Numerical study of Fisher's reaction–diffusion equation by the Sinc collocation method

Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by the Sinc collocation method. The derivatives and integrals are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The error in the approximation of the solution is shown to converge at an exponential rate. Numerical examples are given to illustrate the accuracy and the implementation of the method, the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are independent of the initial values.     

Studies in oxidation: Oxidation of some aliphatic secondary alcohols with chromium (VI) oxide—Part I

Proceedings - Mathematical Sciences - The oxidation of isopropyl alcohol, butanol-2, pentanol-2 and octanol-2 by chromium (VI) oxide, has been studied in acetic acid-water mixtures under conditions...     

Note on the stability of reaction–diffusion systems with delays by Lyapunov functional

This paper aims to investigate the stability of reaction–diffusion equations with delays. We extend a stability theorem on FDEs introduced by Hale to reaction–diffusion equations with time delay.     

How to find the volume of a solid of revolution——by using the definite integral

<正>You've already learned that area is one of the many applications of definite integral.This means area is an important application in finding the area in the plane(the two dimensional area).This means the application of integrals to the computation of areas in the plane can be extended to the three dimensional solid.We have another important application is computation of certain volumes in space.     

A computational modeling of the behavior of the two-dimensional reaction–diffusion Brusselator system

This paper studies a meshfree technique for the numerical solution of the two-dimensional reaction–diffusion Brusselator system along with Dirichlet and Neumann boundary conditions. Combination of collocation method using the radial basis functions (RBFs) with first order accurate forward difference approximation is employed for obtaining meshfree solution of the problem. Different types of RBFs are used for this purpose. The method is shown to converge to the only equilibrium point of the system. Performance of the proposed method is successfully tested in terms of various error norms. In the case of non-availability of exact solution, performance of the new method is compared with the results obtained from the existing methods [7] and [8]. The elementary stability analysis is established theoretically and is also supported by numerical results.     

On the solutions of time-fractional reaction–diffusion equations

In this paper, a new application of generalized differential transform method (GDTM) has been used for solving time-fractional reaction–diffusion equations. To illustrate the reliability of the method, some examples are provided.     

Dynamical Systems in the Variational Formulation of the Fokker—Planck Equation by the Wasserstein Metric

R. Jordan, D. Kinderlehrer, and F. Otto proposed the discrete-time approximation of the Fokker—Planck equation by the variational formulation. It is determined by the Wasserstein metric, an energy functional, and the Gibbs—Boltzmann entropy functional. In this paper we study the asymptotic behavior of the dynamical systems which describe their approximation of the Fokker—Planck equation and characterize the limit as a solution to a class of variational problems. Accepted 2 June 2000. Online publication 6 October 2000.     

Mathematical modeling of catalytic wet oxidation in trickle-bed reactors by a diffusion–convection–reaction approach embedded with an interstitial CFD framework

In this work, we propose a diffusion–convection–reaction methodology to gain further insights into the heterogeneous multiphase flow of trickle beds. Our case-study encompasses a multi-fluid model embedded within an interstitial framework on the numerical simulation of continuous catalytic wet oxidation of hazardous compounds. First, with the proviso that phase holdup, pressure drop, and liquid distribution are fundamental criteria for the efficient design of trickle beds, the multiphase flow constitutive equations have been developed and solved by the conservative unstructured finite volume method. Second, several numerical variables were parametrically optimized based on the application of different under-relaxation parameters, mesh densities, and time stepping strategies. The segregated solver has been found to reveal good properties in terms of convergence and stability criteria, which endorsed the further corroboration. Finally, this theoretical probing-sensing scheme enabled the characterization of liquid flow texture accomplished by three-dimensional flow patterns exposing their deviation from ideal plug flow. The diffusion–convection–reaction framework coupled within a CFD model can then be further exploited on the simulation of complex multiphase reactive flows with adjustable parameters.     

On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

On the measure of approximation of the number π by algebraic numbers

By means of the method of the Laurent interpolation determinant, it is proved that, if ζ is an algebraic number, the real numbersd andL satisfy the inequalitiesd≥degζ,L≥L(ζ), andL≥3, and the numberd is sufficiently large, then the inequality

Dynamical properties of the reaction–diffusion type model of fast synaptic transport

The modulation of a signal that is transmitted in the nerve system takes place in chemical synapses. This article focuses on the phenomena undergone in the presynaptic part of the synapse. A diffusion–reaction type model based on the partial differential equation is proposed. Through an averaging procedure this model is reduced to a model based on ordinary differential equations with control, which is then analyzed according to its dynamical properties—controllability, observability and stability. The system is strongly connected to the one introduced by Aristizabal and Glavinovic (2004) [13]. The biological implications of the obtained mathematical results are also discussed.