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  • Articles  (212)
  • Articles: DFG German National Licenses  (212)
  • Mathematics and Statistics  (185)
  • Stability  (27)
  • 2010-2014
  • 1980-1984  (212)
  • 1950-1954
  • Mathematics  (212)
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  • Articles  (212)
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  • Articles: DFG German National Licenses  (212)
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  • 1
    Electronic Resource
    Electronic Resource
    Some perturbation results for the Linear Complementarity Problem (1982)
    Springer
    Mathematical programming 23 (1982), S. 181-192 
    Details
    ISSN: 1436-4646
    Keywords: Linear Complementarity Problem ; Stability ; Classes of Matrices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It has been shown previously that the Linear Complementarity Problem is stable when the defining matrix is positive semidefinite and when (locally) the set of solutions is nonempty and bounded. We enlarge the class of matrices for which this is true and also demonstrate how the boundedness condition leads to other stability type questions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01583787
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  • 2
    Electronic Resource
    Electronic Resource
    Dynamics of a single species in a spatially varying environment: The stabilizing role of high dispersal rates (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 49-55 
    Details
    ISSN: 1432-1416
    Keywords: Stability ; Diffusion ; Parabolic equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Models for a single species that inhabits an environment that is spatially varying are presented. Simple necessary and sufficient conditions for stability, which are independent of the exact details of the dispersal process, are developed in the case of large diffusion rates. The results highlight the important stabilizing nature of diffusion in a spatially varying environment.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275160
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  • 3
    Electronic Resource
    Electronic Resource
    Lotka-Volterra equations: Decomposition, stability, and structure (1980)
    Springer
    Journal of mathematical biology 9 (1980), S. 65-83 
    Details
    ISSN: 1432-1416
    Keywords: Nonnegative equilibria ; Stability ; Decompositions ; Sub-communities ; Structural perturbations ; Connective stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276036
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  • 4
    Electronic Resource
    Electronic Resource
    Equilibrium solutions of age-specific population dynamics of several species (1981)
    Springer
    Journal of mathematical biology 11 (1981), S. 65-84 
    Details
    ISSN: 1432-1416
    Keywords: Population dynamics ; Age-dependent models ; Equilibrium solutions ; Stability ; Evolution equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L 1 of the dynamics and for existence as well as for stability of equilibrium solutions are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275825
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  • 5
    Electronic Resource
    Electronic Resource
    Stability in a class of cyclic epidemic models with delay (1981)
    Springer
    Journal of mathematical biology 11 (1981), S. 95-103 
    Details
    ISSN: 1432-1416
    Keywords: Epidemiology ; SIRS ; Deterministic models ; Distributed delays ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275827
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  • 6
    Electronic Resource
    Electronic Resource
    Stability in a two host epidemic model (1982)
    Springer
    Journal of mathematical biology 14 (1982), S. 71-75 
    Details
    ISSN: 1432-1416
    Keywords: Epidemiology ; Two host models ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An epidemic model is derived for a two host infectious disease. It is shown that if a non-trivial equilibrium solution exists, it is globally stable. This result is also proved for a similar one host model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02154753
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  • 7
    Electronic Resource
    Electronic Resource
    Analysis of a sterile insect release model with predation (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 33-48 
    Details
    ISSN: 1432-1416
    Keywords: Sterile insect release ; Predation ; Stability ; Limit cycles ; Optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model for the sterile insect release method of pest control in which the target species is under predatory or parasitic regulation is analyzed. The equations are nondimensionalized and the rescaled parameters are interpreted. There are four types of equilibria, whose existence and stability depend on which of ten regions of parameter space contain the rescaled parameters, and in turn give minimal release rates to achieve eradication of the pest. In at least one region, Hopf bifurcation theory shows the existence of limit cycles, but they are found to be unstable. In addition, the optimal release rate to minimize a total cost functional for pest control by the sterile release method is studied. Both approaches show that when predation accounts for a large fraction of the natural deaths, the necessary release rate and total cost are higher than for weak predation. If the predators are removed without being replaced by any other source of mortality, the cost rises in all cases but rises much more dramatically for cases with strong predation. A definite danger of the sterile release method when some predatory control exists is that the predators are frequently driven extinct before the prey, so that the target species could explode to much higher levels and be more difficult to eradicate again after the sterile release is terminated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275159
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  • 8
    Electronic Resource
    Electronic Resource
    Models of epidemics and endemicity in genetically variable host populations (1983)
    Springer
    Journal of mathematical biology 17 (1983), S. 289-304 
    Details
    ISSN: 1432-1416
    Keywords: Endemicity ; Epidemics ; Genetics ; Deterministic models ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A discrete time genetics model is developed for populations that are undergoing selection due to infectious disease. It is assumed that the generation time of the host and infectious agent are non-synchronous and that only the host population is evolving. Two classes of epidemic processes are considered. The first class is for infectious agents that confer immunity following infection, while the second class is for those that do not confer immunity. The necessary and sufficient conditions are found in order for the disease to persist in a stable polymorphic host population. These conditions are shown to depend on the density of susceptibles, the selection coefficients, and the severity and class of the disease process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276518
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  • 9
    Electronic Resource
    Electronic Resource
    Constant-rate stocking of predator-prey systems (1981)
    Springer
    Journal of mathematical biology 11 (1981), S. 1-14 
    Details
    ISSN: 1432-1416
    Keywords: Ecological modelling ; Predator-prey systems ; Ordinary non-linear differential equations ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We examine the qualitative effects of constant-rate stocking of either or both species in a predator-prey system. The hypotheses are made as mild as possible so that several types of systems with different qualitative alternatives may be studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275820
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  • 10
    Electronic Resource
    Electronic Resource
    Coexistence properties of some predator-prey systems under constant rate harvesting and stocking (1982)
    Springer
    Journal of mathematical biology 12 (1982), S. 101-114 
    Details
    ISSN: 1432-1416
