Solutions Of Reaction-diffusion Equations Research Articles

Modeling of Core-shell Catalytic Pellets of Inert Shell with Various Morphologies for Series Reactions

Modeling of core-shell catalytic pellets with inert shell was performed assuming three different particle shapes such as sphere, cylinder, and slab. For isothermal irreversible first order reaction, A→B, analytical solutions of intra-particle concentration of reactant were derived by solving reaction-diffusion equations, considering thickness of catalytically active core (r2), when the pellets were immersed in infinitely large medium. Unlike conventional pellets or core-shell pellets with inert core, effectiveness factor (η) was affected by r2 as well as ratio of effective diffusivity in inert shell and active core (Υ), and η could be enhanced by increasing Υ. η increased in the order of sphere > cylinder > slab due to surface area per unit volume of medium, and η increased with increasing Biot number (Bi), because of decreased external film resistance. For series reaction, A→B→C, transient change of the concentration of starting reactant (CA) and intermittent product (CB) in bulk fluid of reactor could be predicted by solving material balance equations assuming pseudo-steady state approximation and equal value of Thiele modulus for both reactions in batch reactor and CSTR. When B is desirable product, CB increased by increasing Υ and decreasing r2. Similar to conventional pellets, CB in batch reactor decreased by increase of distribution coefficient of B on surface of pellet, KB, due to increased adsorption, and increased in the order of sphere < cylinder < slab, similar to conventional pellets. In CSTR, CB at steady state could be enhanced by decreasing amount of catalyst and Thiele modulus (Φ), implying that performance of reactor can be controlled by adjusting various reaction parameters. From numerical solutions of reaction diffusion equations for series reactions, the effect of r2 and Υ were also confirmed from transient value of CA and CB in exit stream calculated by finite element method.

Modeling and Solution of Reaction-Diffusion Equations by Using the Quadrature and Singular Convolution Methods.

In the present work, polynomial, discrete singular convolution and sinc quadrature techniques are employed as the new techniques to derive accurate and efficient numerical solutions for the reaction–diffusion equations. Three models, Fitzhugh-Nagumo, Newell–Whitehead–Segel, and tumor growth models, were presented. The equations of three models are reduced to nonlinear ordinary differential equations by using different quadrature schemes. Then, Runge–Kutta fourth-order method is employed to solve nonlinear ordinary differential equations. In addition, the MATLAB program is used to solve these problems. Comparisons between the new methods and the existing ones are included, demonstrating the ease of implementation and efficiency. Also, the calculated results are supported by four various statistical errors. It is found that the rate of error reaches ≤ 10–6 in discrete singular convolution depending on regularized Shannon kernel which is better than others. Further, a parametric analysis is presented to discuss the influence of diffusion and reaction parameters on the solution.

Integrodifference master equation describing actively growing blood vessels in angiogenesis

Abstract We study a system of particles in a two-dimensional geometry that move according to a reinforced random walk with transition probabilities dependent on the solutions of reaction-diffusion equations (RDEs) for the underlying fields. A birth process and a history-dependent killing process are also considered. This system models tumor-induced angiogenesis, the process of formation of blood vessels induced by a growth factor (GF) released by a tumor. Particles represent vessel tip cells, whose trajectories constitute the growing vessel network. New vessels appear and may fuse with existing ones during their evolution. Thus, the system is described by tracking the density of active tips, calculated as an ensemble average over many realizations of the stochastic process. Such density satisfies a novel discrete master equation with source and sink terms. The sink term is proportional to a space-dependent and suitably fitted killing coefficient. Results are illustrated studying two influential angiogenesis models.

Solutions of reaction-diffusion equations using similarity reduction and HSSOR iteration

&lt;p&gt;Similarity method is used in finding the solutions of partial differential equation (PDE) in reduction to the corresponding ordinary differential equation (ODE) which are not easily integrable in terms of elementary or tabulated functions. Then, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method is applied in solving the sparse linear system which is generated from the discretization process of the corresponding second order ODEs with Dirichlet boundary conditions. Basically, this ODEs has been constructed from one-dimensional reaction-diffusion equations by using wave variable transformation. Having a large-scale and sparse linear system, we conduct the performances analysis of three iterative methods such as Full-sweep Gauss-Seidel (FSGS), Full-sweep Successive Over-Relaxation (FSSOR) and HSSOR iterative methods to examine the effectiveness of their computational cost. Therefore, four examples of these problems were tested to observe the performance of the proposed iterative methods. Throughout implementation of numerical experiments, three parameters have been considered which are number of iterations, execution time and maximum absolute error. According to the numerical results, the HSSOR method is the most efficient iterative method in solving the proposed problem with the least number of iterations and execution time followed by FSSOR and FSGS iterative methods.&lt;/p&gt;

