Steady-state Reaction-diffusion Research Articles

Modeling catalyst effectiveness factor with space-fractional derivative using Haar wavelet collocation method

Abstract Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive.

Σ-Shaped Bifurcation Curves

AbstractWe study positive solutions to the steady state reaction diffusion equation of the form:−Δu=λf(u); Ω∂u∂η+λu=0; ∂Ω$$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$whereλ> 0 is a positive parameter,Ωis a bounded domain in ℝNwhenN> 1 (with smooth boundary∂ Ω) orΩ= (0, 1), and∂u∂η$\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$is the outward normal derivative ofu. Heref(s) =ms+g(s) wherem≥ 0 (constant) andg∈C2[0,r) ∩C[0, ∞) for somer> 0. Further, we assume thatgis increasing, sublinear at infinity,g(0) = 0,g′(0) = 1 andg″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges ofλleading to the occurrence ofΣ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.

Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition

We study positive solutions to steady state reaction diffusion equations of the form:{−Δu=λf(u);Ω,∂u∂η+μ(λ)u=0;∂Ω, where λ>0, Ω is a bounded domain in RN; N≥1 with smooth boundary ∂Ω, ∂u∂η is the outward normal derivative of u, μ∈C([0,∞)) is strictly increasing such that μ(0)≥0 and f∈C2([0,r0)) with 0<r0≤∞. If r0<∞ we assume f∈C2([0,r0]) with f(r0)=0 and f(s)≤0 for s∈(r0,∞), and if r0=∞ we assume lims→∞⁡f(s)>0 and lims→∞⁡f(s)s=0 (sublinear at ∞). Note here that the parameter λ influences both the equation and the boundary condition. We discuss existence, nonexistence, multiplicity and uniqueness results for the cases when (A) f(0)=0, (B) f(0)<0, and (C) f(0)>0. We obtain existence and multiplicity results by the method of sub-super solutions and uniqueness results by comparison principles and a priori estimates.

An inverse problem for a system of steady-state reaction-diffusion equations on a porous domain using a collage-based approach

We consider an inverse problem for a system of steady-state reaction-diffusions acting on a perforated domain. We establish several results that connect the parameter values for the problem on the perforated domain with the corresponding problem on the related unperforated of solid domain. This opens the possibility of estimating a solution to the inverse problem on the perforated domain by instead working with the easier-to-solve inverse problem on the solid domain. We illustrate the results with an example.

A New Spectral Approach on Steady-State Concentration of Species in Porous Catalysts Using Wavelets.

In this paper, a mathematical model of steady-state reaction-diffusion (RD) model for estimating the concentration of species is discussed. We have applied a new wavelet-based operational matrix of derivative method to obtain the approximate solutions for nonlinear RDEs. The proposed method is a powerful and easy-to-use analytical tool for linear and nonlinear problems. Some illustrative examples are given to validate our results with exact solutions. Satisfactory agreement with the exact solution is noticed. Moreover, the use of Legendre wavelets is found to be simple, accurate, efficient and requires small computation costs.

A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball

We consider steady state reaction diffusion equations on the exterior of a ball, namely, boundary value problems of the form:{−Δpu=λK(|x|)f(u) in ΩE,u=0 on |x|=r0,u→0 when |x|→∞, where Δpz:=div(|∇z|p−2∇z), 1<p<n, λ is a positive parameter, r0>0 and ΩE:={x∈Rn | |x|>r0}. Here the weight function K∈C1[r0,∞) satisfies K(r)>0 for r≥r0, limr→∞⁡K(r)=0, and the reaction term f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies f(0)<0 (semipositone), limsups→0+sf′(s)<∞, lims→∞⁡f(s)=∞, lims→∞⁡f(s)sp−1=0 and f(s)sq is nonincreasing on [a,∞) for some a>0 and q∈(0,p−1). For a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball, and exists for λ≫1. We establish the uniqueness of this positive radial solution for λ≫1.

An efficient Chebyshev wavelet based analytical algorithm to steady state reaction–diffusion models arising in mathematical chemistry

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equation containing a nonlinear term related to non-Michaelis–Menten kinetics of the reaction. In this paper, an efficient Chebyshev wavelet based approximation method is introduced for solving the steady state reaction–diffusion equations. Illustrative examples are given to demonstrate the validity and applicability of the method. The power of the manageable method is confirmed. Moreover the use of Chebyshev wavelet method is found to be simple, flexible, efficient, small computation costs and computationally attractive.

