**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1918

# Search results for: reaction diffusion equation

##### 1918 Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method

**Authors:**
Gülnihal Meral

**Abstract:**

**Keywords:**
Density Dependent Nonlinear Reaction-Diffusion Equation,
Differential Quadrature Method,
Implicit Euler Method.

##### 1917 Modeling and Simulating Reaction-Diffusion Systems with State-Dependent Diffusion Coefficients

**Authors:**
Paola Lecca,
Lorenzo Dematte,
Corrado Priami

**Abstract:**

The present models and simulation algorithms of intracellular stochastic kinetics are usually based on the premise that diffusion is so fast that the concentrations of all the involved species are homogeneous in space. However, recents experimental measurements of intracellular diffusion constants indicate that the assumption of a homogeneous well-stirred cytosol is not necessarily valid even for small prokaryotic cells. In this work a mathematical treatment of diffusion that can be incorporated in a stochastic algorithm simulating the dynamics of a reaction-diffusion system is presented. The movement of a molecule A from a region i to a region j of the space is represented as a first order reaction Ai k- ! Aj , where the rate constant k depends on the diffusion coefficient. The diffusion coefficients are modeled as function of the local concentration of the solutes, their intrinsic viscosities, their frictional coefficients and the temperature of the system. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method the simulation results of the reaction-diffusion system of chaperoneassisted protein folding in cytoplasm are shown.

**Keywords:**
Reaction-diffusion systems,
diffusion coefficient,
stochastic simulation algorithm.

##### 1916 Septic B-spline Collocation Method for Solving One-dimensional Hyperbolic Telegraph Equation

**Authors:**
Marzieh Dosti,
Alireza Nazemi

**Abstract:**

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper, a numerical solution for the one-dimensional hyperbolic telegraph equation by using the collocation method using the septic splines is proposed. The scheme works in a similar fashion as finite difference methods. Test problems are used to validate our scheme by calculate L2-norm and L∞-norm. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact solutions.

**Keywords:**
B-spline,
collocation method,
second-order hyperbolic telegraph equation,
difference schemes.

##### 1915 A Study of Numerical Reaction-Diffusion Systems on Closed Surfaces

**Authors:**
Mei-Hsiu Chi,
Jyh-Yang Wu,
Sheng-Gwo Chen

**Abstract:**

**Keywords:**
Close surfaces,
high-order approach,
numerical solutions,
reaction-diffusion systems.

##### 1914 Simulation of a Multi-Component Transport Model for the Chemical Reaction of a CVD-Process

**Abstract:**

In this paper we present discretization and decomposition methods for a multi-component transport model of a chemical vapor deposition (CVD) process. CVD processes are used to manufacture deposition layers or bulk materials. In our transport model we simulate the deposition of thin layers. The microscopic model is based on the heavy particles, which are derived by approximately solving a linearized multicomponent Boltzmann equation. For the drift-process of the particles we propose diffusionreaction equations as well as for the effects of heat conduction. We concentrate on solving the diffusion-reaction equation with analytical and numerical methods. For the chemical processes, modelled with reaction equations, we propose decomposition methods and decouple the multi-component models to simpler systems of differential equations. In the numerical experiments we present the computational results of our proposed models.

**Keywords:**
Chemical reactions,
chemical vapor deposition,
convection-diffusion-reaction equations,
decomposition methods,
multi-component transport.

