1,580 research outputs found
Diffusion, Fragmentation and Coagulation Processes: Analytical and Numerical Results
We formulate dynamical rate equations for physical processes driven by a
combination of diffusive growth, size fragmentation and fragment coagulation.
Initially, we consider processes where coagulation is absent. In this case we
solve the rate equation exactly leading to size distributions of Bessel type
which fall off as for large -values. Moreover, we provide
explicit formulas for the expansion coefficients in terms of Airy functions.
Introducing the coagulation term, the full non-linear model is mapped exactly
onto a Riccati equation that enables us to derive various asymptotic solutions
for the distribution function. In particular, we find a standard exponential
decay, , for large , and observe a crossover from the Bessel
function for intermediate values of . These findings are checked by
numerical simulations and we find perfect agreement between the theoretical
predictions and numerical results.Comment: (28 pages, 6 figures, v2+v3 minor corrections
Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling
The objective of this paper is to complete certain issues from our recent
contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random
non-autonomous second order linear differential equations: mean square analytic
solutions and their statistical properties, Advances in Difference Equations,
2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with
the homogeneous case, so that the hypotheses are clearer and also easier to
check in applications. Another novelty is that we tackle the non-homogeneous
equation with a theorem of existence of mean square analytic solution and a
numerical example. We also prove the uniqueness of mean square solution via an
habitual Lipschitz condition that extends the classical Picard Theorem to mean
square calculus. In this manner, the study on general random non-autonomous
second order linear differential equations with analytic data processes is
completely resolved. Finally, we relate our exposition based on random power
series with polynomial chaos expansions and the random differential transform
method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table
Control Strategies for the Fokker-Planck Equation
Using a projection-based decoupling of the Fokker-Planck equation, control
strategies that allow to speed up the convergence to the stationary
distribution are investigated. By means of an operator theoretic framework for
a bilinear control system, two different feedback control laws are proposed.
Projected Riccati and Lyapunov equations are derived and properties of the
associated solutions are given. The well-posedness of the closed loop systems
is shown and local and global stabilization results, respectively, are
obtained. An essential tool in the construction of the controls is the choice
of appropriate control shape functions. Results for a two dimensional double
well potential illustrate the theoretical findings in a numerical setup
An exactly solvable self-convolutive recurrence
We consider a self-convolutive recurrence whose solution is the sequence of
coefficients in the asymptotic expansion of the logarithmic derivative of the
confluent hypergeometic function . By application of the Hilbert
transform we convert this expression into an explicit, non-recursive solution
in which the th coefficient is expressed as the th moment of a
measure, and also as the trace of the th iterate of a linear operator.
Applications of these sequences, and hence of the explicit solution provided,
are found in quantum field theory as the number of Feynman diagrams of a
certain type and order, in Brownian motion theory, and in combinatorics
Adaptive high-order splitting schemes for large-scale differential Riccati equations
We consider high-order splitting schemes for large-scale differential Riccati
equations. Such equations arise in many different areas and are especially
important within the field of optimal control. In the large-scale case, it is
critical to employ structural properties of the matrix-valued solution, or the
computational cost and storage requirements become infeasible. Our main
contribution is therefore to formulate these high-order splitting schemes in a
efficient way by utilizing a low-rank factorization. Previous results indicated
that this was impossible for methods of order higher than 2, but our new
approach overcomes these difficulties. In addition, we demonstrate that the
proposed methods contain natural embedded error estimates. These may be used
e.g. for time step adaptivity, and our numerical experiments in this direction
show promising results.Comment: 23 pages, 7 figure
An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
We describe a method for the rapid numerical evaluation of the Bessel
functions of the first and second kinds of nonnegative real orders and positive
arguments. Our algorithm makes use of the well-known observation that although
the Bessel functions themselves are expensive to represent via piecewise
polynomial expansions, the logarithms of certain solutions of Bessel's equation
are not. We exploit this observation by numerically precomputing the logarithms
of carefully chosen Bessel functions and representing them with piecewise
bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions
of orders between and 1\sep,000\sep,000\sep,000 at essentially any
positive real argument. In that regime, it is competitive with existing methods
for the rapid evaluation of Bessel functions and has several advantages over
them. First, our approach is quite general and can be readily applied to many
other special functions which satisfy second order ordinary differential
equations. Second, by calculating the logarithms of the Bessel functions rather
than the Bessel functions themselves, we avoid many issues which arise from
numerical overflow and underflow. Third, in the oscillatory regime, our
algorithm calculates the values of a nonoscillatory phase function for Bessel's
differential equation and its derivative. These quantities are useful for
computing the zeros of Bessel functions, as well as for rapidly applying the
Fourier-Bessel transform. The results of extensive numerical experiments
demonstrating the efficacy of our algorithm are presented. A Fortran package
which includes our code for evaluating the Bessel functions as well as our code
for all of the numerical experiments described here is publically available
One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
We study the one-dimensional Schr\"odinger equation with a disordered
potential of the form where
is a Gaussian white noise with mean and variance , and
is a random superposition of delta functions distributed uniformly on the real
line with mean density and mean strength . Our study is motivated by
the close connection between this problem and classical diffusion in a random
environment (the Sinai problem) in the presence of random absorbers~:
models the force field acting on the diffusing particle and models
the absorption properties of the medium in which the diffusion takes place. The
focus is on the calculation of the complex Lyapunov exponent , where is the integrated density of
states per unit length and the reciprocal of the localisation length.