Transition semi-wave solutions of reaction diffusion equations with free boundaries

In this paper, we define the transition semi-wave solution (cf. Definition 1.1) of the following reaction diffusion equation with free boundaries(1){ut=uxx+f(t,x,u),t∈R,x<h(t),u(t,h(t))=0,t∈R,h′(t)=−μux(t,h(t)),t∈R. In the homogeneous case, i.e., f(t,x,u)=f(u), under the hypothesisf(u)∈C1([0,1]),f(0)=f(1)=0,f′(1)<0,f(u)<0foru>1, we prove that the semi-wave connecting 1 and 0 of (1) is unique provided it exists. Furthermore, we prove that any bounded transition semi-wave connecting 1 and 0 is exactly the semi-wave.In the cases where f is KPP-Fisher type and almost periodic in time (space), i.e., f(t,x,u)=u(c(t)−u) (resp. u(a(x)−u)) with c(t) (resp. a(x)) being almost periodic, applying totally different method, we also prove any bounded transition semi-wave connecting the unique almost periodic positive solution of ut=u(c(t)−u) (resp. uxx+u(a(x)−u)=0) and 0 is exactly the unique almost periodic semi-wave of (1). Finally, we provide an example of the heterogeneous equation to show the existence of the transition semi-wave without any global mean speeds.

Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions

We devote to investigating the following nonlinear reaction diffusion equations with nonlocal boundary conditions{ut=∇⋅(ρ(u)∇u)+k1(t)f(u)inD×(0,t⁎),∂u∂ν=k2(t)∫Dg(u)dxon∂D×(0,t⁎),u(x,0)=u0(x)≥0inD‾, where D is a bounded convex region in Rn(n≥2), and the boundary ∂D is smooth. By constructing some auxiliary functions and using differential inequality technique, we derive that the solution blows up at some finite time. Moreover, upper and lower bounds of the blow-up time are obtained.

Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

AbstractIn this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem.

Distributed order reaction-diffusion systems associated with Caputo derivatives

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. [“The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus Appl. Anal. 4, 153–202 (2001); Mainardi et al. “Fox H-functions in fractional diffusion,” J. Comput. Appl. Math. 178, 321–331 (2005)] for the fundamental solution of the space-time fractional equation, including Haubold et al. [“Solutions of reaction-diffusion equations in terms of the H-function,” Bull. Astron. Soc. India 35, 681–689 (2007)] and Saxena et al. [“Fractional reaction-diffusion equations,” Astrophys. Space Sci. 305, 289–296 (2006a)] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion, and time-fractional diffusion, see Schneider and Wyss [“Fractional diffusion and wave equations,” J. Math. Phys. 30, 134–144 (1989)]. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables given in Appendix B of this paper. Fractional reaction-diffusion equations are of specific interest in physics for non-Gaussian, non-Markovian, and non-Fickian phenomena.

An Efficient Wavelet Based Approximation Method to Steady State Reaction–Diffusion Model Arising in Mathematical Chemistry

The mathematical model of Rahamathunissa and Rajendran (J Math Chem 44:849-861, 2008) in an amperometric biosensor response is discussed. In this paper, we have applied the shifted second kind Chebyshev wavelets (CW) to obtain the numerical solutions of reaction-diffusion equations containing a nonlinear term related to Michaelis-Menton kinetics of the enzymatic reaction. The application of the shifted second kind CW operational matrices for solving initial and boundary value problems is presented. The obtained numerical results demonstrate efficient and applicability of the proposed method. The power of the manageable method is confirmed. Moreover the use of shifted second kind CW method is found to be simple, efficient, accurate, small computation cost, and computationally attractive.

Patrones de Turing sobre superficies sometidas a deformación: un acercamiento desde el método lagrangiano total

In this work we have developed several numerical examples of reaction-diffusion equations with growing domain. For this purpose we have used the Schnakenberg reaction model with parameters in space Turing. Therefore numerical tests are performed on the appearance of Turing patterns on surfaces that have high deformation rate. For the solution of reaction diffusion equations is presented a solution method on surfaces in three dimensions using the finite element method under the use of the total Lagrangian formulation. The results show that the formation of Turing patterns depends on the features of surface deformation and the rate at which change in position of each point of the domain. These results can explain some phenomena of change of pattern on the surface of the skin of animals that exhibit characteristic spots.