    Keywords: Predator-prey systems ; Harvesting ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The global behaviour of a class of predator-prey systems, modelled by a pair of non-linear ordinary differential equations, under constant rate harvesting and/or stocking of both species, is presented. Theoretically possible structures and transitions are developed and validated by computer simulations. The results are presented as transition loci in the F-G (prey harvest rate-predator harvest rate) plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275206
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  • 11
    Electronic Resource
    Electronic Resource
    A predator prey model with age structure (1982)
    Springer
    Journal of mathematical biology 14 (1982), S. 231-250 
    Details
    ISSN: 1432-1416
    Keywords: Predator-prey ; Age structure ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation periodm of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for smallm or are destabilized asm decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01832847
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  • 12
    Electronic Resource
    Electronic Resource
    Stability of stationary distributions in a space-dependent population growth process (1982)
    Springer
    Journal of mathematical biology 15 (1982), S. 37-50 
    Details
    ISSN: 1432-1416
    Keywords: Reaction-diffusion system ; Stationary solution ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We consider a spatial population growth process which is described by a reaction-diffusion equation c(x)u t = (a 2(x)u x ) x +f(u), c(x) 〉0, a(x) 〉 0, defined on an interval [0, 1] of the spatial variable x. First we study the stability of nonconstant stationary solutions of this equation under Neumann boundary conditions. It is shown that any nonconstant stationary solution (if it exists) is unstable if a xx⩽0 for all xε[0, 1], and conversely ifa xx〉0 for some xε[0, 1], there exists a stable nonconstant stationary solution. Next we study the stability of stationary solutions under Dirichlet boundary conditions. We consider two types of stationary solutions, i.e., a solution u 0(x) which satisfies u 0 x≠0 for all xε[0, 1] (type I) and a solution u 0(x) which satisfies u 0x = 0 at two or more points in [0, 1] (type II). It is shown that any stationary solution of type I [type II] is stable [unstable] if a xx ⩾0 [a xx ⩽0] for all xε[0, 1]. Conversely, there exists an unstable [a stable] stationary solution of type I [type II] if a xx 〈0 [a xx 〉0] for some xε[0, 1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275787
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  • 13
    Electronic Resource
    Electronic Resource
    Effects of behavioral adaptation on a predator-prey model (1982)
    Springer
    Journal of mathematical biology 15 (1982), S. 239-247 
    Details
    ISSN: 1432-1416
    Keywords: Predator-prey model ; Behavioral adaptation ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The Volterra-Lotka predator-prey equations are modified so that the predator's ability to utilize the prey varies in proportion to the average number of encounters between the two species in the past. The behavior of this adaptive system is then described in terms of three parameters — the carrying capacity of the prey, the relative death rate of the predator, and the predator's memoryspan. The most stable situation is shown to occur when the carrying capacity of the prey is large, the predator's death rate is close to zero, and the predator is able to adapt quickly to changing levels of prey density.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275076
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  • 14
    Electronic Resource
    Electronic Resource
    On the stability of delay equation models of simple human stretch reflexes (1983)
    Springer
    Journal of mathematical biology 17 (1983), S. 331-349 
    Details
    ISSN: 1432-1416
    Keywords: Stability ; Delay equations ; Stretch reflexes ; Mathematical studies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Human stretch reflexes (SRs) are often too weak and ineffectual to provide adequate postural regulation or rhythmic movement boosting (e.g. in ankle pushoff at the end of stance phase in fast running). Recent improvements in methods of artificially enhancing skeletomotor responses, especially in therapeutic regimens, should not be widely employed until the clonus-resisting stability properties of SRs are better understood. We formulate an idealized linear servo model of a segmentally-mediated SR system which includes the often ignored electromechanical coupling delay. For typical closed-loop (delay/gain) ratios, the model is shown to be unstable for all values of loop gain when operating as a position servo, but maximally stable when operating as a velocity servo. We claim that the velocity servo or one of its nonlinear relatives is a better model for some well studied SRs than, e.g., Houk's stiff muscle hypothesis. We also present evidence that even feeble and quickly saturating monosynaptic postural servos are always unstable if operated as pure position regulators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276520
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  • 15
    Electronic Resource
    Electronic Resource
    Existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay (1983)
    Springer
    Journal of mathematical biology 18 (1983), S. 93-102 
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    ISSN: 1432-1416
    Keywords: Stability ; Time delay ; Feasible equilibrium ; Partially feasible equilibrium
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A sufficient condition for the existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay is obtained, and some properties of the stable equilibrium are proven. Furthermore, some applications in which asymptotic stability only depends on the sign of the coefficients are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00280659
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  • 16
    Electronic Resource
    Electronic Resource
    Effects of mean current and shear on stability of depth distributions of marine phytoplankton (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 33-51 
    Details
    ISSN: 1432-1416
    Keywords: Plankton ; Stability ; Perturbations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Linear perturbation theory is used to examine the stability of steady-state distributions of marine phytoplankton in the presence of a mean current with shear. Solutions are obtained for the general initial-value problem and it is found that all distributions are asymptotically stable so long as the rate of shear is greater than the local production. On the other hand, the early time behavior indicates that the system can be altered, if accomplished soon enough, depending upon a relative combination of diffusion, advection and production. Quantitative assessments are made where data are available.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276394
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  • 17
    Electronic Resource
    Electronic Resource
    Persistence of predator-prey systems in an uncertain environment (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 65-77 
    Details
    ISSN: 1432-1416
    Keywords: Predator-prey ; Persistence ; Stability ; Differential inequalities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary The time derivatives of prey and predator populations are assumed to satisfy a set of inequalities, instead of a precise differential equation, reflecting an uncertain environmental and/or lack of knowledge by the modeler. A system of differential equations is found whose solution gives the boundary of a persistent set, which is positive flow invariant for any system satisfying the inequalities. Conditions are given for the persistent set to be bounded away from both axes, which show that resonance effects cannot drive either predator or prey to extinction if that does not happen for an autonomous system satisfying the inequalities. In general predator-prey systems are more persistent when there is strong asymptotic stability, when there is correlation between prey and predator dynamics, when the effect of perturbations is density dependent, and are more persistent under perturbations of the prey than of the predator.
