105 research outputs found
A nonlocal eigenvalue problem and the stability of spikes for reaction-diffusion systems with fractional reaction rates
We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction-diffusion systems with
fractional reaction rates such as the Sel'kov model, the
Gray-Scott system, the hypercycle Eigen and Schuster, angiogenesis, and the generalized Gierer-Meinhardt
system.
We give some sufficient and explicit conditions for stability
by studying the corresponding nonlocal eigenvalue problem in a new
range of parameters
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence
We consider the closure problem for turbulence in the dry convective atmospheric boundary
layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large
plumes in the well mixed middle part up to the inversion that separates the CBL from the
stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF
approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02)
that additionally includes a term for background turbulence. Thus an exact solution is derived
and all higher order moments (HOMs) are explained by second order moments, correlation
coefficients and the skewness. The solution provides a proof of the extended universality
hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi-
normality of FOM). This refined hypothesis states that CBL turbulence can be considered as
result of a linear interpolation between the Gaussian and the very skewed turbulence regimes.
Although the extended universality hypothesis was confirmed by results of field
measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained
unexplained. These are now answered by the new model including the reasons of the
universality of the functional form of the HOMs, the significant scatter of the values of the
coefficients and the source of the magic of the linear interpolation. Finally, the closures
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predicted by the model are tested against measurements and LES data. Some of the other
issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area
coverage parameters of plumes (so called filling factors) with HOM will be discussed also
Hopf bifurcation from fronts in the Cahn-Hilliard equation
We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard
equation. The primary front is induced by a moving source term. Models of this
form have been used to study a variety of physical phenomena, including pattern
formation in chemical deposition and precipitation processes. Technically, we
study bifurcation in the presence of essential spectrum. We contribute a simple
and direct functional analytic method and determine bifurcation coefficients
explicitly. Our approach uses exponential weights to recover Fredholm
properties and spectral flow ideas to compute Fredholm indices. Simple mass
conservation helps compensate for negative indices. We also construct an
explicit, prototypical example, prove the existence of a bifurcating front, and
determine the direction of bifurcation
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