- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 09:18:51
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=6&iss=4&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2004
4
A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence
F.
Benth
University of Oslo, OSLO, NORWAY
Kenneth
Karlsen
University of Oslo, OSLO, NORWAY
K.
Reikvam
University of Oslo, OSLO, NORWAY
American option, semilinear Black and Scholes partial differential equation, viscosity solution, approximate solutions, numerical schemes, convergence
In \cite{BKR:AmOpI}, we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of 'easy to implement' numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of 'predictor-corrector type' for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor--corrector scheme respect the early exercise constraint and that they converge uniformly to the the American option value. A numerical example is also presented. Besides the predictor--corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed as well.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
379
404
10.4171/IFB/106
http://www.ems-ph.org/doi/10.4171/IFB/106
Existence of self-similar evolution of crystals grown from supersaturated vapor
Yoshikazu
Giga
University of Tokyo, TOKYO, JAPAN
Piotr
Rybka
Warsaw University, WARSZAWA, POLAND
Crystal growth, diffusion field, self-similarity, interfacial energy
We study a cylindrical crystalline flow in three dimensions coupled to a diffusion field. This system arises in modeling crystals grown from supersaturated vapor. We show existence of self-similar solutions to this system under a special choice of interfacial energy and kinetic coefficients.
Partial differential equations
Numerical analysis
Biology and other natural sciences
Mechanics of particles and systems
405
421
10.4171/IFB/107
http://www.ems-ph.org/doi/10.4171/IFB/107
Metastable behavior of premixed gas flames in rectangular channels
Henry
Berestycki
Ecole des hautes études en sciences sociales, PARIS Cedex 13, FRANCE
Leonid
Kagan
Tel-Aviv University, TEL-AVIV, ISRAEL
Shoshana
Kamin
Tel-Aviv University, TEL-AVIV, ISRAEL
Gregory
Sivashinsky
Tel-Aviv University, TEL-AVIV, ISRAEL
flame propagation, flame interface
A two-dimensional model for the upward propagating flame in a vertical square channel is explored. It is proved that under certain special initial conditions, the point where the flame interface attains its maximum, stays off the boundary (channel's wall) for an exponentially long period of time. The proof is an extension of the analysis developed previously for the one-dimensional version of the problem.
Partial differential equations
Classical thermodynamics, heat transfer
General
423
438
10.4171/IFB/108
http://www.ems-ph.org/doi/10.4171/IFB/108
The energy density of martensitic thin films via dimension reduction
Lorenzo
Freddi
Università di Udine, UDINE, ITALY
Roberto
Paroni
Università di Sassari, ALGHERO, ITALY
martensitic thin films, membrane microstructure, variational limit, Young measure, dimension reduction
A variational limit defined on the space of bi-dimensional gradient Young measures is obtained from three-dimensional elasticity via dimension reduction. The obtained limit problem uniquely determines the energy density of the thin film. Our result might be used to compute the microstructure in membranes made of phase transforming material.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
439
459
10.4171/IFB/109
http://www.ems-ph.org/doi/10.4171/IFB/109
On the stability of non-symmetric equilibrium figures of a rotating viscous incompressible liquid
Vsevolod
Solonnikov
Russian Acadademy of Sciences, ST. PETERSBURG, RUSSIAN FEDERATION
rotating liquid, evolution problem, periodic solution, energy functional
We consider a classical problem of stability of equilibrium figures of a liquid rotating uniformly as a rigid body about a fixed axis. We connect the problem of stability with the behavior for large $t$ of solutions of an evolution problem governing a motion of an isolated liquid mass whose initial data are slight perturbations of the regime of rigid rotation. The main attention is given to the case when the figure is not rotationally symmetric; in this case the regime of a rigid rotation defines a periodic solution of the above-mentioned non-stationary problem. It is proved that the sufficient condition of stability is the positivity of the second variation of the energy functional in an appropriate space of functions.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
461
492
10.4171/IFB/110
http://www.ems-ph.org/doi/10.4171/IFB/110