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The Klein–Gordon equation is one of the highly studied partial differential equations in condensed matter physics. Typically, the nonlinear functions associated with the Klein–Gordon equation are in the form of sinusoidal functions or polynomials. However, when looking at phenomena where logarithmic quantum mechanics is involved, various forms of logarithmic nonlinearities can be incorporated within the Klein–Gordon equation. In this paper, we consider three different forms of logarithmic nonlinearity with the goal of obtaining exact traveling wave solutions. By employing the auxiliary equation method, we are able to obtain real-valued exact solutions that are bounded kinks, periodic and bell-shaped solutions or singular solutions.

In this paper, the Fisher’s equation is studied with three different forms of nonlinear diffusion. When studying population problems, various forms of nonlinear diffusion can capture the effects of crowding or aggregation processes. Exact solutions for such nonlinear problems can be extremely useful to practitioners in the field. The Riccati–Bernoulli sub-ODE method is employed to obtain the exact traveling wave solutions for our nonlinear diffusion equation. The solutions that we find are new and to our knowledge, have not been reported in the literature.

We present a stochastic multiscale method for modeling heterogeneous catalysis at the nanoscale. The system is decomposed into the fluid domain and the catalyst-fluid interface. We implemented the fluctuating hydrodynamics framework to model the diffusion of the chemical species in the fluid domain, and the chemical master equation to describe the catalytic activity at the interface. The coupling between the domains occurs simultaneously. Using a simple one-dimensional (1D) linear model, we showed that the predictions of our scheme are in excellent agreement with deterministic simulations. The method was specifically developed to model the spatially asymmetric catalysis on the surface of self-propelled nanoswimmers. Numerical simulations showed that our approach can estimate the uncertainty in the swimming velocity resulting from inherent stochastic nature of the chemical reactions at the catalytic interface. Although the method has been applied to simple 1D and 2D models, it can be generalized to handle different geometries and more sophisticated chemical reactions. Therefore, it can serve as a practical mathematical tool to study how the efficiency of chemically powered nanomachines is affected by the interplay between structural complexity, nonlinear reactivity, and nonequilibrium fluctuations.

We consider a theoretical model for the spread of avian influenza in a poultry population. An avian influenza epidemic model incorporating spatial spread as a diffusive process is discussed, where the infected individuals are restricted from moving to prevent spatial transmission but infection occurs when susceptible individuals come into contact with infected individuals or the virus is contracted from the contaminated environment (e.g. through water or food). The infection is assumed to spread radially and isotropically. After a stability and phase plane analysis of the equivalent system of ordinary differential equations, it is shown that an analytical solution can be obtained in the form of a travelling wave. We outline the methodology for finding such analytical solutions using a travelling wave coordinate when the wave is assumed to move at constant speed. Numerical simulations also produce the travelling wave solution, and a comparison is made with some predictions based on empirical data reported in the literature.

Fuzzy cell mapping is a novel computational technique that combines fuzzy logic and a simple cell mapping method. In a simple cell mapping method, the information about mapping locations of image cells is never incorporated into the method. This limits the usage of a simple cell mapping method. In our fuzzy cell mapping method, we account for the mapping locations of image cells and incorporate the information by employing triangular membership functions. This paper demonstrates the use of fuzzy cell mapping on nonlinear dynamical systems. Our results indicate that fuzzy cell mapping can give good estimates on the global behavior of autonomous dynamical systems that posses limit cycles and strange attractors. Key words: Fuzzy cell mapping, nonlinear dynamical systems, limit cycle, strange attractor.

This paper shows the usefulness of the exponential upwinding technique in convection diffusion computations. In particular, it is demonstrated that, even when convection is dominant, if exponential upwinding is employed in conjunction with either the Jacobi or the Gauss-Seidel iteration process, one can obtain computed solutions that are accurate and free of unphysical oscillations

A mathematical model of a valve-regulated lead-acid battery under discharge is presented as simplified from a standard electrodynamics model. This nonlinear reaction-diffusion model of a battery cell is solved using an operator splitting method to quickly and accurately simulate sulfuric acid concentration. This splitting method incorporates one-sided approximation schemes to preserve continuity over material interfaces encompassing discontinuous parameters. Numerical results are compared with measured data by calculating battery voltage from modeled acid concentration as derived from the Nernst equation.