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    URL: http://dx.doi.org/10.1007/BF00276396
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  • 18
    Electronic Resource
    Electronic Resource
    One-dimensional λ-ω target patterns: Empirical stability tests (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 97-100 
    Details
    ISSN: 1432-1416
    Keywords: Reaction diffusion equations ; Wave-trains ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary Stability of a class of solutions to reaction-diffusion equations is studied numerically. It is found that the solutions do not persist under a variety of boundary conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276399
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  • 19
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    Stability analysis for models of diseases without immunity (1981)
    Springer
    Journal of mathematical biology 13 (1981), S. 185-198 
    Details
    ISSN: 1432-1416
    Keywords: Epidemiology ; Deterministic models ; Distributed delays ; Thresholds ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A cyclic, constant parameter epidemiological model is described for a closed population divided into susceptible, exposed and infectious classes. Distributed delays are introduced and the model is formulated as two coupled Volterra integral equations. The delays do not change the general nature of thresholds or asymptotic stability; in all cases considered the disease either dies out, or approaches an endemic steady state.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275213
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  • 20
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    The community matrix in three species community models (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 25-31 
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    ISSN: 1432-1416
    Keywords: Stability ; Community matrix ; Lotka-Volterra models
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The explicit function of the community matrix of a three dimensional Lotka-Volterra model is delineated by a set of necessary and sufficient conditions for a positive equilibrium to be asymptotically stable. In the special case that the community matrix is quasi weakly diagonally dominant, it is shown that a positive determinant for the community matrix is not only necessary but is also sufficient for stability. The results are specific to three dimensional models and do not extend to communities of dimension greater than three.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275158
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  • 21
    Electronic Resource
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    Stability results for delayed-recruitment models in population dynamics (1984)
    Springer
    Journal of mathematical biology 19 (1984), S. 147-156 
    Details
    ISSN: 1432-1416
    Keywords: Stability ; delay ; difference equations ; whale models
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.
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    URL: http://dx.doi.org/10.1007/BF00275937
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  • 22
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    Integral equation models for endemic infectious diseases (1980)
    Springer
    Journal of mathematical biology 9 (1980), S. 37-47 
    Details
    ISSN: 1432-1416
    Keywords: Epidemiology ; Endemic infectious diseases ; Deterministic models ; Thresholds ; Distributed delays ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (births and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.
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    URL: http://dx.doi.org/10.1007/BF00276034
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  • 23
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    The existence of globally stable equilibria of ecosystems of the generalized Volterra type (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 401-415 
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    ISSN: 1432-1416
    Keywords: Stability ; Volterra ecosystems ; Linear complementarity theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper, global asymptotic stability of ecosystems of the generalized Volterra type $$dx_i /dt = x_i \left( {b_{i - } \mathop \sum \limits_{j = 1}^n a_{ij} x_j } \right),{\text{ }}i = 1,...,n,$$ is investigated. We obtain the conditions for the existence of a nonnegative and stable equilibrium point of the system by applying a result of linear complementarity theory. The results of this paper show that there exists a class of systems that do not have multiple domains of attractions. This class is defined in terms of the species interactions alone, and does not involve carrying capacities or species net birth rates.
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    URL: http://dx.doi.org/10.1007/BF00276098
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  • 24
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    Clines in the presence of asymmetric migration (1981)
    Springer
    Journal of mathematical biology 11 (1981), S. 207-233 
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    ISSN: 1432-1416
    Keywords: Clines ; Population genetics ; Nonlinear diffusion problems ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract If a population, which consists of individuals having genetic variation at one locus, with two alleles A and a, evolves under the influence of migration and selection, gradients in the distribution of alleles may arise. We consider the effect of asymmetry in the migration and spatial dependence of the selection process, upon the emergence and stability of such gradients.
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    URL: http://dx.doi.org/10.1007/BF00275443
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  • 25
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    The diffusive Lotka-Volterra system with three species can have a stable non-constant equilibrium solution (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 103-112 
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    ISSN: 1432-1416
    Keywords: Bifurcation ; Competition ; Diffusive Lotka-Volterra system ; Predator-prey interaction ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Three examples of the diffusive 3-species Lotka-Volterra system with constant interaction parameters are given, and by bifurcation techniques shown to have stable spatially non-constant equilibrium solutions. One example is competitive; the second one predator-two-competing prey and the third involves two predators and a single prey.
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    URL: http://dx.doi.org/10.1007/BF00275163
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  • 26
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    Existence and stability of travelling wave solutions of a nonlinear integral operator (1983)
    Springer
    Journal of mathematical biology 16 (1983), S. 199-220 
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    ISSN: 1432-1416
    Keywords: Nonlinear integral operator ; Travelling wave ; Wave speed ; Asymptotic speed of propagation ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper, we establish the existence and stability property of travelling wave solutions of a nonlinear integral operator in the inferior case.
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    URL: http://dx.doi.org/10.1007/BF00276502
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  • 27
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    Stable spatio-temporal oscillations of diffusive Lotka-Volterra system with three or more species (1983)
    Springer
    Journal of mathematical biology 18 (1983), S. 213-221 
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    ISSN: 1432-1416
    Keywords: Diffusive Lotka-Volterra system ; Hopf-bifurcation ; Spatiotemporal oscillation ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stability condition for Hopf-bifurcating solutions from the uniform equilibrium of clasical Lotka-Volterra interaction-diffusion equations is presented. Using this condition, it is shown that stable spatio-temporal oscillations exist in the framework of such equations.
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    URL: http://dx.doi.org/10.1007/BF00276088
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  • 28
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    The effect of a thin inclusion of high rigidity in an elastic body (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 251-270 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study an elastic body comprising an inclusion whose thickness is 2∊ and for which Lamé's constants are λ∊ and μ. We are interested in the limit behaviour of the inclusion, when ∊ → 0 and μ∊ → ∊; that is when the inclusion becomes thiner and more rigid. Different behaviours are possible, according to the rate at which μ∊ converges to infinity; the limit inclusion may “vanish” if it is not very rigid, but also it may be entirely solid.
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    URL: http://dx.doi.org/10.1002/mma.1670020302
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    Nonselfadjoint operators in diffraction and scattering (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 327-346 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670020307
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    Partial asymptotic stability via limiting equations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 516-522 
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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 the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.