Towards a Unified Method for Exact Solutions of Evolution Equations. An Application to Reaction Diffusion Equations with Finite Memory Transport

We present a brief report on the different methods for finding exact solutions of nonlinear evolution equations. Explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones. It is shown that most of methods that exist in the literature are equivalent to the “generalized mapping method” that unifies them. By using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained. Attention is focused to the finite-memory-transport-Fisher and Nagumo equations.

Stability Analysis of Reaction-Diffusion Equations with Double Diffusivity System

12 2 1 12 D k aL k bL k ba L ba L0 '' ' ' '' +++-> ABSTRACT Stability analysis for steady state solution of reaction-diff usion equations with double diffusivity discuss and arise in the solution of problems of flow of homogeneous liquids and heat conduction involving air-temperature ) , ( t x u and a grain-temperat ure

Solutions of the Coupled Reaction and Diffusion Equations within Polymer-Modified Ultramicroelectrodes

In this work, we have theoretically analyzed the non-steady-state current response witnessed at conducting polymer-modified ultramicroelectrodes when the electrode is operating under batch amperometric conditions. Furthermore, the effect of substrate diffusion in the solution adjacent to the polymer film on both the concentration profiles and current response was examined. Analytical expressions pertaining to substrate concentration, mediator concentration, and current response pertaining to all values of diffusion parameters are presented. In addition, we have employed "reduction of order method" to evaluate the approximate analytical solutions of reaction diffusion equations.

Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u ∈ C 2 ( R n × ( 0 , ∞ ) ) ∩ C ( R n × [ 0 , ∞ ) ) to the parabolic reaction diffusion equation u t = Lu + f ( x , u ) , ( x , t ) ∈ R n × ( 0 , ∞ ) , u ( x , 0 ) = g ( x ) ⩾ 0 , x ∈ R n , u ⩾ 0 , where L = ∑ i , j = 1 n a i , j ( x ) ∂ 2 ∂ x i ∂ x j + ∑ i = 1 n b i ( x ) ∂ ∂ x i is a nondegenerate elliptic operator, g ∈ C ( R n ) and the reaction term f converges to - ∞ at a super-linear rate as u → ∞ . The first result in this paper is a parabolic Osserman–Keller type estimate. We give a sharp minimal growth condition on f, independent of L, in order that there exist a universal, a priori upper bound for all solutions to the above Cauchy problem—that is, in order that there exist a finite function M ( x , t ) on R n × ( 0 , ∞ ) such that u ( x , t ) ⩽ M ( x , t ) , for all solutions to the Cauchy problem. Assuming now in addition that f ( x , 0 ) = 0 , so that u ≡ 0 is a solution to the Cauchy problem, we show that under a similar growth condition, an intimate relationship exists between two seemingly disparate phenomena—namely, uniqueness for the Cauchy problem with initial data g = 0 and the nonexistence of unbounded, stationary solutions to the corresponding elliptic problem. We also give a generic sufficient condition guaranteeing uniqueness for the Cauchy problem.

MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION

The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region. Finally, a few examples of application are given.

Nonlocal symmetry and generation of solutions for reaction-diffusion equations

A class of second-order nonlinear evolution equations is considered. A new equation of this class which can be linearized has been found. Invariant nonlocal transformations and nonlocal superposition principle have been constructed for this equation. Nonlocal invariant transformations for some equations of this class also were studied. There are shown examples of generation of partial solutions.

Spatial inhomogenity due to turing bifurcation in a system of Gierer-Meinhardt type

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction-diffusion equations of Gierer-Meinhardt type, subject to Neumann boundary data. The domains in which diffusion takes place are of three types: a regular hexagon, a rectangle and an isosceles rectangular triangle. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven instability occurs and Turing bifurcation takes place: a pattern emerges.

Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain

The paper is concerned with stable subharmonic solutions of timeperiodic spatially inhomogeneous reaction-diffusion equations. We show that such solutions exist on any spatial domain, provided the nonlinearity is chosen suitably. This contrasts with our previous results on spatially homogeneous equations that admit stable subharmonic solutions on some, but not on arbitrary domains.

Finite difference solutions of reaction diffusion equations with continuous time delays

This paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays. The system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either Dirichlet or Neumann-Robin boundary conditions. Various monotone iterative schemes, which depend on the property of the nonlinear reaction mechanism, are developed for the computation of numerical solutions. It is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Applications are given to two model problems known as the diffusive logistic equation and the Fisher's diffusion equation in population genetics.

Traveling wave speed and solution in reaction-diffusion equation in one dimension

By Painleve analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.