##### 1913 The Effects of Tissue Optical Parameters and Interface Reflectivity on Light Diffusion in Biological Tissues

**Authors:**
MA. Ansari

**Abstract:**

**Keywords:**
Diffusion equation,
boundary element method,
refractive index

##### 1912 Stochastic Simulation of Reaction-Diffusion Systems

**Authors:**
Paola Lecca,
Lorenzo Dematte

**Abstract:**

Reactiondiffusion systems are mathematical models that describe how the concentration of one or more substances distributed in space changes under the influence of local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose general form is ÔêétX(x, t) = DΔX(x, t), where X(x, t) is the state vector, D is the matrix of the diffusion coefficients and Δ is the Laplace operator. If the solute move in an homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and of solutes and on local temperature of the medium. In this paper a new stochastic reaction-diffusion model in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces of solvent and solute is presented. Such a model provides a more realistic description of the molecular kinetics in non-homogenoeus and highly structured media as the intra- and inter-cellular spaces. The movement of a molecule A from a region i to a region j of the space is described as a first order reaction Ai k- → Aj , where the rate constant k depends on the diffusion coefficient. Representing the diffusional motion as a chemical reaction allows to assimilate a reaction-diffusion system to a pure reaction system and to simulate it with Gillespie-inspired stochastic simulation algorithms. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the specific speed of reaction and diffusion events. Redi is the software tool, developed to implement the model of reaction-diffusion kinetics and dynamics. It is a free software, that can be downloaded from http://www.cosbi.eu. To demonstrate the validity of the new reaction-diffusion model, the simulation results of the chaperone-assisted protein folding in cytoplasm obtained with Redi are reported. This case study is redrawing the attention of the scientific community due to current interests on protein aggregation as a potential cause for neurodegenerative diseases.

**Keywords:**
Reaction-diffusion systems,
Fick's law,
stochastic simulation algorithm.

##### 1911 Stability Analysis of Impulsive BAM Fuzzy Cellular Neural Networks with Distributed Delays and Reaction-diffusion Terms

**Authors:**
Xinhua Zhang,
Kelin Li

**Abstract:**

In this paper, a class of impulsive BAM fuzzy cellular neural networks with distributed delays and reaction-diffusion terms is formulated and investigated. By employing the delay differential inequality and inequality technique developed by Xu et al., some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM fuzzy cellular neural networks with distributed delays and reaction-diffusion terms are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters, diffusion effect and impulsive disturbed intention. It is believed that these results are significant and useful for the design and applications of BAM fuzzy cellular neural networks. An example is given to show the effectiveness of the results obtained here.

**Keywords:**
Bi-directional associative memory,
fuzzy cellular neuralnetworks,
reaction-diffusion,
delays,
impulses,
global exponentialstability.

##### 1910 Haar Wavelet Method for Solving Fitz Hugh-Nagumo Equation

**Authors:**
G.Hariharan,
K.Kannan

**Abstract:**

In this paper, we develop an accurate and efficient Haar wavelet method for well-known FitzHugh-Nagumo equation. The proposed scheme can be used to a wide class of nonlinear reaction-diffusion equations. The power of this manageable method is confirmed. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.

**Keywords:**
FitzHugh-Nagumo equation,
Haar wavelet method,
adomain decomposition method,
computationally attractive.

##### 1909 Mathematical Modelling of Transport Phenomena in Radioactive Waste-Cement-Bentonite Matrix

**Authors:**
Ilija Plecas,
Uranija Kozmidis-Luburic,
Radojica Pesic

**Abstract:**

The leaching rate of 137Cs from spent mix bead (anion and cation) exchange resins in a cement-bentonite matrix has been studied. Transport phenomena involved in the leaching of a radioactive material from a cement-bentonite matrix are investigated using three methods based on theoretical equations. These are: the diffusion equation for a plane source an equation for diffusion coupled to a firstorder equation and an empirical method employing a polynomial equation. The results presented in this paper are from a 25-year mortar and concrete testing project that will influence the design choices for radioactive waste packaging for a future Serbian radioactive waste disposal center.

**Keywords:**
bentonite,
cement ,
radioactive waste,
composite,
disposal,
diffusion

##### 1908 A Comparison of Recent Methods for Solving a Model 1D Convection Diffusion Equation

**Authors:**
Ashvin Gopaul,
Jayrani Cheeneebash,
Kamleshsing Baurhoo

**Abstract:**

In this paper we study some numerical methods to solve a model one-dimensional convection–diffusion equation. The semi-discretisation of the space variable results into a system of ordinary differential equations and the solution of the latter involves the evaluation of a matrix exponent. Since the calculation of this term is computationally expensive, we study some methods based on Krylov subspace and on Restrictive Taylor series approximation respectively. We also consider the Chebyshev Pseudospectral collocation method to do the spatial discretisation and we present the numerical solution obtained by these methods.

**Keywords:**
Chebyshev Pseudospectral collocation method,
convection-diffusion equation,
restrictive Taylor approximation.