In this paper we describe and test a sub-model that integrates the cycling of carbon (C), nitrogen (N) and phosphorus (P) in the Soil Water Assessment Tool (SWAT) watershed model. The core of the sub-model is a multi-layer, one-pool soil organic carbon ( S C ) algorithm, in which the decomposition rate of S C and input rate to S C (through decomposition and humification of residues) depend on the current size of S C . The organic N and P fluxes are coupled to that of C and depend on the available mineral N and P, and the C:N and N:P ratios of the decomposing pools. Tillage explicitly affects the soil organic matter turnover rate through tool-specific coefficients. Unlike most models, the turnover of soil organic matter does not follow first order kinetics. Each soil layer has a specific maximum capacity to accumulate C or C saturation ( S x ) that depends on texture and controls the turnover rate. It is shown in an analytical solution that S x is a parameter with major influence in the model C dynamics. Testing with a 65-yr data set from the dryland wheat growing region in Oregon shows that the model adequately simulates the S C dynamics in the topsoil (top 0.3 m) for three different treatments. Three key model parameters, the optimal decomposition and humification rates and a factor controlling the effect of soil moisture and temperature on the decomposition rate, showed low uncertainty as determined by generalized likelihood uncertainty estimation. Nonetheless, the parameter set that provided accurate simulations in the topsoil tended to overestimate S C in the subsoil, suggesting that a mechanism that expresses at depth might not be represented in the current sub-model structure. The explicit integration of C, N, and P fluxes allows for a more cohesive simulation of nutrient cycling in the SWAT model. The sub-model has to be tested in forestland and rangeland in addition to agricultural land, and in diverse soils with extreme properties such high or low pH, an organic horizon, or volcanic soils.

In this paper we study a population model that incorporates the efficiency of resource utilization. We carry out a phase plane analysis of themodel and look at the solution behavior when the model is posed as an initial boundary value problem. The study involves using MATLAB and doing numerical simulations employing a finite difference scheme. The results are contrasted to the logistic population model.

This paper presents a sunlight-dependent algae growth model. Driven by the circumstances surrounding Lake Chapala, Mexico, this theoretical model is an endeavor to understand the resilient sustainability of algae that threatens the area’s ecosystem. In this paper, free-floating algae (phytoplankton) are treated as two distinct populations according to their location in the body of water: the vibrant sunlit upper region and the stagnate lower region where photosynthesis is not possible. The numerical solution for the model is analyzed and results are discussed in light of previous studies and the state of Lake Chapala.

Allostery through DNA is increasingly recognized as an important modulator of DNA functions. Here, we show that the coalescence of protein-induced DNA bubbles can mediate allosteric interactions that drive protein aggregation. We propose that such allostery may regulate DNA's flexibility and the assembly of the transcription machinery. Mitochondrial transcription factor A (TFAM), a dual-function protein involved in mitochondrial DNA (mtDNA) packaging and transcription initiation, is an ideal candidate to test such a hypothesis owing to its ability to locally unwind the double helix. Numerical simulations demonstrate that the coalescence of TFAM-induced bubbles can explain experimentally observed TFAM oligomerization. The resulting melted DNA segment, approximately 10 base pairs long, around the joints of the oligomers act as flexible hinges, which explains the efficiency of TFAM in compacting DNA. Since mitochondrial polymerase (mitoRNAP) is involved in melting the transcription bubble, TFAM may use the same allosteric interaction to both recruit mitoRNAP and initiate transcription.

In this paper we study a free boundary problem modeling a phase-change process by using microwave heating. The mathematical model consists of Maxwell's equations coupled with nonlinear heat conduction with a phase-change. The enthalpy form is used to characterize the phase-change process in the model. It is shown that the problem has a global solution.