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    On spherical spline interpolation and approximation (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 551-575 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Spherical spline functions are introduced by use of Green's surface functions with respect to the (Laplace-)Beltrami operator of the (unit) sphere. Natural (spherical) spline functions are used to interpolate data discretely given on the sphere. A method is presented that allows the smoothing of irregularities in measured values or experimental data. Extensions of Peano's theorem and Sard's theory of best approximation to the spherical case are given by integral formulas. Schoenberg's theorem is transcribed into spherical nomenclature.
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    Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3 (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 1-14 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p 〉 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm rot }\upsilon {\rm = }\omega {\rm, div }\upsilon = 0,\upsilon \cdot \bar n||_{\partial \Omega } = 6 $$\end{document} where v - velocity vector, ω - vorticity vector and n is an unit outward vector normal to the boundary,(2) the existence of solutions of the following evolution problem for given velocity vector \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} \omega _t + \upsilon ^\kappa \omega _x \kappa - \omega ^\kappa \upsilon _x \kappa = F \equiv {\rm rot }f \\ \omega |_{t = 0} = \omega _0,\omega |_{s1} = \eta \\ \end{array} $$\end{document}(3) the method of successive approximations, using solvability of problems (1) and (2).
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    Galerkin approximations for the equation Lu + Nu = f (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 286-290 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Introducing the concept of a supercontinuous operator, we obtain a general convergence theorem for Galerkin approximations. Under the stronger assumption that N is a monotone operator with N(0) = 0, we show norm convergence of the unique Galerkin approximations to the unique solution.
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    Biorthogonal sequences of solutions of the generalized feller equation (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 317-353 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.
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    Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 354-381 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The problem of stress determination in the area of cut-outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part Ct1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part Ct2. Starting from the Betti-Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions.The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo-differential operators, the Galerkin method converges. Assuming that curves Ct1 and Ct1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.
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    Zum 80. Geburtstag von Wolfgang Haack (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 451-453 
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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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    Zur theorie rotierender und schwingender schaufelkränze in einer unterschallstromung durch einen ringkanalDie vorliegende Arbeit wurde von der Deutschen Forschungsgemeinschaft unter der Numraer Me 261/3 unterstützt. (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 454-479 
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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 the three-dimensional perturbation flow induced by a rotating and oscillating blade row which operates in a subsonic flow in axial direction of an annular channel is studied. The velocity potential is reduced to the infinite Hilbert space vector of Fourier coefficients of an eigen-function expansion with respect to vanishing normal derivatives on both cylinder walls. These coefficients satisfy an infinite set of ordinary differential equations of second order after an application of a one-dimensional Fourier transform in axial direction.Several canonical two-part mixed boundary value problems are then investigated by reduction to “infinite two-by-two-Wiener-Hopf functional systems”. In case of strong factorizability of certain matrix-operator-valued functions on the line these systems may be solved explicitely. Criteria for the factorization are not given here.
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    Über die existenz und eindeutigkeit einer lösung für ein randwertproblem für eine gleichung vom gemischten typ in einem rechteck (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 510-528 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the paper a boundary value problem is studied for the equation of mixed typek(y)uxx + uyy + r(x, y)u = f(x, y) in the rectangular domain {(x, y)| -1 〈 x 〈 +1, yc 〈 y 〈 yH} with yc 〈 0, yH 〉 0, k(y) = sign y|y|m, m 〉 0 (and more generally for a function k = k(y) with k(O) = 0, k(y)y 〉 O for y ≠ O). Specific for the stated problem is that no data are prescribed on the line {(x, yc), -1 〈 x 〈 +1}. It is proved that the formulated problem is well-posed in the sense that there is at most one quasi-regular solution and that a generalized solution exists. The energy-integral-(abc-)method is used to show uniqueness and to obtain an apriori estimate for the solution of the adjoint problem whence the existence statement follows.
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    Rounding errors in numerical solutions of two linear equations in two unknownsDedicated to Professor Wolfgang Haack on the occasion of his eightieth birthday. (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 549-571 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: New condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.
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    The sommerfeld half-plane problem revisited, II the factoring of a matrix of analytic functionsThese results were announced at the XXth General Assembly of the U.R.S.I. held in Washington, D.C., August 1981. (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 14-21 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A half-plane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary condition on the other. These boundary conditions contrast the ones used by A. Sommerfeld in his classical paper. The present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method suggested by A. E. Heins in 1950.
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    Asymptotic equipartition of energy for equations of elasticity (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 186-194 
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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 give a review of the equipartition of energy results of Goldstein and Sandefur [3], [4], [5] as well as proving a new result in the case of a particular fourth order equation. These results are then applied to the equations of elasticity to give a weak asymptotic orthogonality for the shear and pressure waves. In the case of boundary value problems in the interior of a bounded domain we get weak asymptotic orthogonality in the average.
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    Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 530-543 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The purpose of this paper is to introduce and study a new type of derivative - the variational gradient - for a functional on Cn[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cn[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations.Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.
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    Asymptotic study of the macroscopic behaviour of a solid-fluid mixture (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 1-11 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the framework of homogenization theory we study a mixture of an elastic solid and a viscous compressible fluid with periodic structure and its limit behaviour as the period tends to zero Existence, uniqueness and convergence theorems are given. The limit behaviour is viscoelastic.
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    Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung ΔU + k2U = 0 (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 48-67 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k2U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L2(∂G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L2(∂G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G.The well-known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.
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    A singular perturbation approach for a reaction diffusion system arising in nuclear reactor theory (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 91-107 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The qualitative behavior of the solutions of a reaction-diffusion system arising in the theory of nuclear reactors is investigated by means of singular perturbation techniques, the small parameter ∊ representing the inverse of neutron velocity. At the lowest approximation for small ∊ the solutions exhibit oscillations about the unique strictly positive equilibrium, much as in the case of the associated lumped parameter system. A two-times representation for solutions of small amplitudes is also given.
