Solutions of an Allen-Cahn Model Equation

2003 ◽  
pp. 245-256
Author(s):  
Paul H. Rabinowitz ◽  
Ed Stredulinsky
Keyword(s):  
Model Equation

Nitrification in bulk water and biofilms of algae wastewater stabilization ponds

2007 ◽  
Vol 55 (11) ◽  
pp. 93-101 ◽  
Author(s):  
M.A. Babu ◽  
M.M. Mushi ◽  
N.P. van der Steen ◽  
C.M. Hooijmans ◽  
H.J. Gijzen
Keyword(s):  
Model Equation ◽  
Bulk Water ◽  
Balance Model ◽  
Nitrification Rate ◽  
Relative Contribution ◽  
Activity Test ◽  
Stabilization Ponds ◽  
Wide Range ◽  
Water Columns ◽  
Wastewater Stabilization Ponds

Nitrogen removal in wastewater stabilization ponds is poorly understood and effluent monitoring data show a wide range of differences in ammonium. For effluent discharge into the environment, low levels of nitrogen are recommended. Nitrification is limiting in facultative wastewater stabilization ponds. The reason why nitrification is considered to be limiting is attributed to low growth rate and wash out of the nitrifiers. Therefore to maintain a population, attached growth is required. The aim of this research is to study the relative contribution of bulk water and biofilms with respect to nitrification. The hypothesis is that nitrification can be enhanced in stabilization ponds by increasing the surface area for nitrifier attachment. In order to achieve this, transparent pond reactors representing water columns in algae WSP have been used. To discriminate between bulk and biofilm activity, 5-day batch activity tests were carried out with bulk water and biofilm sampled. The observed value for Rnitrbulk was 2.7 × 10−1 mg-N L−1 d−1 and for Rbiofilm was 1,495 mg-N m−2 d −1. During the 5 days of experiment with the biofilm, ammonia reduction was rapid on the first day. Therefore, a short-term biofilm activity test was performed to confirm this rapid decrease. Results revealed a nitrification rate, Rbiofilm, of 2,125 mg-N m−2 d−1 for the first 5 hours of the test, which is higher than the 1,495 mg-N m−2 d−1, observed on the first day of the 7-day biofilm activity test. Rbiofilm and Rnitrbulk values obtained in the batch activity tests were used as parameters in a mass balance model equation. The model was calibrated by adjusting the fraction of the pond volume and biofilm area that is active (i.e. aerobic). When assuming a depth of 0.08 m active upper layer, the model could describe well the measured effluent values for the pond reactors. The calibrated model was validated by predicting effluent Kjeldahl nitrogen of algae ponds in Palestine and Colombia. The model equation predicted well the effluent concentrations of ponds in Palestine.

Spatial structure of a nonlinear wave of stratification

1998 ◽  
Vol 60 (2) ◽  
pp. 215-228 ◽  
Author(s):  
M. ROTTMANN ◽  
K. H. SPATSCHEK
Keyword(s):  
Nonlinear Wave ◽  
Model Equation ◽  
Positive Column ◽  
Landau Equation ◽  
Reductive Perturbation Method ◽  
Ginzburg Landau ◽  
Subcritical Regime ◽  
Ginzburg Landau Equation ◽  
Argon Discharge ◽  
Radial Structure

The positive column of a gas discharge is investigated in the subcritical regime when no self-excited nonlinear ionization wave exists. We derive a model equation for the so-called wave of stratification, which is an externally excited envelope wave. The main point of this work is that, in contrast to previous papers, we take into account the radial structure of the homogeneous column. The mathematical description of the wave of stratification in the form of a complex Ginzburg–Landau equation follows by a systematic reductive perturbation method. The coefficients in the Ginzburg–Landau equation are first calculated in general and are then evaluated explicitly for a low-pressure argon discharge. The results are compared with those obtained from a frequently used simpler model that neglects any radial structure. Finally, the dynamics of the nonlinear wave of stratification is demonstrated via numerical simulations.

scholarly journals Quasilinear generalized parabolic Anderson model equation

2018 ◽  
Vol 7 (1) ◽  
pp. 40-63 ◽  
Author(s):  
I. Bailleul ◽  
A. Debussche ◽  
M. Hofmanová
Keyword(s):  
Anderson Model ◽  
Model Equation ◽  
Parabolic Anderson Model

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

1987 ◽  
Vol 76 (3) ◽  
pp. 265-275
Author(s):  
I. H. Herron ◽  
S. A. Maslowe ◽  
S. Melkonian
Keyword(s):  
Model Equation ◽  
Shear Flows ◽  
Long Waves ◽  
Subcritical Instability

Exact Solitary wave and Soliton solutions of the fifth order model equation

2002 ◽  
Vol 22 (1) ◽  
pp. 138-144 ◽  
Author(s):  
Zhibin Li ◽  
Yinping Liu ◽  
Mingliang Wang
Keyword(s):  
Solitary Wave ◽  
Model Equation ◽  
Soliton Solutions ◽  
Order Model ◽  
Fifth Order

Continuous precipitation of monodispersed yttrium basic carbonate powders: Part III. Kinetics

1992 ◽  
Vol 7 (8) ◽  
pp. 2269-2272 ◽  
Author(s):  
Y-S. Her ◽  
E. Matijević ◽  
W.R. Wilcox
Keyword(s):  
Experimental Data ◽  
Reaction Time ◽  
Linear Function ◽  
Model Equation ◽  
Conversion Yield ◽  
Continuous Precipitation ◽  
Empirical Constant ◽  
Continuous Processes ◽  
Acidic Solutions ◽  
Basic Carbonate

The reaction kinetics involved in the continuous precipitation of monodispersed spherical yttrium basic carbonate particles was studied. The amount of powder produced was a linear function of H2CO3 generated by the decomposition of urea in mildly acidic solutions. A model equation, containing an empirical constant F, was derived to describe the conversion yield as a function of reactant concentrations, reaction time, and temperature. The calculated and experimental data agree well for both the batch and continuous processes.

GLOBAL REGULARITY FOR YANG–MILLS FIELDS IN R1+5

2010 ◽  
Vol 07 (03) ◽  
pp. 433-470 ◽  
Author(s):  
ATANAS STEFANOV
Keyword(s):  
Model Equation ◽  
Global Regularity ◽  
Gauge Condition ◽  
Coulomb Gauge ◽  
Important Case ◽  
Small Data ◽  
Yang Mills

We show global persistence of solutions with small data for the model equation □u = u⋅∇u + u3, on R 1+d, d ≥ 5, subject to the Coulomb gauge condition [Formula: see text]. In particular, this covers the important case of the Yang–Mills problem.

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