Macropore transport of chemicals in soil often causes unexpected contamination of groundwater. The effect of soil structure on the functions of various sized macropores was assessed, investigating transport of nonreactive bromide (Br) under matric heads of 0, � 2, � 5 and � 10 cm using undisturbed soil columns from A, Bw and E horizons of a Thatuna silt loam soil (fine-silty, mixed, mesic Xeric Argialbolls). The experimental breakthrough curves (BTC) for Br were described with a two-region physical nonequilibrium model. Greatest macroporosity occurred in the A horizon and lowest in the E horizon. The measured pore water velocity m under saturated conditions ranged from 18.92 cm day � 1 in the E horizon to 64.28 cm day � 1 in the A horizon. While the greatest dispersivity k occurred in the Bw horizon due to medium subangular blocky and prizmatic aggregates, the lowest dispersivity occurred in the E horizon due to its low macroporosity and massive structure. The fitted mobile water partitioning coefficient b ranged from 0.30 in the A horizon under 0 cm matric head to 0.93 in the E horizon under 0 cm matric head. The calculated values of rate of diffusive mass exchange a decreased with decreasing matric head in A and Bw horizons, and slightly increased and then decreased in the E horizon. The difference among each of the values of the parameters m, b, a and k for the A, Bw and E horizons was greatest under saturated conditions. However, gradually decreasing matric head until about � 3 cm decreased the difference among the values for a particular parameter for different horizons, sharply. The difference remained fairly unchanged with further decreases in the matric head, suggesting that most of the variability in macropore transport of bromide for these horizons caused by pores with radii larger than about 0.5 mm. In A and Bw horizons, there was a sudden change in soil solution movement between � 2 and � 5 cm matric head, indicating that macropore flow generally occurred at matric heads greater than

We study a contaminant transport model with a cubic sorption isotherm. It is shown that aqueous concentration profiles can be obtained exactly in the form of homoclinic and heteroclinic waves. We describe the method of obtaining such exact solutions and present the solutions along with the corresponding phase plane portraits. This paper generalizes our earlier work [1].

Tetrachloroethylene (PCE) and trichloroethylene (TCE) exhibited significant nonlinear sorption to nonreactive sites when exposed to four cast irons. Cast iron is a reactive material that promotes reductive dechlorination and has recently been used for \Iin-situ\N remediation of chlorinated solvent contminated ground water. Comparisons between PCE sorption to cast iron, graphite, and iron-containing minerals indicate that nonreactive soprtion is due to exposed graphite inclusions in the cast iron. Sorption of the homologous series of chloroethenes to a cast iron adheres to Traube’s rule; thus, the extent of sorption is related primarily to compound hydrophobicity. An analytical model incorporating rate-limited sorption/desorption to nonreactive sites was used to assess sorption nonequilibrium. Effective sorption and desorption rate coefficients determined how significant mass transfer limitations to nonreactive sorption sites exist for PCE and not for TCE. The nonreactive sorption observed indicates that flow-through cast iron treatment systems will exhibit significant delayed attainment of steady-state conditions for chlorinated ethenes, particularly PCE and TCE.

Conjugate gradient type methods with preconditionings based on incompleteLU factorization (ILU) are known to be very useful for solving nonsymmetric linear systems of equations. In particular, the effectiveness of such iterative methods has been demonstrated on sparse linear systems which arise from discretizing non-self adjoint elliptic problems. In this paper, we present a simplifiedILU preconditioning coupled to a variational iterative method (Orthomin) on a model problem. It is shown that this simplified version ofILU is easy to implement and when coupled to Orthomin method, is as competitive or better than the standardILU with Orthomin method.

We study a contaminant transport model with Langmuir sorption under nonequilibrium conditions and show that the aqueous phase concentration profile can be obtained exactly in the form of a traveling wave front. The methodology of finding the exact solution is outlined, and an illustrative example is presented.

We consider a two-dimensional diffusive transport model with convection. This model is studied numerically using an alternating direction implicit (ADI) method with exponential upwinding. It is known that the exponential upwinding technique can perform really well on one-dimensional steady transport problems. So, we are interested to find out how good exponential upwinding will be on a two-dimensional problem when coupled to locally one-dimensional methods, i.e., ADI methods. Our numerical simulations show that an ADI method with exponential upwinding can be a very useful method even in two dimensions. Especially, when there are no systematic ways of choosing the upwinding parameters in the conventional upwinding methods, one can confidently employ an ADI method with exponential upwinding on a convection dominated diffusion problem and expect good answers.

Contaminant transport with nonlinear sorption under nonequilibrium conditions is studied with the objective of finding exact formulae for solute concentrations. We introduce the concept of linear “caricature” isotherm and obtain the concentration profiles as travelling waves. The solution procedure does not make any simplifying assumptions such as the coupled effect of dispersion and nonequilibrium is negligible or local equilibrium is valid.