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    On uniqueness of Neumann-Tricomi problem in R2 (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 168-177 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the equation of mixed type (k(y) ⋛ 0 whenever y ⋛ 0) in a region G which is bounded by the curves: A piecewise smooth curve Γ lying in the half-plane y 〉 0 which intersects the line y = 0 at the points A(-1, 0) and B(0, 0). For y 〈 0 by a piecewise smooth curve Γ through A which meets the characteristic of (1) issued from B at the point P and the curve Γ which consists of the portion PB of the characteristic through B. We obtain sufficient conditions for the uniqueness of the solution of the problem L[u] = f, dnu: = k(y)uxdy - uydx|γ0 = = Ψ(s) for a “general” function k(y), when r(x, y) is not necessarily zero and Γ1 is of a more general form then in the papers of V. P. Egorov [6], [7].
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    Numerical solution of Hopf bifurcation problems (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 178-190 
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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 two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ∊. It has isolated solutions in the ∊-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.
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    Zur L∞-Konvergenz finiter Elemente bei parabolischen Differentialgleichungen (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 221-234 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: For a parabolic model problem it is shown that the convergence rate of higher order finite element approximation is quasioptimal in L∞. Moreover, the estimate does not depend on the interval [0, T]. The essential tool of the proof is a modified weighted norm technique.
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    Scattering of elastic waves through a heterogeneous medium (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 271-287 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Topics: Mathematics
    Notes: In this paper we study the elastic wave diffraction in R3 through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.
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    Masthead (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980) 
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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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    Über die Gaußsche Methode zur angenäherten Berechnung von IntegralenMeinem verehrten Lehrer, Herrn Prof. Dr. Cl. Müller, in Dankbarkeit zum 60. Geburtstag gewidmet. (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 397-409 
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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: The purpose of this paper is the application of Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.
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    Zur numerischen Lösung gewöhnlicher Differential-gleichungen mit Splines in einem SonderfallHerrn Claus Müller zum 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 419-428 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].
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    Some remarks on the horizontal line methodHerrn Claus Müller zum 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 471-479 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In the following paper a methodological survey with respect to applications of the horizontal line method (Rothe's method) to a class of initial boundary value problems is given. By means of results from abstract perturbation theory, convergence results and error estimates are established for several special initial boundary value problems of mathematical physics.
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    Elektrodynamik mit SpinHerrn Prof. Dr. Claus Müller zu seinem 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 457-470 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Atomistic equations of the electromagnetic field for a particle with spin are derived from a Lagrangian. These equations are consistent with the equations of motion for such a particle. The resulting phenomenological equations are the well-known equations of Maxwell for the electromagnetic field in matter. The atomistic field equations for a particle with spin and magnetic moment give a dipole field. This result and the corresponding quantum mechanics for a particle with spin are applied to compute the hyperfine structure of the hydrogen atom by perturbation theory.
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    On the boundary value problem of the biharmonic operator on domains with angular corners (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 556-581 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.
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    Bifurcation results for a class of periodic quasi-linear parabolic equations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 11-20 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the problem where a and f are 1-periodic in t, a is positive, f satisfies appropriate decreasing conditions; smoothness of a, f, ∂Ω is also assumed. Denote by λ0 the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non-negative periodic solution exists but zero, and any solution with continuous non-negative initial datum converges to zero uniformly as t → ∞; (b) if ε 〉 0, then a unique non trivial non-negative 1-periodic solution u* exists, and any solution with continuous, non-negative not identically zero initial datum approaches uniformly u* as t → ∞.
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    Free boundary problems for the laplace equation: A priori estimates, existence theorems, asymptotic behaviours (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 38-69 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: This work is devoted to Free Boundary Problems with Laplace equation Δu = f in the domain and the two conditions on the Free Boundary u = 1 and |grad u| = λ = const. In the model problems which we study, three cases arise: 1) f = 0, λ is given, 2) f = 0, λ is unknown and the length of the Free Boundary is given, 3) f ≠ 0, λ = 0 (Obstacle Problem).
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    A linear Uzawa-algorithm for the steady-state Navier-Stokes problem (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 115-120 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: An Uzawa algorithm of Temam is improved so that the Navier-Stokes boundary value problem is reduced to a sequence of linear Poisson problems. A simple finite element method is shown to give a convergent sequence of approximate solutions. Finally a penalty variant is discussed.
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    Eine Integralgleichungsmethode für akustische Reflexionsprobleme an lokal gestörten Ebenen (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 128-144 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we consider the reflection of acoustic waves at an unbounded surface which coincides with a plane outside a sufficiently large sphere. We prove uniqueness and existence theorems for the corresponding boundary value problems for the reduced wave equation with Dirichlet and Neumann data by employing integral equation methods.
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    A singular perturbation problem for linear operators with an application to electrostatic and magnetostatic boundary and transmission problemsPart of this work was carried out while both authors were at the Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA. (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 249-274 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The treatment of certain electro- and magnetostatic boundary and transmission problems by boundary integral equations leads to parameter-dependent integral equations of the second kind. The integral operators involved have the property that the dimension of their nullspaces changes between two nonzero values (depending on the geometry of the problem) as the parameter tends to zero. We investigate the continuous dependence of solutions to these equations on the parameter. To this end, we treat the problem of continuous dependence of solutions to parameter-dependent linear operator equations of the second kind in a Banach space in the framework of generalized inverses.
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    Über das asymptotische Verhalten thermoelastischer Wellen im R3Diese Arbeit entstand mit Unterstützung des durch die Deutsche Forschungsgemeinschaft getragenen Sonderforschungsbereiches 72 an der Universität Bonn. (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 312-317 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In [L] the author provided existence theorems for initial-boundary-value problems in thermoelasticity for unbounded domains. Continuing this work the asymptotic behaviour of solutions in R3 with respect to t → ∞ will be described in this paper.
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    Stable solutions of some ill-posed problems (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 336-363 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Stable solutions of equations of the first kind and equations of the second kind at a characteristic value are given. Iterative processes for solutions are constructed. Extension of operators is used to turn an ill-posed problem into a well-posed one.
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    Mathematische Behandlung ernes Modells der Haemodialyse (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 393-404 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In this paper a simple mathematical model for the process of hemodialysis is presented. This model is based on a system of two linear differential equations of first order with partly discontinuous coefficients that describe the time-development of the concentrations of a certain toxin (like urea) in the intra- and extracellular part of the human body. The main result is the existence of periodic positive solutions of this system under the natural assumption that the generation of the toxin and its removal by hemodialysis are periodic processes. These periodic positive solutions are also computed numerically for a realistic choice of the coefficients of the modelling differential equations.
