174,589 research outputs found
Operator splittings and spatial approximations for evolution equations
The convergence of various operator splitting procedures, such as the
sequential, the Strang and the weighted splitting, is investigated in the
presence of a spatial approximation. To this end a variant of Chernoff's
product formula is proved. The methods are applied to abstract partial delay
differential equations.Comment: to appear in J. Evol. Equations. Reviewers comments are incorporate
Regular order reductions of ordinary and delay-differential equations
We present a C program to compute by successive approximations the regular
order reduction of a large class of ordinary differential equations, which
includes evolution equations in electrodynamics and gravitation. The code may
also find the regular order reduction of delay-differential equations.Comment: 4 figure
A fast and precise method to solve the Altarelli-Parisi equations in x space
A numerical method to solve linear integro-differential equations is
presented. This method has been used to solve the QCD Altarelli-Parisi
evolution equations within the H1 Collaboration at DESY-Hamburg. Mathematical
aspects and numerical approximations are described. The precision of the method
is discussed.Comment: 18 pages, 4 figure
Deterministic Equations for Stochastic Spatial Evolutionary Games
Spatial evolutionary games model individuals who are distributed in a spatial
domain and update their strategies upon playing a normal form game with their
neighbors. We derive integro-differential equations as deterministic
approximations of the microscopic updating stochastic processes. This
generalizes the known mean-field ordinary differential equations and provide a
powerful tool to investigate the spatial effects in populations evolution. The
deterministic equations allow to identify many interesting features of the
evolution of strategy profiles in a population, such as standing and traveling
waves, and pattern formation, especially in replicator-type evolutions
Enzyme kinetics far from the standard quasi-steady-state and equilibrium approximations
Analytic approximations of the time-evolution of the single enzyme-substrate reaction are valid for all but a small region of parameter space in the positive initial enzyme-initial substrate concentration plane. We find velocity equations for the substrate decomposition and product formation with the aid of the total quasi-steady-state approximation and an aggregation technique for cases where neither the more normally employed standard nor reverse quasi-steady-state approximations are valid. Applications to determining reaction kinetic parameters are discussed
Non-linear QCD evolution with improved triple-pomeron vertices
In a previous publication, we have constructed a set of non-linear evolution
equations for dipole scattering amplitudes in QCD at high energy, which extends
the Balitsky-JIMWLK hierarchy by including the effects of fluctuations in the
gluon number in the target wavefunction. In doing so, we have relied on the
color dipole picture, valid in the limit where the number of colors is large,
and we have made some further approximations on the relation between scattering
amplitudes and dipole densities, which amount to neglecting the non-locality of
the two-gluon exchanges. In this Letter, we relax the latter approximations,
and thus restore the correct structure of the `triple-pomeron vertex' which
describes the splitting of one BFKL pomeron into two within the terms
responsible for fluctuations. The ensuing triple-pomeron vertex coincides with
the one previously derived by Braun and Vacca within perturbative QCD. The
evolution equations can be recast in a Langevin form, but with a multivariable
noise term with off-diagonal correlations. Our equations are shown to be
equivalent with the modified version of the JIMWLK equation recently proposed
by Mueller, Shoshi, and Wong.Comment: 15 page
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