##### 1907 Stability Analysis of Impulsive Stochastic Fuzzy Cellular Neural Networks with Time-varying Delays and Reaction-diffusion Terms

**Authors:**
Xinhua Zhang,
Kelin Li

**Abstract:**

In this paper, the problem of stability analysis for a class of impulsive stochastic fuzzy neural networks with timevarying delays and reaction-diffusion is considered. By utilizing suitable Lyapunov-Krasovskii funcational, the inequality technique and stochastic analysis technique, some sufficient conditions ensuring global exponential stability of equilibrium point for impulsive stochastic fuzzy cellular neural networks with time-varying delays and diffusion are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters, diffusion effect and impulsive disturbed intention. It is believed that these results are significant and useful for the design and applications of fuzzy neural networks. An example is given to show the effectiveness of the obtained results.

**Keywords:**
Exponential stability,
stochastic fuzzy cellular neural networks,
time-varying delays,
impulses,
reaction-diffusion terms.

##### 1906 Basket Option Pricing under Jump Diffusion Models

**Authors:**
Ali Safdari-Vaighani

**Abstract:**

**Keywords:**
Radial basis function,
basket option,
jump diffusion,
RBF-PUM.

##### 1905 Closed Form Solution to problem of Calcium Diffusion in Cylindrical Shaped Neuron Cell

**Authors:**
Amrita Tripathi,
Neeru Adlakha

**Abstract:**

**Keywords:**
Laplace Transform,
Modified Bessel function,
reaction diffusion equation,
diffusion coefficient,
excess buffer,
calcium influx

##### 1904 A Finite Point Method Based on Directional Derivatives for Diffusion Equation

**Authors:**
Guixia Lv,
Longjun Shen

**Abstract:**

**Keywords:**
Finite point method,
directional derivatives,
diffusionequation,
method for selecting neighbor point set.

##### 1903 On Diffusion Approximation of Discrete Markov Dynamical Systems

**Authors:**
Jevgenijs Carkovs

**Abstract:**

**Keywords:**
Markov dynamical system,
diffusion approximation,
equilibrium stochastic stability.

##### 1902 Numerical Studies of Galerkin-type Time-discretizations Applied to Transient Convection-diffusion-reaction Equations

**Authors:**
Naveed Ahmed,
Gunar Matthies

**Abstract:**

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numerical results which shows that the cGP(k) and dG(k)- methods are accurate of order k +1, respectively, in the whole time interval. Moreover, the cGP(k)-method is superconvergent of order 2k and dG(k)-method is of order 2k +1 at the discrete time points. Furthermore, the dependence of the results on the choice of the stabilization parameter are discussed and compared.

**Keywords:**
Convection-diffusion-reaction equations,
stabilized finite elements,
discontinuous Galerkin,
continuous Galerkin-Petrov.

##### 1901 Formation of Chemical Compound Layer at the Interface of Initial Substances A and B with Dominance of Diffusion of the A Atoms

**Authors:**
Pavlo Selyshchev,
Samuel Akintunde

**Abstract:**

A theoretical approach to consider formation of chemical compound layer at the interface between initial substances *A* and *B* due to the interfacial interaction and diffusion is developed. It is considered situation when speed of interfacial interaction is large enough and diffusion of *A*-atoms through *AB*-layer is much more then diffusion of *B*-atoms. Atoms from *A*-layer diffuse toward *B*-atoms and form *AB*-atoms on the surface of *B*-layer. *B*-atoms are assumed to be immobile. The growth kinetics of the *AB*-layer is described by two differential equations with non-linear coupling, producing a good fit to the experimental data. It is shown that growth of the thickness of the *AB*-layer determines by dependence of chemical reaction rate on reactants concentration. In special case the thickness of the *AB*-layer can grow linearly or parabolically depending on that which of processes (interaction or the diffusion) controls the growth. The thickness of *AB*-layer as function of time is obtained. The moment of time (transition point) at which the linear growth are changed by parabolic is found.

**Keywords:**
Phase formation,
Binary systems,
Interfacial Reaction,
Diffusion,
Compound layers,
Growth kinetics.

##### 1900 A New Time Discontinuous Expanded Mixed Element Method for Convection-dominated Diffusion Equation

**Authors:**
Jinfeng Wang,
Yuanhong Bi,
Hong Li,
Yang Liu,
Meng Zhao

**Abstract:**

In this paper, a new time discontinuous expanded mixed finite element method is proposed and analyzed for two-order convection-dominated diffusion problem. The proofs of the stability of the proposed scheme and the uniqueness of the discrete solution are given. Moreover, the error estimates of the scalar unknown, its gradient and its flux in the L1( ¯ J,L2( )-norm are obtained.