Galerkin Solution of Stochastic Reaction-Diffusion Problems

In this paper, the Galerkin method is used to obtain numerical solutions to two-dimensional steady-state reaction-diffusion problems. Uncertainties in reaction and diffusion coefficients are modeled using parameterized stochastic processes. A stochastic version of the Lax–Milgram lemma is used in order to guarantee existence and uniqueness of the theoretical solutions. The space of approximate solutions is constructed by tensor product between finite dimensional deterministic functional spaces and spaces generated by chaos polynomials, derived from the Askey–Wiener scheme. Performance of the developed Galerkin scheme is evaluated by comparing first and second order moments and probability histograms obtained from approximate solutions with the corresponding estimates obtained via Monte Carlo simulation. Results for three example problems show very fast convergence of the approximate Galerkin solutions. Results also show that complete probability densities (histograms) of the responses are correctly approximated by the developed Galerkin basis.

Analytical expression of the concentration of species and effectiveness factors in porous catalysts using the Adomian decomposition method

A mathematical model by Hayes et al. [10] of porous catalysts that have non-linear reactions kinetic is discussed. The model involves the non-linear steady-state reaction-diffusion equation. The analytical solution for the concentration of species is obtained using the Adomian decomposition method. Simple and an approximate polynomial expressions for concentration and effectiveness factors are derived for general non-linear Langmiur-Hinshelwood-Haugen-Watson (LHHW) type models which has variety of real rate function. Comparison of the analytical approximation and numerical simulation is also presented. A good agreement between theoretical predictions and numerical results is observed. The concentration and the effectiveness factors are also computed for the limiting cases of LHHW type models.

Developmental origin of patchy axonal connectivity in the neocortex: a computational model.

Injections of neural tracers into many mammalian neocortical areas reveal a common patchy motif of clustered axonal projections. We studied in simulation a mathematical model for neuronal development in order to investigate how this patchy connectivity could arise in layer II/III of the neocortex. In our model, individual neurons of this layer expressed the activator–inhibitor components of a Gierer–Meinhardt reaction–diffusion system. The resultant steady-state reaction–diffusion pattern across the neuronal population was approximately hexagonal. Growth cones at the tips of extending axons used the various morphogens secreted by intrapatch neurons as guidance cues to direct their growth and invoke axonal arborization, so yielding a patchy distribution of arborization across the entire layer II/III. We found that adjustment of a single parameter yields the intriguing linear relationship between average patch diameter and interpatch spacing that has been observed experimentally over many cortical areas and species. We conclude that a simple Gierer–Meinhardt system expressed by the neurons of the developing neocortex is sufficient to explain the patterns of clustered connectivity observed experimentally.

Analytical Expressions for the Steady-State Concentrations of Glucose, Oxygen and Gluconic Acid in a Composite Membrane for Closed-Loop Insulin Delivery

The mathematical model of Abdekhodaie and Wu (J Membr Sci 335:21-31, 2009) of glucose-responsive composite membranes for closed-loop insulin delivery is discussed. The glucose composite membrane contains nanoparticles of an anionic polymer, glucose oxidase and catalase embedded in a hydrophobic polymer. The model involves the system of nonlinear steady-state reaction-diffusion equations. Analytical expressions for the concentration of glucose, oxygen and gluconic acid are derived from these equations using the Adomian decomposition method. A comparison of the analytical approximation and numerical simulation is also presented. An agreement between analytical expressions and numerical results is observed.

A nonlinearity lagging for the solution of nonlinear steady state reaction diffusion problems

This paper concerns with the analysis of the iterative procedure for the solution of a nonlinear reaction diffusion equation at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. This procedure, called Lagged Diffusivity Functional Iteration (LDFI)-procedure, computes the solution by “lagging” the diffusion term. A model problem is considered and a finite difference discretization for that model problem is described. Furthermore, properties of the finite difference operator are proved. Then, sufficient conditions for the convergence of the LDFI-procedure are given. At each stage of the LDFI-procedure a weakly nonlinear algebraic system has to be solved and the simplified Newton–Arithmetic Mean (Newton–AM) method is used. This method is particularly well suited for implementation on parallel computers. Numerical studies show the efficiency, for different test functions, of the LDFI-procedure combined with the simplified Newton–AM method. Better results are obtained when in the reaction diffusion equation also a convection term is present.

