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    Singularities of the Laplacian at corners and edges of three-dimensional domains and their treatment with finite element methods (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 339-350 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Three-dimensional Poisson problems containing boundary singularities are treated. The forms of the solutions for certain problems of this type are derived, where the domains of the problems can be represented in terms either of spherical- or of cylindrical-polar co-ordinates. These singular forms are used to augment the basis of a standard piecewise polynomial Galerkin space, thus producing an augmented Galerkin technique which is suited to the context of a problem involving a singularity. Error estimates are derived.
    Additional Material: 6 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670100309
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    Interpolation of coefficients and transformation of the dependent variable in finite element methods for the non-linear heat equation (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 105-124 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Error estimates are shown for some spatially discrete Galerkin finite element methods for a non-linear heat equation. The approximation schemes studied are based on the introduction of the enthalpy as a new dependent variable, and also on the application of the Kirchhoff transformation and on interpolation of the non-linear coefficients into standard Lagrangian finite element spaces.
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    URL: http://dx.doi.org/10.1002/mma.1670110108
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    On the instability of resonances in parallel-plane waveguides (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 279-315 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: It has been observed13 that the propagation of acoustic waves in the region Ω0= ∝2 × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown9 in the case of Dirichlet's boundary condition U = 0 on ∂Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω0. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω0 -B, where B is a smooth bounded domain with B̄⊂Ω0. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n·x′ ≤ 0 on ∂ B, where x′ = (x1, x2, 0) and n is the normal unit vector pointing into the interior B of ∂ B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω0 will be discussed in a subsequent paper.
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    URL: http://dx.doi.org/10.1002/mma.1670110301
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    On properties of the iterative maximum likelihood reconstruction method (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 331-342 
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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 continue our investigations6 on the iterative maximum likelihood reconstruction method applied to a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Equations of this type often occur in connection with the determination of density functions from measured data. There are certain relations between the directed Kullback-Leibler divergence and the iterative maximum likelihood reconstruction method some of which were already observed by other authors. Using these relations, further properties of the iterative scheme are shown and, in particular, a new short and elementary proof of convergence of the iterative method is given for the discrete case. Numerical examples have already been given in References 6. Here, an example is considered which can be worked out analytically and which demonstrates fundamental properties of the algorithm.
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    URL: http://dx.doi.org/10.1002/mma.1670110303
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    Finite element solution for the transcritical shallow-water equations (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 503-524 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new solution of the two-dimensional shallow-water equations, using a finite element method is described. The formulation is based on the velocity and height variables and follows two steps. In the first step, the convective terms are solved by a characteristic method and in the second step the propagative and diffusion terms are taken into account. The close relation between the latter operator and a Stokes penalized problem is shown; an algorithm of Uzawa type is then used for its solution. Acceleration convergence is obtained by preconditioning, and new methods are presented, the quality of the convergence being shown on application to some sample tests. The overall method has revealed quite efficient and application to industrial cases is already planned.
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    URL: http://dx.doi.org/10.1002/mma.1670110407
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    Bifurcation from the essential spectrum for some non-compact non-linearities (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 525-542 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Conditions ensuring bifurcation from the infimum of the essential spectrum are given. The main result concerns non-linear eigenvalue problems with variational structure in a Hilbert space. It extends previous results of a similar nature by admitting situations where the non-linearity is not compact with respect to the graph norm of the linearization. The general result is applied to non-linear elliptic equations on RN.
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    URL: http://dx.doi.org/10.1002/mma.1670110408
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    Stability of a smooth flow in a Laval nozzle (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 559-572 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle.The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.
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    URL: http://dx.doi.org/10.1002/mma.1670110501
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    On a free boundary value problem of physical geodesy, I (uniqueness)This paper is part of the author's “Habilitationsschrift”, accepted by the “Fachbereich 6 der Universität - Gesamthochschule - Essen”. (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 269-289 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The determination of the figure of the earth is considered as a local free boundary value problem of potential theory: the shape of a slowly rotating heavy body has to be found from the boundary data of the attracting force and its potential outside of the body, provided that an approximation to the solution is already known. In this paper a uniqueness result is proved which is local with respect to the C1-topology.
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    URL: http://dx.doi.org/10.1002/mma.1670070118
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    Feynman-diagramme als vektorsysteme invariantentheoretisch behandelt (compton-streuung, elektron-positron-vernichtungDiese Arbeit wurde aus Mitteln der Deutschen Forschungsgemeinschaft gefördert. (1985)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 7 (1985), S. 309-330 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Employing a special contact transformation devised by S. Lie, which takes spheres into lines, we interpret the Feynman diagrams of photon electron scattering in terms of vector systems. This gives a nice kinematic model of Compton scattering. We further compute in detail the transition probabilities of the Compton scattering process by making use of the calculus of chains of complexes from classical invariant theory rather than applying the usual Dirac-matrix technique. In the final paragraph of this paper an application of our calculations to the treatment of myon decay is indicated.
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    URL: http://dx.doi.org/10.1002/mma.1670070120
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