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  • 1
    Electronic Resource
    Electronic Resource
    Existenz und erste Einschließung positiver Lösungen bei superlinearen Randwertaufgaben zweiter Ordnung (1976)
    Springer
    Numerische Mathematik 26 (1976), S. 421-428 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Some classes of second-order superlinear boundary value problems are considered. Using M. A. Krasnoselskii's fixed point theorem for operators expanding a cone in a real Banach space we can prove new existence theorems for positive solutions for nonlinearities not necessarily satisfying a condition of Nagumo-type. At the same time, we give first bounds for the solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01409963
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  • 2
    Electronic Resource
    Electronic Resource
    Pointwise inclusions of fixed points by finite dimensional iteration schemes (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 381-392 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65R05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Finite dimensional iteration schemes which provide pointwise bounds for the solutions of nonlinear integral equations are considered. The method is based on a discretization technique which takes advantage of apriori known structure properties of the solutions. The resulting iteration can be carried out on a computer for as many steps as desired. Its high degree of accuracy is shown by numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01401042
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  • 3
    Electronic Resource
    Electronic Resource
    Exact bounds for the solution branches of nonlinear eigenvalue problems (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 29-40 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N25 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=λTu. The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463996
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  • 4
    Electronic Resource
    Electronic Resource
    Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 759-778 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au23Cu30Zn47 demonstrate the quality of the discrete model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385653
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  • 5
    Electronic Resource
    Electronic Resource
    On the uniqueness of conduction in relaxation-case semiconductors (1981)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 32 (1981), S. 344-346 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung In dieser Arbeit wird bewiesen, daß eine vereinfachte Version des Differentialgleichungssystems, durch das Relaxationshalbleiter modelliert werden, höchstens eine Lösung besitzen kann. Dieses Resultat zeigt, daß das vereinfachte Modell die physikalische Erscheinung des „Switching“ in Halbleitern nicht erklären kann.
    Notes: Abstract In this paper it is proved that a certain simplified version of the system of nonlinear differential equations which describe the relaxation regime in a semiconductor has at most one solution. This uniqueness result shows that the simplified system does not model the physical phenomenon of “switching” in semiconductors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00946649
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  • 6
    Electronic Resource
    Electronic Resource
    Hysteresis operators in phase-field models of Penrose-fife type (1998)
    Springer
    Applications of mathematics 43 (1998), S. 207-222 
    Details
    ISSN: 1572-9109
    Keywords: Phase-field systems ; phase transitions ; hysteresis operators ; well-posedness of parabolic systems ; thermodynamic consistency ; Penrose-Fife model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1023276524286
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity (1998)
    Springer
    Applications of mathematics 43 (1998), S. 173-205 
    Details
    ISSN: 1572-9109
    Keywords: thermoplasticity ; viscoelasticity ; hysteresis ; Prandtl-Ishlinskii operator ; PDEs with hysteresis ; thermodynamical consistency
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we develop a thermodynamically consistent description of the uniaxial behavior of thermovisco-elastoplastic materials for which the total stress σ contains, in addition to elastic, viscous and thermic contributions, a plastic component σ p of the form σp(x,t)=Ρ[ε, θ(x,t)](x,t). Here ∈ and θ are the fields of strain and absolute temperature, respectively, and {Ρ[·, θ]}θ〉0 denotes a family of (rate-independent) hysteresis operators of Prandtl-Ishlinskii type, parametrized by the absolute temperature. The system of momentum and energy balance equations governing the space-time evolution of the material forms a system of two highly nonlinearly coupled partial differential equations involving partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a unique global strong solution which depends continuously on the data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1023224507448
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  • 8
    Electronic Resource
    Electronic Resource
    On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type (1995)
    Springer
    Annali di matematica pura ed applicata 169 (1995), S. 269-289 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arise naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with nonconserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01759357
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    Optimal control of thermomechanical phase transitions in shape memory alloys: Necessary conditions of optimality (1991)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 14 (1991), S. 265-280 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The necessary conditions for the optimality of an optimal control problem associated with soiid-solid phase transitions in shape memory alloys are established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670140404
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  • 10
    Electronic Resource
    Electronic Resource
    Global Solutions to a Penrose-Fife Phase-field Model Under Flux Boundary Conditions for the Inverse Temperature (1996)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 19 (1996), S. 1053-1072 
    Details
    ISSN: 0170-4214
    Keywords: Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper, we study an initial-boundary value problem for a system of phase-field equations arising from the Penrose-Fife approach to model the kinetics of phase transitions. In contrast to other recent works in the field, the correct form of the boundary condition for the temperature field is assumed which leads to additional difficulties in the mathematical treatment. It is demonstrated that global existence of strong solutions can be shown under essentially the same assumptions on the data as in the previous papers where a simplified boundary condition for the heat exchange with the surrounding medium has been used.
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
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