2,108 research outputs found
Diffusion in a crowded environment
We analyze a pair of diffusion equations which are derived in the infinite
system--size limit from a microscopic, individual--based, stochastic model.
Deviations from the conventional Fickian picture are found which ultimately
relate to the depletion of resources on which the particles rely. The
macroscopic equations are studied both analytically and numerically, and are
shown to yield anomalous diffusion which does not follow a power law with time,
as is frequently assumed when fitting data for such phenomena. These anomalies
are here understood within a consistent dynamical picture which applies to a
wide range of physical and biological systems, underlining the need for clearly
defined mechanisms which are systematically analyzed to give definite
predictions.Comment: 4 pages, 3 figures, minor change
Reaction-Diffusion Process Driven by a Localized Source: First Passage Properties
We study a reaction-diffusion process that involves two species of atoms,
immobile and diffusing. We assume that initially only immobile atoms, uniformly
distributed throughout the entire space, are present. Diffusing atoms are
injected at the origin by a source which is turned on at time t=0. When a
diffusing atom collides with an immobile atom, the two atoms form an immobile
stable molecule. The region occupied by molecules is asymptotically spherical
with radius growing as t^{1/d} in d>=2 dimensions. We investigate the survival
probability that a diffusing atom has not become a part of a molecule during
the time interval t after its injection and the probability density of such a
particle. We show that asymptotically the survival probability (i) saturates in
one dimension, (ii) vanishes algebraically with time in two dimensions (with
exponent being a function of the dimensionless flux and determined as a zero of
a confluent hypergeometric function), and (iii) exhibits a stretched
exponential decay in three dimensions.Comment: 7 pages; version 2: section IV is re-written, references added, 8
pages (final version
Role of proton irradiation and relative air humidity on iron corrosion
This paper presents a study of the effects of proton irradiation on iron
corrosion. Since it is known that in humid atmospheres, iron corrosion is
enhanced by the double influence of air and humidity, we studied the iron
corrosion under irradiation with a 45% relative humidity. Three proton beam
intensities (5, 10 and 20 nA) were used. To characterise the corrosion layer,
we used ion beam methods (Rutherford Backscattering Spectrometry (RBS), Elastic
Recoil Detection Analysis (ERDA)) and X-ray Diffraction (XRD) analysis. The
corrosion kinetics are plotted for each proton flux. A diffusion model of the
oxidant species is proposed, taking into account the fact that the flux through
the surface is dependent on the kinetic factor K. This model provides evidence
for the dependence of the diffusion coefficient, D, and the kinetic factor, K,
on the proton beam intensity. Comparison of the values for D with the diffusion
coefficients for thermal oxygen diffusion in iron at 300 K suggests an
enhancement due to irradiation of 6 orders of magnitude
2D pattern evolution constrained by complex network dynamics
Complex networks have established themselves along the last years as being
particularly suitable and flexible for representing and modeling several
complex natural and human-made systems. At the same time in which the
structural intricacies of such networks are being revealed and understood,
efforts have also been directed at investigating how such connectivity
properties define and constrain the dynamics of systems unfolding on such
structures. However, lesser attention has been focused on hybrid systems,
\textit{i.e.} involving more than one type of network and/or dynamics. Because
several real systems present such an organization (\textit{e.g.} the dynamics
of a disease coexisting with the dynamics of the immune system), it becomes
important to address such hybrid systems. The current paper investigates a
specific system involving a diffusive (linear and non-linear) dynamics taking
place in a regular network while interacting with a complex network of
defensive agents following Erd\"os-R\'enyi and Barab\'asi-Albert graph models,
whose nodes can be displaced spatially. More specifically, the complex network
is expected to control, and if possible to extinguish, the diffusion of some
given unwanted process (\textit{e.g.} fire, oil spilling, pest dissemination,
and virus or bacteria reproduction during an infection). Two types of pattern
evolution are considered: Fick and Gray-Scott. The nodes of the defensive
network then interact with the diffusing patterns and communicate between
themselves in order to control the spreading. The main findings include the
identification of higher efficiency for the Barab\'asi-Albert control networks.Comment: 18 pages, 32 figures. A working manuscript, comments are welcome
Solving two-phase freezing Stefan problems: Stability and monotonicity
[EN] The two-phase Stefan problems with phase formation and depletion are special
cases ofmoving boundary problemswith interest in science and industry. In this
work, we study a solidification problem, introducing a front-fixing transformation.