**Keywords:**
Convection-dominated diffusion equation,
expanded mixed method,
time discontinuous scheme,
stability,
error estimates.

##### 1899 An Optimal Control of Water Pollution in a Stream Using a Finite Difference Method

**Authors:**
Nopparat Pochai,
Rujira Deepana

**Abstract:**

**Keywords:**
Finite difference,
One-dimensional,
Steady state,
Waterpollution control,
Optimization,
Convection-diffusion equation.

##### 1898 Finite Volume Model to Study The Effect of Voltage Gated Ca2+ Channel on Cytosolic Calcium Advection Diffusion

**Authors:**
Brajesh Kumar Jha,
Neeru Adlakha,
M. N. Mehta

**Abstract:**

**Keywords:**
Ca2+ Profile,
Advection Diffusion,
VOC,
FVM.

##### 1897 Effects of Li2O Thickness and Moisture Content on LiH Hydrolysis Kinetics in Slightly Humidified Argon

**Authors:**
S. Xiao,
M. B. Shuai,
M. F. Chu

**Abstract:**

The hydrolysis kinetics of polycrystalline lithium hydride (LiH) in argon at various low humidities was measured by gravimetry and Raman spectroscopy with ambient water concentration ranging from 200 to 1200 ppm. The results showed that LiH hydrolysis curve revealed a paralinear shape, which was attributed to two different reaction stages that forming different products as explained by the 'Layer Diffusion Control' model. Based on the model, a novel two-stage rate equation for LiH hydrolysis reactions was developed and used to fit the experimental data for determination of Li2O steady thickness Hs and the ultimate hydrolysis rate vs. The fitted data presented a rise of Hs as ambient water concentration cw increased. However, in spite of the negative effect imposed by Hs increasing, the upward trend of vs remained, which implied that water concentration, rather than Li2O thickness, played a predominant role in LiH hydrolysis kinetics. In addition, the proportional relationship between vsHs and cw predicted by rate equation and confirmed by gravimetric data validated the model in such conditions.

**Keywords:**
Hydrolysis kinetics,
‘Layer Diffusion Control’ model,
Lithium hydride

##### 1896 A Model to Study the Effect of Excess Buffers and Na+ Ions on Ca2+ Diffusion in Neuron Cell

**Authors:**
Vikas Tewari,
Shivendra Tewari,
K. R. Pardasani

**Abstract:**

Calcium is a vital second messenger used in signal transduction. Calcium controls secretion, cell movement, muscular contraction, cell differentiation, ciliary beating and so on. Two theories have been used to simplify the system of reaction-diffusion equations of calcium into a single equation. One is excess buffer approximation (EBA) which assumes that mobile buffer is present in excess and cannot be saturated. The other is rapid buffer approximation (RBA), which assumes that calcium binding to buffer is rapid compared to calcium diffusion rate. In the present work, attempt has been made to develop a model for calcium diffusion under excess buffer approximation in neuron cells. This model incorporates the effect of [Na+] influx on [Ca2+] diffusion,variable calcium and sodium sources, sodium-calcium exchange protein, Sarcolemmal Calcium ATPase pump, sodium and calcium channels. The proposed mathematical model leads to a system of partial differential equations which have been solved numerically using Forward Time Centered Space (FTCS) approach. The numerical results have been used to study the relationships among different types of parameters such as buffer concentration, association rate, calcium permeability.

**Keywords:**
Excess buffer approximation,
Na+ influx,
sodium calcium exchange protein,
sarcolemmal calcium atpase pump,
forward time centred space.

##### 1895 Membrane Distillation Process Modeling: Dynamical Approach

**Authors:**
Fadi Eleiwi,
Taous Meriem Laleg-Kirati

**Abstract:**

This paper presents a complete dynamic modeling of a membrane distillation process. The model contains two consistent dynamic models. A 2D advection-diffusion equation for modeling the whole process and a modified heat equation for modeling the membrane itself. The complete model describes the temperature diffusion phenomenon across the feed, membrane, permeate containers and boundary layers of the membrane. It gives an online and complete temperature profile for each point in the domain. It explains heat conduction and convection mechanisms that take place inside the process in terms of mathematical parameters, and justify process behavior during transient and steady state phases. The process is monitored for any sudden change in the performance at any instance of time. In addition, it assists maintaining production rates as desired, and gives recommendations during membrane fabrication stages. System performance and parameters can be optimized and controlled using this complete dynamic model. Evolution of membrane boundary temperature with time, vapor mass transfer along the process, and temperature difference between membrane boundary layers are depicted and included. Simulations were performed over the complete model with real membrane specifications. The plots show consistency between 2D advection-diffusion model and the expected behavior of the systems as well as literature. Evolution of heat inside the membrane starting from transient response till reaching steady state response for fixed and varying times is illustrated.