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    On the convexity of equilibrium plasma configurations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 435-443 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We investigate the problem of plasma confinement by a magnetic field in an infinite cylinder. We show that if the cylinder has convex cross-section, then there exists an equilibrium plasma configuration with convex cross-section.
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    The impedance boundary value problem for the time harmonic Maxwell equations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 475-487 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The existence of a unique solution to Maxwell's equations defined in an exterior domain with impedance boundary condition is established for all frequencies. This is accomplished by reducing this problem to that of solving a system of singular integral equations and then regularizing this system such that the Riesz theory is applicable. We also consider the inverse problem in which it is desired to determine the impedance from a knowledge of the far field pattern. By restricting the impedance to lie a priori in a compact set results are obtained on the existence, uniqueness, and stability of the solution to this inverse scattering problem.
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    Study of motions of a vibrating string subject to solid friction (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 287-300 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we consider a string moving in its plane and subject to solid friction (Coulomb's law). It is known that when the time increases indefinitely the string reaches an equilibrium position and by analogy with the case of a mass point we ask if the equilibrium is reached after a finite time. We prove that this is the case when the string is initially at rest and 1. when the initial shape possesses a second derivative bounded by certain limits, or 2. when the initial shape is formed by two straight line segments. In the last section we obtain some partial results when the string is initially at rest in the shape of a polygonal line. The case of an arbitrary initial position is still an open problem.
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    On complete recovery of geophysical data (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 33-73 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: One considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x2, x3)δ(t) for x1 = 0 are applied with displacement responses ui = gi(t, x2, x3) at x1 = 0 (i = 1, 2, 3). Let vi(t, x1) = ∫∫uidx2dx3 (i = 1, 2 is enough) and set w(t, x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations for v1, v2, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled vi can then be solved to produce 2 functions A1 and A2 involving (Q, λ, μ) as functions of “bound” variables y1 and y2 containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x1. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.
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    Crank-nicolsen-Galerkin approximation for Maxwell's equations (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 123-130 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The Maxwell equations are formulated as an evolution equation in a suitable chosen Hilbert space involving a densely defined closed skew-Hermitian operator which generates a unitary group. A Crank-Nicolsen-Galerkin approximation is then established and convergence is shown by arguments from the theory of approximation of groups of operators.
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    On the transition from supercritical to subcritical Hopf bifurcations (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 143-163 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Topics: Mathematics
    Notes: By using the method of averaging, we give a complete Hopf bifurcation diagram when the differential equation has more than one parameter. The method is applied to a system of parabolic equations with nonlinear coupling on the boundary. This set of equations is a mathematical model which describes the normalized concentrations of substances in a solution produced by two enzymes.
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    Bifurcation problems for discrete variational inequalities (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 243-258 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The buckling of a beam or a plate which is subject to obstacles is typical for the variational inequalities that are considered here. Birfurcation is known to occur from the first eigenvalue of the linearized problem. For a discretization the bifurcation point and the bifurcating branches may be obtained by solving a constrained optimization problem. An algorithm is proposed and its convergence is proved. The buckling of a clamped beam subject to point obstacles is considered in the continuous case and some numerical results for this problem are presented.
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    Monotone schemes for semilinear elliptic systems related to ecology (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 272-285 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Semilinear elliptic systems of partial differential equations related to ecology are studied, with Dirichlet boundary conditions. Monotone sequences of functions which satisfy scalar equations are constructed so that they will converge to upper and lower bounds for the solutions of the systems. In case a related system has a unique positive solution, then these sequences will converge to the solution of the original system. Applications of the monotone sequences to uniqueness and stability are also given.
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    Spectral properties of certain stiff problems in elasticity and acoustics (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 291-306 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We consider two vibration problems containing a small parameter → 0: a) Vibration of an elastic, slightly compressible body, and b) acoustic vibration of a slightly viscous compressible barotropic fluid in a vessel.The asymptotics of eigenvalues for problem a) is studied by using a uniformly convergent expansion of the stiff type. After a re-scaling of the spectral parameter, the problem b) reduces to an analogous problem, and we prove that, as ε → 0, infinitely many eigenvalues converge to 0 (which is an eigenvalue of infinite multiplicity of the corresponding inviscid acoustic problem).
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    Spline methods in geodetic approximation problems (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 382-396 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: This paper is concerned with spline methods in a reproducing kernel Hilbert space consisting of functions defined and harmonic in the outer space of a regular surface (e.g. sphere, ellipsoid, telluroid, geoid, (regularized) earth's surface). Spline methods are used to solve interpolation and smoothing problems with respect to a (fundamental) system of linear functional giving information about earth's gravity field. Best approximations to linear functionals are discussed. The spline of interpolation is characterized as the spline of best approximation in the sense of an appropriate (energy) norm.
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    A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 415-424 
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    Notes: The theory of Rayleigh functionals for non-linear eigenvalue problems T(λ) u = 0 is extended to cases where the functional is defined only on a proper subset. The theory applies to problems which do not satisfy an overdamping condition and yields a minimax characterization of eigenvalues. Applications to damped free vibrations of an elastic body are discussed.
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    Applications of bloch expansion to periodic elastic and viscoelastic media (1982)
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    Mathematical Methods in the Applied Sciences 4 (1982), S. 433-449 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We consider an unbounded, elastic non homogeneous material with periodic structure. It is shown that the solution of the equations of motion can be expanded in eigenfunctions of periodic operators. The Bloch expansion is used to prove that when the wave-length is long compared to the period of the structure (related to a small parameter ε), the first term of the exact solution expansion, in powers of ε, is the solution of the equations obtained in homogenization theory. The case of a long memory linear viscoelastic material is then studied.
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    Ein nichtlineares Glättungsverfahren zur rekonstruktion von flugbahnen aus radardaten (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 497-509 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Data processing is an important tool in airspace surveillance and air traffic control today. In this paper the problem is treated how to reconstruct a flown trajectory from its correlated radar plots subject to a certain knowledge of the aircraft manoeuverability and the radar measurement statistics. A variational approach leads to a generalized smoothing spline method. Simulation results are presented.