Existence of Alternate Steady States in a Phosphorous Cycling Model

We analyze the positive solutions to the steady-state reaction diffusion equation with Dirichlet boundary conditions of the form: . Here, is the Laplacian of , is the diffusion coefficient, and are positive constants, and is a smooth bounded region with in . This model describes the steady states of phosphorus cycling in stratified lakes. Also, it describes the colonization of barren soils in drylands by vegetation. In this paper, we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of subsuper solutions.

Lagged diffusivity fixed point iteration for solving steady-state reaction diffusion problems

The paper concerns with the computational algorithms for a steady-state reaction diffusion problem. A lagged diffusivity iterative algorithm is proposed for solving resulting system of quasilinear equations from a finite difference discretization. The convergence of the algorithm is discussed and the numerical results show the efficiency of this algorithm.

Analytical expressions of steady-state concentrations of species in potentiometric and amperometric biosensor

A mathematical model of potentiometric and amperometric enzyme electrodes is discussed. The model is based on the system of non-linear steady-state coupled reaction diffusion equations for Michaelis-Menten formalism that describe the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate, product and corresponding flux response have been derived for all values of parameters using Homotopy analysis method. The obtained solution allow a full characterization of the response curves for only two kinetic parameters (The Michaelis constant and the ratio of overall reaction and the diffusion rates). A simple relation between the concentration of substrate and products for all values of parameter is also reported. All the analytical results are compared with simulation results (Scilab/Matlab program). The simulated results are agreed with the appropriate theories. The obtained theoretical results are valid for the whole solution domain.

Analytical Expressions of Concentrations inside the Cationic Glucose-Sensitive Membrane

A mathematical model of Wu et al. [J. Membr. Sci 254 (2005) 119-127] of a cationic glucose-sensitive membrane is discussed. The model involves the system of non-linear steady-state reaction-diffusion equations. Analytical expres-sions pertaining to concentration of oxygen, glucose, and gluconic acid for all values of parameters are presented. We have employed Homotopy analysis method to evaluate the approximate analytical solutions of the non-linear boundary value problem. A comparison of the analytical approximation and numerical simulation is also presented. A good agreement between theoretical predictions and numerical results is observed.

Chemical interdiffusion in binary systems; interface barriers and phase competition

The problem of simultaneous growth and competition of intermediate phases during reactive diffusion is formulated and solved. In this paper, we compare existing models of steady state reaction diffusion and introduce the new one basing on the bi-velocity method. We extend old problem and propose method based on material (lattice) fixed frame of reference. It allows computing the material velocity in the reacting system in which reactions at several moving interfaces occur. All reactions lead to the lattice shift due to the difference of intrinsic diffusivities and different molar volumes. The following peculiarities are taken into account: (1) the deviation from local equilibrium at all interfaces; (2) the mobilities of the components in the bulk and interphase zone; and (3) the molar volumes of the components. We show the kinetic of the reactions, the non parabolic regime, the multiphase scale growth and present the practical application of the method.

An ecological model with a [formula omitted]-shaped bifurcation curve

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: { − u ″ = λ [ u − u 2 K − c u 2 1 + u 2 − ϵ ] , x ∈ ( 0 , 1 ) u ( 0 ) = 0 = u ( 1 ) . Here 1 λ is the diffusion coefficient and K , c and ϵ are positive constants. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. In this paper, we discuss the occurrence of a Σ -shaped bifurcation diagram for positive solutions. In particular, for certain parameter values of c , K , ϵ and the diffusion coefficient, we prove the existence of at least four positive solutions. We prove our results by the quadrature method.

Analytical expression of the steady-state catalytic current of mediated bioelectrocatalysis and the application of He’s Homotopy perturbation method

A mathematical model of mediated bioelectrocatalysis is restudied in this paper. Here He’s Homotopy perturbation method is implemented to give approximate and analytical solutions of steady-state non-linear reaction diffusion equation containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. Approximate analytical expressions for mediator concentration and current have been derived for all values of saturation parameter α and reaction diffusion parameter k for the various types of boundary conditions. The Homotopy perturbation method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with numerical results (Matlab program) and are found to be in good agreement.

S-shaped bifurcation curves in ecosystems

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: { − Δ u = λ [ u − u 2 K − c u 2 1 + u 2 ] , x ∈ Ω , u = 0 , x ∈ ∂ Ω . Here Δ u = div ( ∇ u ) is the Laplacian of u, 1 λ is the diffusion coefficient, K and c are positive constants and Ω ⊂ R N is a smooth bounded region with ∂ Ω in C 2 . This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. In this paper we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of sub-supersolutions.