**Keywords:**
Membrane distillation,
Dynamical modeling,
Advection-diffusion equation,
Thermal equilibrium,
Heat
equation.

##### 1894 A Nonlinear Parabolic Partial Differential Equation Model for Image Enhancement

**Authors:**
Tudor Barbu

**Abstract:**

**Keywords:**
Image denoising and restoration,
nonlinear PDE
model,
anisotropic diffusion,
numerical approximation scheme,
finite
differences.

##### 1893 Finite Volume Model to Study the Effect of Buffer on Cytosolic Ca2+ Advection Diffusion

**Authors:**
Brajesh Kumar Jha,
Neeru Adlakha,
M. N. Mehta

**Abstract:**

**Keywords:**
Ca2+ profile,
buffer,
Astrocytes,
Advection diffusion,
FVM

##### 1892 Nitrogen Effects on Ignition Delay Time in Supersonic Premixed and Diffusion Flames

**Authors:**
A. M. Tahsini

**Abstract:**

Computational study of two dimensional supersonic reacting hydrogen-air flows is performed to investigate the nitrogen effects on ignition delay time for premixed and diffusion flames. Chemical reaction is treated using detail kinetics and the advection upstream splitting method is used to calculate the numerical inviscid fluxes. The results show that just in stoichiometric condition for both premixed and diffusion flames, there is monotone dependency of the ignition delay time to the nitrogen addition. In other situations, the optimal condition from ignition viewpoint should be found using numerical investigations.

**Keywords:**
Diffusion flame,
Ignition delay time,
Mixing layer,
Numerical simulation,
Premixed flame,
Supersonic flow.

##### 1891 The Relationship between Fugacity and Stress Intensity Factor for Corrosive Environment in Presence of Hydrogen Embrittlement

**Authors:**
A. R. Shahani,
E. Mahdavi,
M. Amidpour

**Abstract:**

Hydrogen diffusion is the main problem for corrosion fatigue in corrosive environment. In order to analyze the phenomenon, it is needed to understand their behaviors specially the hydrogen behavior during the diffusion. So, Hydrogen embrittlement and prediction its behavior as a main corrosive part of the fractions, needed to solve combinations of different equations mathematically. The main point to obtain the equation, having knowledge about the source of causing diffusion and running the atoms into materials, called driving force. This is produced by either gradient of electrical or chemical potential. In this work, we consider the gradient of chemical potential to obtain the property equation. In diffusion of atoms, some of them may be trapped but, it could be ignorable in some conditions. According to the phenomenon of hydrogen embrittlement, the thermodynamic and chemical properties of hydrogen are considered to justify and relate them to fracture mechanics. It is very important to get a stress intensity factor by using fugacity as a property of hydrogen or other gases. Although, the diffusive behavior and embrittlement event are common and the same for other gases but, for making it more clear, we describe it for hydrogen. This considering on the definite gas and describing it helps us to understand better the importance of this relation.

**Keywords:**
Hydrogen embrittlement,
Fracture mechanics,
Thermodynamic,
Stress intensity factor.

##### 1890 An Efficient Backward Semi-Lagrangian Scheme for Nonlinear Advection-Diffusion Equation

**Authors:**
Soyoon Bak,
Sunyoung Bu,
Philsu Kim

**Abstract:**

In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results is in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes.

**Keywords:**
Semi-Lagrangian method,
Iteration free method,
Nonlinear advection-diffusion equation.

##### 1889 The Dividend Payments for General Claim Size Distributions under Interest Rate

**Authors:**
Li-Li Li,
Jinghai Feng,
Lixin Song

**Abstract:**

**Keywords:**
Dividend payout,
Integro-differential equation,
Jumpdiffusion model,
Volterra equation