A contaminant transport model is studied with nonlinear sorption. Using the Melnikov method, it is shown that homoclinic concentration waves exist under certain conditions. We obtain the analytical form of such a concentration wave.

Comparisons are made between the results obtained by the application of various methods of weakly nonlinear stability analysis to several prototype reaction-diffusion equations which among them reproduce most of the salient features representative of such investigations of model systems for dissipative phenomena. Emphasis is placed upon the subtleties involved in scaling these equations and deducing asymptotically valid solutions. Phenomenological interpretations of the bifurcation behavior of the model equations are offered, based on the pattern formational aspects of convective instabilities in hydrodynamics, morphological instabilities in solidification, and diffusive instabilities in ecology.

Sorption and reduction kinetics of trichloroethylene (TCE) and tetrachloroethylene (PCE) with metallic (zero-valent) iron were determined in a closed, well-mixed, anaerobic batch system by measuring aqueous and total system concentrations of the respective chlorinated solvent as a function of time. The reaction orders with respect to TCE and PCE total system concentrations were 2.7 and 1.3, respectively, indicating that the reaction mechanisms are complex. Both compounds exhibited nonlinear sorption behavior and could be fitted by the generalized Langmuir isotherm expression. After accounting for the mass sorbed to the iron, the reduction rates of PCE and TCE are first-order. This indicates that the bulk of sorption is to nonreactive sites. Competitive sorption was observed when both PCE and TCE were present ; however, no competition for reaction was detected. The design and study of treatment systems for chlorinated solvents using metallic iron requires consideration of sorption processes.

Fuzzy logic in control systems is becoming increasingly popular as a method of handling non-linear control problems. The design of fuzzy systems is generally undertaken using a trial-and-error approach. A system so developed may be less than optimum and often this is the case for known examples of control systems. Currently, there is no procedure for optimization which is appropriate for every system. In this paper a method to design and to optimize a fuzzy control system is developed. The results of computer simulations of a system constructed using this technique are presented and discussed.

In a convection-diffusion equation, if the convection term is very dominant, the linear system of equations which result from either finite-differencing or finite elementing will not have a strictly diagonally dominant coefficient matrix. So, if one tries a standard iteration method (Jacobi or Gauss-Seidel) to solve the linear system of equations, the iteration matrix may not satisfy the spectral radius condition for convergence and hence no converging solution may be obtained. The problem can be overcome, under certain conditions, if one uses the two step iteration/spectrum enveloping procedure of de Pillis in the format of a Jacobi method. This paper extends the two step iteration/spectrum enveloping idea to a Gauss-Seidel format for a model convection-diffusion problem

Porcine reproductive and respiratory syndrome (PRRS) has devastated pig industries worldwide for many years. It is caused by a small RNA virus (PRRSV), which targets almost exclusively pig monocytes or macrophages. In the present study, five SAGE (serial analysis of gene expression) libraries derived from 0 hour mock-infected and 6, 12, 16 and 24 hours PRRSV-infected porcine alveolar macrophages (PAMs) produced a total 643,255 sequenced tags with 91,807 unique tags. Differentially expressed (DE) tags were then detected using the Bayesian framework followed by gene/mRNA assignment, arbitrary selection and manual annotation, which determined 699 DE genes for reactome analysis. The DAVID, KEGG and REACTOME databases assigned 573 of the DE genes into six biological systems, 60 functional categories and 504 pathways. The six systems are: cellular processes, genetic information processing, environmental information processing, metabolism, organismal systems and human diseases as defined by KEGG with modification. Self-organizing map (SOM) analysis further grouped these 699 DE genes into ten clusters, reflecting their expression trends along these five time points. Based on the number one functional category in each system, cell growth and death, transcription processes, signal transductions, energy metabolism, immune system and infectious diseases formed the major reactomes of PAMs responding to PRRSV infection. Our investigation also focused on dominant pathways that had at least 20 DE genes identified, multi-pathway genes that were involved in 10 or more pathways and exclusively-expressed genes that were included in one system. Overall, our present study reported a large set of DE genes, compiled a comprehensive coverage of pathways, and revealed system-based reactomes of PAMs infected with PRRSV. We believe that our reactome data provides new insight into molecular mechanisms involved in host genetic complexity of antiviral activities against PRRSV and lays a strong foundation for vaccine development to control PRRS incidence in pigs.