The resulting non-linear partial differential system involves singularities,
both at the beginning of the freezing process and when the depletion is complete,
that are treated with special attention in the numerical modelling. The
problem is decomposed in three stages, in which implicit and explicit finite
difference schemes are used. Numerical analysis reveals qualitative properties
of the numerical solution spatial monotonicity of both solid and liquid temperatures
and the evolution of the solidification front. Numerical experiments
illustrate the behaviour of the temperatures profiles with time, as well as the
dynamics of the solidification front.Ministerio de Ciencia, Innovacion y Universidades, Grant/Award Number: MTM2017-89664-P.Piqueras, MA.; Company Rossi, R.; Jódar Sánchez, LA. (2020). Solving two-phase freezing Stefan problems: Stability and monotonicity. Mathematical Methods in the Applied Sciences. 43(14):7948-7960. https://doi.org/10.1002/mma.5787S794879604314Schmidt, A. (1996). Computation of Three Dimensional Dendrites with Finite Elements. Journal of Computational Physics, 125(2), 293-312. doi:10.1006/jcph.1996.0095Singh, S., & Bhargava, R. (2014). Simulation of Phase Transition During Cryosurgical Treatment of a Tumor Tissue Loaded With Nanoparticles Using Meshfree Approach. Journal of Heat Transfer, 136(12). doi:10.1115/1.4028730Company, R., Egorova, V. N., & Jódar, L. (2014). Solving American Option Pricing Models by the Front Fixing Method: Numerical Analysis and Computing. Abstract and Applied Analysis, 2014, 1-9. doi:10.1155/2014/146745Griewank, P. J., & Notz, D. (2013). Insights into brine dynamics and sea ice desalination from a 1-D model study of gravity drainage. Journal of Geophysical Research: Oceans, 118(7), 3370-3386. doi:10.1002/jgrc.20247Javierre, E., Vuik, C., Vermolen, F. J., & van der Zwaag, S. (2006). A comparison of numerical models for one-dimensional Stefan problems. Journal of Computational and Applied Mathematics, 192(2), 445-459. doi:10.1016/j.cam.2005.04.062Briozzo, A. C., Natale, M. F., & Tarzia, D. A. (2007). Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases. Journal of Mathematical Analysis and Applications, 329(1), 145-162. doi:10.1016/j.jmaa.2006.05.083Caldwell, J., & Chan, C.-C. (2000). Spherical solidification by the enthalpy method and the heat balance integral method. Applied Mathematical Modelling, 24(1), 45-53. doi:10.1016/s0307-904x(99)00031-1Chantasiriwan, S., Johansson, B. T., & Lesnic, D. (2009). The method of fundamental solutions for free surface Stefan problems. Engineering Analysis with Boundary Elements, 33(4), 529-538. doi:10.1016/j.enganabound.2008.08.010Hon, Y. C., & Li, M. (2008). A computational method for inverse free boundary determination problem. International Journal for Numerical Methods in Engineering, 73(9), 1291-1309. doi:10.1002/nme.2122RIZWAN-UDDIN. (1999). A Nodal Method for Phase Change Moving Boundary Problems. International Journal of Computational Fluid Dynamics, 11(3-4), 211-221. doi:10.1080/10618569908940875Caldwell, J., & Kwan, Y. Y. (2003). On the perturbation method for the Stefan problem with time-dependent boundary conditions. International Journal of Heat and Mass Transfer, 46(8), 1497-1501. doi:10.1016/s0017-9310(02)00415-5Stephan, K., & Holzknecht, B. (1976). Die asymptotischen lösungen für vorgänge des erstarrens. International Journal of Heat and Mass Transfer, 19(6), 597-602. doi:10.1016/0017-9310(76)90042-9Savović, S., & Caldwell, J. (2003). Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. International Journal of Heat and Mass Transfer, 46(15), 2911-2916. doi:10.1016/s0017-9310(03)00050-4Kutluay, S., Bahadir, A. R., & Özdeş, A. (1997). The numerical solution of one-phase classical Stefan problem. Journal of Computational and Applied Mathematics, 81(1), 135-144. doi:10.1016/s0377-0427(97)00034-4Asaithambi, N. S. (1997). A variable time step Galerkin method for a one-dimensional Stefan problem. Applied Mathematics and Computation, 81(2-3), 189-200. doi:10.1016/0096-3003(95)00329-0Landau, H. G. (1950). Heat conduction in a melting solid. Quarterly of Applied Mathematics, 8(1), 81-94. doi:10.1090/qam/33441Churchill, S. W., & Gupta, J. P. (1977). Approximations for conduction with freezing or melting. International Journal of