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Reaction–diffusion System
Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form :\partial_t \boldsymbol = \underline \,\nabla^2 \boldsymbol + \boldsymbol(\boldsymbol), where represents the unknown vector function, is a diagonal matrix of diffusion coefficients, and accounts for all local reactions. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Maini
Philip Kumar Maini (born 16 October 1959 in Magherafelt, Northern Ireland) is a Northern Irish mathematician. Since 1998, he has been the Professor of Mathematical Biology at the University of Oxford and is the director of the Wolfson Centre for Mathematical Biology in the Mathematical Institute. Personal Life Philip Maini is the son of Panna Lal Maini and Satya Wati Bhandari. Panna Lal and Satya Wati were from Punjab in North West India. Panna Lal traveled to Northern Ireland in 1954. He had sailed to London on the ship Maloja of the Peninsula and Orient Steam Navigation Company arriving there on 18 February 1954. Satya Wati and Philip's elder brother Arvind did not arrive in Northern Ireland until 1957. Education Maini was educated at Rainey Endowed School in County Londonderry and Balliol College, Oxford where he was awarded a BA in 1982 and a DPhil in 1985, the latter for a thesis modelling morphogenetic pattern formation supervised by James D. Murray Research and career ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as Df = \lambda f for some scalar eigenvalue \lambda. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector. Eigenfunctions In general, an eigenvector of a linear operator ''D'' defined on some vector space is a nonzero vector in the domain of ''D'' that, when ''D'' acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where ''D'' is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function ''f'' is an eigenfunction of ''D'' if it satisfies the equation where λ is a scalar. The solutions to Equation may also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Travelling Wave For Fisher Equation
Travel is the movement of people between distant geographical locations. Travel can be done by foot, bicycle, automobile, train, boat, bus, airplane, ship or other means, with or without luggage, and can be one way or round trip. Travel can also include relatively short stays between successive movements, as in the case of tourism. Etymology The origin of the word "travel" is most likely lost to history. The term "travel" may originate from the Old French word ''travail'', which means 'work'. According to the Merriam-Webster dictionary, the first known use of the word ''travel'' was in the 14th century. It also states that the word comes from Middle English , (which means to torment, labor, strive, journey) and earlier from Old French (which means to work strenuously, toil). In English, people still occasionally use the words , which means struggle. According to Simon Winchester in his book ''The Best Travelers' Tales (2004)'', the words ''travel'' and ''travail'' both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion does not always result in fire, because a flame is only visible when substances undergoing combustion vaporize, but when it does, a flame is a characteristic indicator of the reaction. While the activation energy must be overcome to initiate combustion (e.g., using a lit match to light a fire), the heat from a flame may provide enough energy to make the reaction self-sustaining. Combustion is often a complicated sequence of elementary radical reactions. Solid fuels, such as wood and coal, first undergo endothermic pyrolysis to produce gaseous fuels whose combustion then supplies the heat required to produce more of them. Combustion is often hot enough that incandescent light in the form of either glowing or a flame is produced. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeldovich Number
The Zel'dovich number is a dimensionless number which provides a quantitative measure for the activation energy of a chemical reaction which appears in the Arrhenius exponent, named after the Russian scientist Yakov Borisovich Zel'dovich, who along with David A. Frank-Kamenetskii, first introduced in their paper in 1938. In 1983 ICDERS meeting at Poitiers, it was decided to name after Zel'dovich.Clavin, P. (1985). Dynamic behavior of premixed flame fronts in laminar and turbulent flows. Progress in energy and combustion science, 11(1), 1-59. It is defined as :\beta = \frac \cdot \frac where *E_a is the activation energy of the reaction *R is the universal gas constant *T_b is the burnt gas temperature *T_u is the unburnt mixture temperature. In terms of heat release parameter \alpha, it is given by :\beta = \frac \alpha For typical combustion phenomena, the value for Zel'dovich number lies in the range \beta\approx 8-20. Activation energy asymptotics Activation energy asymp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ZFK Equation
ZFK equation, abbreviation for Zeldovich–Frank-Kamenetskii equation, is a reaction–diffusion equation that models premixed flame propagation. The equation is named after Yakov Zeldovich and David A. Frank-Kamenetskii who derived the equation in 1938 and is also known as the Nagumo equation. The equation is analogous to KPP equation except that is contains an exponential behaviour for the reaction term and it differs fundamentally from KPP equation with regards to the propagation velocity of the traveling wave. In non-dimensional form, the equation reads :\frac = \frac + \omega(\theta) with a typical form for \omega given by :\omega =\frac \theta(1-\theta) e^ where \theta\in ,1/math> is the non-dimensional dependent variable (typically temperature) and \beta is the Zeldovich number. In the ZFK regime, \beta\gg 1. The equation reduces to Fisher's equation for \beta\ll 1 and thus \beta\ll 1 corresponds to KPP regime. The minimum propagation velocity U_ (which is usually the lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh–Bénard Convection
In fluid thermodynamics, Rayleigh–Bénard convection is a type of natural convection, occurring in a planar horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems. Buoyancy, and hence gravity, are responsible for the appearance of convection cells. The initial movement is the upwelling of less-dense fluid from the warmer bottom layer. This upwelling spontaneously organizes into a regular pattern of cells. Physical processes The features of Bénard convection can be obtained by a simple experiment first conducted by Henri Bénard, a French physicist, in 1900. Development of convection The experimental set-up uses a layer of liquid, e.g. water, between ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a census, a process of collecting, analysing, compiling, and publishing data regarding a population. Perspectives of various disciplines Social sciences In sociology and population geography, population refers to a group of human beings with some predefined criterion in common, such as location, race, ethnicity, nationality, or religion. Demography is a social science which entails the statistical study of populations. Ecology In ecology, a population is a group of organisms of the same species who inhabit the same particular geographical area and are capable of interbreeding. The area of a sexual population is the area where inter-breeding is possible between any pair within the area and more probable than cross-breeding with in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher's Equation
In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher) also known as the Kolmogorov–Petrovsky–Piskunov equation (named after Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov), KPP equation or Fisher–KPP equation is the partial differential equation: : \frac - D \frac = r u(1-u).\, It is a kind of reaction–diffusion system that can be used to model population growth and wave propagation. Details Fisher's equation belongs to the class of reaction–diffusion equation: in fact, it is one of the simplest semilinear reaction-diffusion equations, the one which has the inhomogeneous term : f(u,x,t) = r u (1-u),\, which can exhibit traveling wave solutions that switch between equilibrium states given by f(u) = 0. Such equations occur, e.g., in ecology, physiology, combustion, crystallization, plasma physics, and in general phase transition problems. Fisher proposed this equation in his 1937 paper ''The wave of advance of advanta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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