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    Masthead (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983) 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Half-stable solution branches for ordinary bifurcation problemsThis paper is an extended version of a part of the author's work [2]. (1983)
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    Mathematical Methods in the Applied Sciences 5 (1983), S. 1-13 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: It is shown that second order bifurcation problems with a positive, autonomous nonlinearity have a smooth branch of positive solutions which tends to infinity. Moreover, this branch satisfies a stability rule saying that the solutions are stable if the branch turns to the right and unstable if it turns to the left.
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    A critical case for the solvability of stefan-like problemsWork performed under the auspices of the Italian C.N.R., G.N.F.M. (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 84-96 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We consider a one-phase one-dimensional Stefan problem with general data with the aim to investigate some open questions on existence of classical solutions. We show how existence and nonexistence are discriminated by the behavior of the initial datum in the neighborhood of the starting point of the free boundary.
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    Periodic solutions of u'(t) = -f(u(t - 1)): Multiplicity results and small period solutions (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 162-175 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we are concerned with the differential delay equationFor odd and continuous nonlinearities f: R → R. We want to prove existence and multiplicity criteria for so called “slowly oscillating” periodic solutions of (A).The oddness condition is of course a rather restrictive one, but due to results of Nussbaum [9], [10] and Peters [13],[14] and numerical studies of Jürgens, Peitgen and Saupe [7] and Hadeler [6] we know that even for odd nonlinearities f (A) may display a complicated dynamical behaviour. On the other hand the oddness condition allows a classification of symmetry properties of periodic solutions of (A).
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    Lösungen der poissongleichung und harmonische vektorfelder in unbeschränkten gebieten (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 233-255 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of Rn(n ≥ 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R3 with vanishing Dirichlet or Neumann boundary condition.
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    URL: http://dx.doi.org/10.1002/mma.1670050117
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    Gruppentheoretische aspekte der signalübertragung und der kardinalen interpolationssplines I (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 195-215 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: As is well known the real Heisenberg nilpotent group A(R) constitutes the group-theoretic embodiment of the Heisenberg canonical commutation relations (CCR) of classical quantum mechanics. In this connection, quantum mechanics stands for the quantum-mechanical description, at a given instant of time, of a non-relativistic microparticle moving in the one-dimensional configuration space R and having the plane R2 as its (flat) phase space. In fact, the (nilpotent) Lie algebra n of Ã(R) reflects the Weyl equations which are global versions of the Heisenberg CCR. Unfortunately, the subject of Heisenberg nilpotent groups is outside quantum mechanics and all the more outside mathematical physics not as commonly known as it should be considering its wide range of applications in a variety of different fields. The present paper which has two parts aims to develop a central topic of nilpotent harmonic analysis, to wit, the microparticle model, the lattice model which will be realized on the Heisenberg compact nilmanifold, and the complex wave model (or Bargmann-Fock-Segal model) of the linear Schrödinger representation of Ã(R) in order to examine geometrically several applications which are governed by the real Heisenberg nilpotent group Ã(R). These applications are in Part I the classical Whittaker-Shannon sampling theorem which is of basic importance in signal processing, to wit, for the transmission of digital signals as well as analog signals, and in Part II the Subbotin-Schoenberg existence and uniqueness theorem of cardinal spline interpolation. Moreover, Part II indicates briefly some connections of the aforementioned models to the Wigner phase-space quasiprobability density function of quantum statistical mechanics via the Schwartz kernels theory on unimodular Lie groups, an approach to the cross- and autoambiguity functions of radar synthesis, and to the Zak transform of solid state physics. The second part also points out some relations of harmonic analysis of the finite nilpotent group A(Z/NZ) to periodic spline interpolants admitting N equidistant knots on the one-dimensional compact torus group T. These last examples should serve mainly as hints for some further lines of investigations in the field of applications of nilpotent harmonic analysis.
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    URL: http://dx.doi.org/10.1002/mma.1670050115
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    Das asymptotische verhalten der greenschen funktion N-irregulaärer eigenwertprobleme mit zerfallenden randbedingungen (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 331-345 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove asymptotic estimates for the Green's function of N-irregular eigenvalue problems My = λNγ with splitting boundary conditions.In contrast to the N-regular case the Green's function G(x,ζ,λ) grows exponentially for |λ| → ∞ if x 〉 ζ.These estimates are fundamental for the expansion of functions into a series of eigenfunctions of N-irregular eigenvalue problems. In a subsequent paper it will be shown that this irregular behavior of G(x,ζ,λ) implies that only a very small class of functions can be expanded into a series of eigenfunctions of such problems.
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    URL: http://dx.doi.org/10.1002/mma.1670050122
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    Approximation of the signorini problem with friction, obeying the coulomb law (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 422-437 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The approximation of the Signorini problem with friction by mixed finite element method is studied. The relation between the continuous case and its finite dimensional discretization is analyzed.
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    URL: http://dx.doi.org/10.1002/mma.1670050127
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    Free boundary problem for a two-layer inviscid incompressible fluid (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 476-490 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.
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    URL: http://dx.doi.org/10.1002/mma.1670050129
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    Masthead (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984) 
    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.1670060101
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    Solvability of an initial boundary value problem for the euler equations in twodimensional domain with corners (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 1-22 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ⊂ R2 with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three-dimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2]-[6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.
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    Faedo-galerkin approximations in equations of evolution (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 41-54 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the evolution equation u' = Au + Nu u(0) = φu is a function on a real interval [0, T] with values in a Hilbert space H, A is a linear operator in H generating a continuous semigroup eAt and N is a nonlinear operator in H.We show that 1Existence of the exact solution implies existence of the Faedo-Galerkin Approximations.2Existence of the Faedo-Galerkin Approximations implies existence of the exact solution.3Uniform convergence of the Faedo-Galerkin Approximations to the exact solution.The Paper consists of two parts. In the first five sections we require that A possesses a complete orthonormal system of eigenfunctions, in section 6 we drop this requirement.