Heat and Mass Transfer, 20(11), 1251-1253. doi:10.1016/0017-9310(77)90134-xKutluay, S., & Esen, A. (2004). An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition. Applied Mathematics and Computation, 150(1), 59-67. doi:10.1016/s0096-3003(03)00197-8Esen, A., & Kutluay, S. (2004). A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method. Applied Mathematics and Computation, 148(2), 321-329. doi:10.1016/s0096-3003(02)00846-9Mitchell, S. L., & Vynnycky, M. (2016). On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. Journal of Computational and Applied Mathematics, 300, 259-274. doi:10.1016/j.cam.2015.12.021Meek, P. C., & Norbury, J. (1984). Nonlinear Moving Boundary Problems and a Keller Box Scheme. SIAM Journal on Numerical Analysis, 21(5), 883-893. doi:10.1137/0721057Tarzia, D. (2017). Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions. Thermal Science, 21(1 Part A), 187-197. doi:10.2298/tsci140607003tPlemmons, R. J. (1977). M-matrix characterizations.I—nonsingular M-matrices. Linear Algebra and its Applications, 18(2), 175-188. doi:10.1016/0024-3795(77)90073-8Axelsson, O. (1994). Iterative Solution Methods. doi:10.1017/cbo978051162410
Dissolution in a field
We study the dissolution of a solid by continuous injection of reactive
``acid'' particles at a single point, with the reactive particles undergoing
biased diffusion in the dissolved region. When acid encounters the substrate
material, both an acid particle and a unit of the material disappear. We find
that the lengths of the dissolved cavity parallel and perpendicular to the bias
grow as t^{2/(d+1)} and t^{1/(d+1)}, respectively, in d-dimensions, while the
number of reactive particles within the cavity grows as t^{2/(d+1)}. We also
obtain the exact density profile of the reactive particles and the relation
between this profile and the motion of the dissolution boundary. The extension
to variable acid strength is also discussed.Comment: 6 pages, 6 figures, 2-column format, for submission to PR
From the solutions of diffusion equation to the solutions of subdiffusive one
Starting with the Green's functions found for normal diffusion, we construct
exact time-dependent Green's functions for subdiffusive equation (with
fractional time derivatives), with the boundary conditions involving a linear
combination of fluxes and concentrations. The method is particularly useful to
calculate the concentration profiles in a multi-part system where different
kind of transport occurs in each part of it. As an example, we find the
solutions of subdiffusive equation for the system composed from two parts with
normal diffusion and subdiffusion, respectively.Comment: 11 pages, 2 figure
Realistic boundary conditions for stochastic simulations of reaction-diffusion processes
Many cellular and subcellular biological processes can be described in terms
of diffusing and chemically reacting species (e.g. enzymes). Such
reaction-diffusion processes can be mathematically modelled using either
deterministic partial-differential equations or stochastic simulation
algorithms. The latter provide a more detailed and precise picture, and several
stochastic simulation algorithms have been proposed in recent years. Such
models typically give the same description of the reaction-diffusion processes
far from the boundary of the simulated domain, but the behaviour close to a
reactive boundary (e.g. a membrane with receptors) is unfortunately
model-dependent. In this paper, we study four different approaches to
stochastic modelling of reaction-diffusion problems and show the correct choice
of the boundary condition for each model. The reactive boundary is treated as
partially reflective, which means that some molecules hitting the boundary are
adsorbed (e.g. bound to the receptor) and some molecules are reflected. The
probability that the molecule is adsorbed rather than reflected depends on the
reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical
reaction and on the number of available receptors), and on the stochastic model
used. This dependence is derived for each model.Comment: 24 pages, submitted to Physical Biolog
Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations
Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation
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