The sterile insect release method (SIRM) for pest control is modeled by a reaction-diffusion equation. Conditions on the strip size of the favorable environment and the initial population to avoid an outbreak state are obtained by using a combination of analytical and numerical techniques.

The solitary‐wave interaction mechanism for the good Boussinesq equation is investigated and found to be far more complicated than was previously thought. Three salient features are that solitary waves only exist for a finite range of velocities, that large solitons can turn into so‐called antisolitons, and that it is possible for solitons to merge and split. Small solitons, however, appear to be stable. The existence of a potential well is linked to the different behaviors observed between small and large initial conditions.

During the aggregation phase of their life cycle, Dictyostelium discoideum amoebae communicate with each other by traveling waves of cyclic AMP (cAMP). These waves often have the geometry of rotating spirals. We calculate the properties of these spiral waves, their rotation period, wavespeed, and shape, from a model of the synthesis and degradation of cAMP by Dictyostelium cells. Our model is based on kinetic rate laws developed by Martiel and Goldbeter to account for oscillations and signal relaying by Dictyostelium amoebae in well-stirred cell suspensions. We show that the model also describes in quantitative detail experimental observations of rotating spiral waves of cAMP in fields of amoebae distributed over an agar surface. Furthermore, our numerically calculated spiral waves agree quantitatively with the singular perturbation theory of rotating spirals developed by Keener and Tyson.

A method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations. The method can be a useful alternative to the usual leapfrog technique, in that it does not suffer from the occurrence of blowup phenomena. Numerical examples concerning the Korteweg-de Vries equation and the nonlinear Schrodinger equation are given.

The kink interactions in the Higgs model are studied numerically using a finite element Galerkin method and a clear picture regarding the ‘resonances’ is presented. It is found that these ‘resonances’ are dependent on spatial increment and time step.

On considere les cas lineaires et non lineaires. Les schemas correspondants sont explicites

The “good” Boussinesq equation is studied numerically using Galerkin methods and soliton solutions are shown to exist. An analytic formula for the two-soliton interaction is presented and verified by numerical experiment.

The complex spatial patterns of feather follicles and scales and osteoderms seen in various species of birds, reptiles and mammals are reproduced via the interaction of two pattern generating mechanisms. For illustrative purposes the two mechanisms we consider are a two-component reaction-diffusion system of epidermal origin coupled to a cell traction model of dermal or mesenchymal origin. Other mechanisms with qualitatively similar dispersion relations to those used here, could also have been chosen. Each mechanism acting alone is capable of generating spatial wave-like patterns with wavelengths within a narrow range of λ n , say, in the dermis and λ v , say, in the epidermis, although neither mechanism acting alone can produce the complex spatial patterns observed. When the two mechanisms interact with each other the spatial patterns which form in both the dermis and epidermis are a complex superposition of the two wave-like patterns with wavelengths λ n and λ v . The complex spatial patterns produced are in qualitative agreement with those observed.

Different types of basis functions are considered for the pseudospectral method applied to the equation u t =Δu+ƒ(u) in 1- and 2-space dimensions. For large N, where N is the number of basis functions, the choice of Cheybyshev polynomials gives much greater accuracy than the eigenfunctions of the Laplacian Δ. However for small N (≲ 7), the eigenfunction expansion can be more accurate even with weakly nonperiodic boundary conditions.

In a convection-diffusion equation, if the convection term is very dominant, the linear system of equations which result from either finite-differencing or finite elementing will not have a strictly diagonally dominant coefficient matrix. So, if one tries a conventional iteration method (Jacobi or Gauss-Seidel) to solve the linear system of equations, the iteration matrix may not satisfy the spectral radius condition for convergence, and hence, no converging solution may be obtained. The problem can be overcome, under certain conditions, if one uses a two-step iteration procedure involving the spectral enveloping ellipse for the iteration matrix. In this paper, we present such a two-step method that combines an Arnoldi-Chebyshev approach for convection-diffusion computations.