By using the continuous version of the Dyson-Schmidt method, we find an exact
formula, in terms of a Hankel function, in the particular case where the
strength of the delta functions is exponentially-distributed with mean .
Building on this result, we then solve the general case -- in the low-energy
limit -- in terms of an infinite sum of Hankel functions. Our main result,
valid without restrictions on the parameters of the model, is that the
integrated density of states exhibits the power law behaviour
N(E) \underset{E\to0+}{\sim} E^\nu \hspace{0.5cm} \mbox{where }
\nu=\sqrt{\mu^2+2\rho/g}\:.
This confirms and extends several results obtained previously by approximate
methods.Comment: LaTeX, 44 pages, 17 pdf figure
Asymptotic Exit Location Distributions in the Stochastic Exit Problem
Consider a two-dimensional continuous-time dynamical system, with an
attracting fixed point . If the deterministic dynamics are perturbed by
white noise (random perturbations) of strength , the system state
will eventually leave the domain of attraction of . We analyse the
case when, as , the exit location on the boundary
is increasingly concentrated near a saddle point of the
deterministic dynamics. We show that the asymptotic form of the exit location
distribution on is generically non-Gaussian and asymmetric,
and classify the possible limiting distributions. A key role is played by a
parameter , equal to the ratio of the stable
and unstable eigenvalues of the linearized deterministic flow at . If
then the exit location distribution is generically asymptotic as
to a Weibull distribution with shape parameter , on the
length scale near . If it is generically
asymptotic to a distribution on the length scale, whose
moments we compute. The asymmetry of the asymptotic exit location distribution
is attributable to the generic presence of a `classically forbidden' region: a
wedge-shaped subset of with as vertex, which is reached from ,
in the limit, only via `bent' (non-smooth) fluctuational paths
that first pass through the vicinity of . We deduce from the presence of
this forbidden region that the classical Eyring formula for the
small- exponential asymptotics of the mean first exit time is
generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy
requests to [email protected] or [email protected]
Multiscale differential Riccati equations for linear quadratic regulator problems
We consider approximations to the solutions of differential Riccati equations
in the context of linear quadratic regulator problems, where the state equation
is governed by a multiscale operator. Similarly to elliptic and parabolic
problems, standard finite element discretizations perform poorly in this
setting unless the grid resolves the fine-scale features of the problem. This
results in unfeasible amounts of computation and high memory requirements. In
this paper, we demonstrate how the localized orthogonal decomposition method
may be used to acquire accurate results also for coarse discretizations, at the
low cost of solving a series of small, localized elliptic problems. We prove
second-order convergence (except for a logarithmic factor) in the
operator norm, and first-order convergence in the corresponding energy norm.
These results are both independent of the multiscale variations in the state
equation. In addition, we provide a detailed derivation of the fully discrete
matrix-valued equations, and show how they can be handled in a low-rank setting
for large-scale computations. In connection to this, we also show how to
efficiently compute the relevant operator-norm errors. Finally, our theoretical
results are validated by several numerical experiments.Comment: Accepted for publication in SIAM J. Sci. Comput. This version differs
from the previous one only by the addition of Remark 7.2 and minor changes in
formatting. 21 pages, 12 figure
Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
The concept of Lyapunov exponent has long occupied a central place in the
theory of Anderson localisation; its interest in this particular context is
that it provides a reasonable measure of the localisation length. The Lyapunov
exponent also features prominently in the theory of products of random matrices
pioneered by Furstenberg. After a brief historical survey, we describe some
recent work that exploits the close connections between these topics. We review
the known solvable cases of disordered quantum mechanics involving random point
scatterers and discuss a new solvable case. Finally, we point out some
limitations of the Lyapunov exponent as a means of studying localisation
properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on
"Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change
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