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    URL: http://dx.doi.org/10.1002/mma.1670060104
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    Functional differential equations with certain monotonicity properties (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 97-103 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The present paper is concerned with the study of retarded differential equations in n-space, which satisfy some monotonicity properties. Sufficient conditions are given guaranteeing that solutions are contained in the probability (n - 1)-simplex. Existence and uniqueness of constant solutions are proved. It is also shown that every solution converges to a constant as t → ∞.
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    URL: http://dx.doi.org/10.1002/mma.1670060108
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    On the transmission problem of the helmholtz equation for quadrantsHerrn Prof. Dr. K. Nickel zum 60. Geburtstag am 9. 2. 1984 gewidmet (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 129-157 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The plane transmission problem of the Helmholtz equation for quadrants is characterized by a one-dimensional singular integral equation, which refers to the Fourier transform of the normal derivative of the solution along the x-axis. It is derived by solving the transmission problem for the upper and the lower half-plane involving a Neumann condition at y = 0. This is done by a two-dimensional Laplace transform technique. The inverse Laplace transform with respect to the second cartesian coordinate and the restriction of this one to y = 0 then lead to the integral equation. Thereby the transmission conditions of the original problem at y = 0 have to be taken into account. The resulting integral equation is of generalized Wiener-Hopf-type. It is solved via the contraction theorem imposing restricting conditions on the wave numbers.
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    URL: http://dx.doi.org/10.1002/mma.1670060110
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    Bifurcation of periodic solutions for wave equations with scalar nonlinearities (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 206-214 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We treat wave equations with “scalar nonlinearities” and demonstrate the connection between the bifurcation theory of an associated system of Hammerstein integral equations and the existence of periodic solutions of the nonlinear wave equation.
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    URL: http://dx.doi.org/10.1002/mma.1670060114
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    Weak solutions of the initial value problem for the unmodified non-linear vlasov equation (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 262-279 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of weak LL-solutions of the initial value problem for Vlasov's equation is proved under rather general assumptions. In the three-dimensional case the solutions exist on the entire time axis.
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    URL: http://dx.doi.org/10.1002/mma.1670060118
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    A generalized jacobi theta function (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 327-344 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The delta function initial condition solution v*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function \documentclass{article}\pagestyle{empty}\begin{document}$ \Theta (x,t) = \upsilon *(x,t;0) + 2\sum\limits_{n = 1}^\infty {v*(x,t;y_n)} $\end{document} for a sufficiently rapidly increasing and unbounded positive sequence {yy}. It is shown that Θ(x,t) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.
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    URL: http://dx.doi.org/10.1002/mma.1670060120
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    Applications of convex analysis and comparison functionals to extremum principles (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 353-370 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In many areas of applied mathematics, the govering equations can be expressed as dual equations which are elements in two vector spaces where a given functional defined on a cartesian product space has zero gradients. If this functional has a global saddle structure then the variational principle can be replaced by dual extremum principles of the Noble and Sewell type. The purpose of the present paper is to investigate how comparison functional which have convex, concave or saddle structure can be used to generate upper and lower bounding principles and critical sequences. The methods are illustrated by applications to the periodic solutions of a non-linear differential equation and the Fredholm integral equation.
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    URL: http://dx.doi.org/10.1002/mma.1670060122
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    Continuous dependence on geometry for the backward heat equation (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 433-448 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we are concerned with the development of criteria for stabilizing inherently unstable initial-boundary value problems under small errors in the geometry of the underlying domain. We consider in particular the initial-boundary-value problem for the backward heat equation assuming that some error has been made in characterizing the geometry of the domain under consideration. It is shown that solutions which belong to an appropriately defined constraint set depend continuously in L2 on errors in the geometry.
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    URL: http://dx.doi.org/10.1002/mma.1670060126
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    Transmutation via the momentum plane (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 467-495 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Transmutation methods are developed for equations of the form x2 ϕ“ + x2(k2” - q̃(x)) ϕ = (v2 - (1/4)) ϕ, with v as spectral variable, which correspond to problems in quantum scattering theory at fixed energy k2 (here v ˜ l + (1/2) with l complex angular momentum). Spectral formulas for transmutation kernels are constructed and the machinery of transmutation theory developed by the author for spectral variable k is shown to have a version here. General Kontrorovič-Lebedev theorems are also proved.
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    URL: http://dx.doi.org/10.1002/mma.1670060129
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    Two remarks on continuation procedures for solving some nonlinear equations (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 512-514 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let F be a mapping from Rn+1 into Rn, x(t) be a parametrization for ∣t - t0 ∣ 〈 c, of a smooth solution branch of the equation F(x) = 0, A(t) be the jacobian (n + 1) associated to a standard continuation method used for computing the solution branch. We study for a class of singularities at x(t0) the behaviour, near t = t0, of the determinant of A(t).
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    URL: http://dx.doi.org/10.1002/mma.1670060131
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    On the Vlasov-Fokker-Planck equation (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 527-538 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study a modification of the Vlasov-Poisson equation, obtained by adding a diffusion term with respect to velocity. It describes, from a physical point of view, a plasma in thermal equilibrium, in a mean field limit situation. We find that the already known results concerning existence and uniqueness of the solutions for the ordinary Vlasov equation (constructive results and counterexamples) translate to our case.
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    URL: http://dx.doi.org/10.1002/mma.1670060134
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    Galerkin's method and stability (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 347-377 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Approximation in least squares by Galerkin's method leads to a consideration of strongly minimal systems. Theorems are derived which permit the recognition of systems which are not strongly minimal from the characteristics of the elements themselves. Normalised systems cannot be strongly minimal without their eigenvalues being bounded above.Speared systems, which have desirable properties, are introduced and their main features determined. Convergence earmarks and error bounds are exposed.A new definition of stability, which is self-checking in a computation, is suggested and its attributes delineated.The extension of the theory to equations involving positive-definite operators is mentioned.
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    URL: http://dx.doi.org/10.1002/mma.1670020308
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    Die Lösung der Prae-Maxwellschen Gleichungen mit Hilfe vollständiger LösungssystemeUnserem verehrten Lehrer, Herrn Prof. Dr. C. Müller, zum 60. Geburtstag gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 410-418 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ⊂ R3. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .
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    URL: http://dx.doi.org/10.1002/mma.1670020404
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