592 research outputs found
Fokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits
It is proved that the distributions of scaling limits of Continuous Time
Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck
Equations for diffusion processes. In contrast to previous such results, it is
not assumed that the underlying process has absolutely continuous laws.
Moreover, governing equations in the backward variables are derived. Three
examples of anomalous diffusion processes illustrate the theory.Comment: in Proceedings of the American Mathematical Society, Published
electronically July 12, 201
Competitive interference of plant species in monocultures and mixed stands
A dynamic model is presented for calculating the dry matter yields of individual plant species in a mixed stand by means of parameters derived from a spacing experiment with species grown at 2 densities and harvested at regular intervals. Results are given of trials with different crops, including tall and dwarf peas. The model is intended for use under conditions of near-optimum supply of water and nutrients, where the principal competition is for light. A simple method for measuring relative light interception by species in mixed stands is also described
Brownian subordinators and fractional Cauchy problems
A Brownian time process is a Markov process subordinated to the absolute
value of an independent one-dimensional Brownian motion. Its transition
densities solve an initial value problem involving the square of the generator
of the original Markov process. An apparently unrelated class of processes,
emerging as the scaling limits of continuous time random walks, involve
subordination to the inverse or hitting time process of a classical stable
subordinator. The resulting densities solve fractional Cauchy problems, an
extension that involves fractional derivatives in time. In this paper, we will
show a close and unexpected connection between these two classes of processes,
and consequently, an equivalence between these two families of partial
differential equations.Comment: 18 pages, minor spelling correction
A Vector-Valued Operational Calculus and Abstract Cauchy Problems.
Initial and boundary value problems for linear differential and integro-differential equations are at the heart of mathematical analysis. About 100 years ago, Oliver Heaviside promoted a set of formal, algebraic rules which allow a complete analysis of a large class of such problems. Although Heaviside\u27s operational calculus was entirely heuristic in nature, it almost always led to correct results. This encouraged many mathematicians to search for a solid mathematical foundation for Heaviside\u27s method, resulting in two competing mathematical theories: (a) Laplace transform theory for functions, distributions and other generalized functions, (b) J. Mikusinski\u27s field of convolution quotients of continuous functions. In this dissertation we will investigate a unifying approach to Heaviside\u27s operational calculus which allows us to extend the method to vector-valued functions. The main components are (a) a new approach to generalized functions, considering them not primarily as functionals on a space of test functions or as convolution quotients in Mikusinski\u27s quotient field, but as limits of continuous functions in appropriate norms, and (b) an asymptotic extension of the classical Laplace transform allowing the transform of functions and generalized functions of arbitrary growth at infinity. The mathematics are based on a careful analysis of the convolution transform This is done via a new inversion formula for the Laplace transform, which enables us to extend Titchmarsh\u27s injectivity theorem and Foias\u27 dense range theorem for the convolution transform to Banach space valued functions. The abstract results are applied to abstract Cauchy problems. We indicate the manner in which the operational methods can be employed to obtain existence and uniqueness results for initial value problems for differential equations in Banach spaces
Boundary Conditions for Fractional Diffusion
This paper derives physically meaningful boundary conditions for fractional
diffusion equations, using a mass balance approach. Numerical solutions are
presented, and theoretical properties are reviewed, including well-posedness
and steady state solutions. Absorbing and reflecting boundary conditions are
considered, and illustrated through several examples. Reflecting boundary
conditions involve fractional derivatives. The Caputo fractional derivative is
shown to be unsuitable for modeling fractional diffusion, since the resulting
boundary value problem is not positivity preserving
Operando characterization of interfacial charge transfer processes
Interface science has become a key aspect for fundamental research questions
and for the understanding, design and optimization of urgently needed energy
and information technologies. As the interface properties change during
operation, e.g. under applied electrochemical stimulus, and because multiple
bulk and interface processes coexist and compete, detailed operando
characterization is needed. In this perspective, I present an overview of the
state-of-the art and challenges in selected X-ray spectroscopic techniques,
concluding that among others, interface-sensitivity remains a major concern in
the available techniques. I propose and discuss a new method to extract
interface-information from nominally bulk sensitive techniques, and critically
evaluate the selection of X-ray energies for the recently developed meniscus
X-ray photoelectron spectroscopy, a promising operando tool to characterize the
solid-liquid interface. I expect that these advancements along with further
developments in time and spatial resolution will expand our ability to probe
the interface electronic and molecular structure with sub-nm depth and complete
our understanding of charge transfer processes during operation.Comment: 44 pages, 8 figure
Space-time duality for fractional diffusion
Zolotarev proved a duality result that relates stable densities with
different indices. In this paper, we show how Zolotarev duality leads to some
interesting results on fractional diffusion. Fractional diffusion equations
employ fractional derivatives in place of the usual integer order derivatives.
They govern scaling limits of random walk models, with power law jumps leading
to fractional derivatives in space, and power law waiting times between the
jumps leading to fractional derivatives in time. The limit process is a stable
L\'evy motion that models the jumps, subordinated to an inverse stable process
that models the waiting times. Using duality, we relate the density of a
spectrally negative stable process with index to the density of
the hitting time of a stable subordinator with index , and thereby
unify some recent results in the literature. These results also provide a
concrete interpretation of Zolotarev duality in terms of the fractional
diffusion model.Comment: 16 page
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