In multigrid methods, it is preferred to employ smoothing techniques which are convergent. In practice, the standard Jacobi and Gauss-Seidel methods are the choices for “smoothers” However, it is known that if the spectral radius condition is violated, then convergence is not guaranteed for these methods. In this paper we develop a simple two-step Jacobi-type method which has better convergence properties and which can be employed as a convergent “smoother” wherever the standard iterative methods fail. We provide the convergence proofs and demonstrate the applicability of the method on a variety of problems.

Summary Mathematical models and controlled experiments have been useful In understanding the mechanisms of various process steps in wastewater treatment. However, it is nevertheless difficult to forecast accurately the output quality of a full-scale treatment plant from knowledge of the input to the plant. This is due to random fluctuations in feed. One possible answer is to use time-series models which could predict the output quality as part of a feed-forward/feed-back control system. However, the predicted effluent quality may not fall within the required target range. Statistical optimisation techniques have been successfully used in Japan to reduce the variability of manufacturing processes. This review discusses the potential use of such techniques to reduce the variability of treatment processes within a desired target-range of output quality. The essence of such techniques is to modify the system to reduce its sensitivity to the causes of variability, rather than attempting to control the actual sources of variability. In wastewater treatment, controlling the sources of variability Is usually impossible or expensive.

Summary Nitrogen compounds found in water can contribute to many mechanisms of environmental pollution. In atmospheric pollution, their effects Include damaging the ozone layer and contributing to the greenhouse effect. In water, nitrogenous pollutants can have toxic effects, including acting as carcinogen precursors, stimulate deoxygenation and eutrophication in rivers and Interfere with purification processes. Nitrogenous pollutants are very common, occurring in domestic, agricultural and industrial wastes, distributed as fertilisers and formed naturally by living organisms and lightning. Techniques for conversion and removal of nitrogen compounds from water and wastewater are well developed. In wastewater treatment, ammonia Is oxidised microbiologically to nitrate, which is then removed by microbiological denltriflcation where sufficient substrate is available. In high-nitrogen industrial wastewaters, substrate may have to be added. A wide range of nutrient wasteproducts and be used for this purpose. In water purification, there may be insufficient dissolved oxygen for ammonia oxidation and/or organic substrate for microbiological denltriflcation. Several physico-chemical methods, such as reverse osmosis, electrodialysis and ion-exchange processes, are commercially avallable.

The Belousov-Zhabotinskii reaction has been modelled by Field and Noyes [5] as a pair of nonlinear parabolic equations. Previous studies of these, both theoretical and numerical, have assumed wave solutions travelling with constant velocity leading to a simplification of the mathematical model in the form of a system of ordinary differential equations. In the present study a finite element Galerkin method is used directly on the original parabolic system for a range of parameter values.

Periodic expanding target patterns of chemical activity are observed in thin unstirred layers of solution containing bromate, malonic acid and ferroin in dilute sulfuric acid [1/2]. Commonly these patterns appear as thin blue rings propagating outward from a central point into red bulk medium. Since the indicator, ferroin (Iron 1,10-phenanthroline), is blue in the oxidized state (Fe3+) and red in the reduced state (Fe2+), the thin blue ring is a zone of oxidation in a predominantly reduced background. The zone of oxidation is delimited by two wave fronts: a wave front of oxidation carrying the medium from the reduced state to the oxidized state, and a wave front (or, we might say, wave “back”) of reduction restoring the medium to the original reduced state. These two waves travel at the same speed, which depends on the exact chemical constitution of the reactive solution but is practically independent of the temporal frequency and spatial wavelength of the target pattern.

Wave solutions to a mechanochemical model for cytoskeletal activity are studied and the results applied to the waves of chemical and mechanical activity that sweep over an egg shortly after fertilization. The model takes into account the calcium-controlled presence of actively contractile units in the cytoplasm, and consists of a viscoelastic force equilibrium equation and a conservation equation for calcium. Using piecewise linear caricatures, we obtain analytic solutions for travelling waves on a strip and demonstrate uiat the full nonlinear system behaves as predicted by the analytic solutions. The equations are solved on a sphere and the numerical results are similar to the analytic solutions. We indicate how the speed of the waves can be used as a diagnostic tool with which the chemical reactivity of the egg surface can be measured.