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Integro-differential Equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function (mathematics), function. General first order linear equations The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form : \fracu(x) + \int_^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0. As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation. Example Consider the following second-order problem, : u'(x) + 2u(x) + 5\int_^u(t)\,dt = \theta(x) \qquad \text \qquad u(0)=0, where : \theta(x) = \left\{ \begin{ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inhibitory Postsynaptic Potential
An inhibitory postsynaptic potential (IPSP) is a kind of synaptic potential that makes a postsynaptic neuron less likely to generate an action potential.Purves et al. Neuroscience. 4th ed. Sunderland (MA): Sinauer Associates, Incorporated; 2008. The opposite of an inhibitory postsynaptic potential is an excitatory postsynaptic potential (EPSP), which is a synaptic potential that makes a postsynaptic neuron ''more'' likely to generate an action potential. IPSPs can take place at all chemical synapses, which use the secretion of neurotransmitters to create cell-to-cell signalling. EPSPs and IPSPs compete with each other at numerous synapses of a neuron. This determines whether an action potential occurring at the presynaptic terminal produces an action potential at the postsynaptic membrane. Some common neurotransmitters involved in IPSPs are GABA and glycine. Inhibitory presynaptic neurons release neurotransmitters that then bind to the postsynaptic receptors; this induces a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrodifference Equation
In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: : n_(x) = \int_ k(x, y)\, f(n_t(y))\, dy, where \\, is a sequence in the function space and \Omega\, is the domain of those functions. In most applications, for any y\in\Omega\,, k(x,y)\, is a probability density function on \Omega\,. Note that in the definition above, n_t can be vector valued, in which case each element of \ has a scalar valued integrodifference equation associated with it. Integrodifference equations are widely used in mathematical biology, especially theoretical ecology Theoretical ecology is the scientific discipline devoted to the study of ecosystem, ecological systems using theoretical methods such as simple conceptual models, mathematical models, computer simulation, computational simulations, and advanced d ..., to model the dispersal and growth of populations. In this case, n_t(x) is the population size or density at location x at time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Equation
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2,x_3,\ldots,x_n) ; I^1 (u), I^2(u), I^3(u), \ldots, I^m(u)) = 0 where I^i(u) is an integral operator acting on ''u.'' Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2,x_3,\ldots,x_n) ; D^1 (u), D^2(u), D^3(u), \ldots, D^m(u)) = 0where D^i(u) may be viewed as a differential operator of order ''i''. Due to this close connection between differential and integral equations, one can often convert between the two. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delay Differential Equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with a functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs: # Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kermack–McKendrick Theory
Kermack–McKendrick theory is a hypothesis that predicts the number and distribution of cases of an immunizing infectious disease over time as it is transmitted through a population based on characteristics of infectivity and recovery, under a strong-mixing assumption. Building on the research of Ronald Ross and Hilda Hudson, A. G. McKendrick and W. O. Kermack published their theory in a set of three articles from 1927, 1932, and 1933. Kermack–McKendrick theory is one of the sources of the SIR model and other related compartmental models. This theory was the first to explicitly account for the dependence of infection characteristics and transmissibility on the age of infection. Because of their seminal importance to the field of theoretical epidemiology, these articles were republished in the '' Bulletin of Mathematical Biology'' in 1991. Epidemic model (1927) In its initial form, Kermack–McKendrick theory is a partial differential-equation model that structures the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Pyramid
A population pyramid (age structure diagram) or "age-sex pyramid" is a graphical illustration of the distribution of a population (typically that of a country or region of the world) by age groups and sex; it typically takes the shape of a pyramid when the population is growing. Males are usually shown on the left and females on the right, and they may be measured in absolute numbers or as a percentage of the total population. The pyramid can be used to visualize the age of a particular population. It is also used in ecology to determine the overall age distribution of a population; an indication of the reproductive capabilities and likelihood of the continuation of a species. Number of people per unit area of land is called population density. Structure A population pyramid often contains continuous stacked-histogram bars, making it a horizontal bar diagram. The population size is shown on the x-axis (horizontal) while the age-groups are represented on the y-axis (vertical). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epidemic
An epidemic (from Greek ἐπί ''epi'' "upon or above" and δῆμος ''demos'' "people") is the rapid spread of disease to a large number of hosts in a given population within a short period of time. For example, in meningococcal infections, an attack rate in excess of 15 cases per 100,000 people for two consecutive weeks is considered an epidemic. Epidemics of infectious disease are generally caused by several factors including a change in the ecology of the host population (e.g., increased stress or increase in the density of a vector species), a genetic change in the pathogen reservoir or the introduction of an emerging pathogen to a host population (by movement of pathogen or host). Generally, an epidemic occurs when host immunity to either an established pathogen or newly emerging novel pathogen is suddenly reduced below that found in the endemic equilibrium and the transmission threshold is exceeded. An epidemic may be restricted to one location; however, if it sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and Risk factor (epidemiology), determinants of health and disease conditions in a defined population, and application of this knowledge to prevent diseases. It is a cornerstone of public health, and shapes policy decisions and evidence-based practice by identifying Risk factor (epidemiology), risk factors for disease and targets for preventive healthcare. Epidemiologists help with study design, collection, and statistical analysis of data, amend interpretation and dissemination of results (including peer review and occasional systematic review). Epidemiology has helped develop methodology used in clinical research, public health studies, and, to a lesser extent, basic research in the biological sciences. Major areas of epidemiological study include disease causation, transmission (medicine), transmission, outbreak investigation, disease surveillance, environmental epidemiology, forensic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitham Equation
In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves. The equation is notated as follows:This integro-differential equation for the oscillatory variable ''η''(''x'',''t'') is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves in 1967. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven. For a certain choice of the kernel ''K''(''x'' − ''ξ'') it becomes the Fornberg–Whitham equation. Water waves Using the Fourier transform (and its inverse), with respect to the space coordinate ''x'' and in terms of the wavenumber ''k'': * For surface gravity waves, the phase speed ''c''(''k'') as a function of wavenumber ''k'' is taken as: :: c_\text(k) = \sqrt, while \alpha_\text = \frac \sqrt, :with ''g'' the gravitational acceleration and ''h'' the mean water depth. The associated kernel ''K''w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
