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Functional Differential Equation
A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values. Functional differential equations find use in mathematical models that assume a specified behavior or phenomenon depends on the present as well as the past state of a system. In other words, past events explicitly influence future results. For this reason, functional differential equations are more applicable than ordinary differential equations (ODE), in which future behavior only implicitly depends on the past. Definition Unlike ordinary differential equations, which contain a function of one variable and its derivatives evaluated with the same input, functional differential equations contain a function and its derivatives evaluated with different input values. *An example of an ordinary differential equation would be f'(x) = 2f(x) +1 *In compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logistic Function
A logistic function or logistic curve is a common S-shaped curve (sigmoid function, sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the graph of f approaching L as x approaches +\infty and approaching zero as x approaches -\infty. The logistic function finds applications in a range of fields, including biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, statistics, and artificial neural networks. A generalization of the logistic function is the hyperbolastic functions, hyperbolastic function of type I. The standard logistic function, where L=1,k=1,x_0=0, is sometimes simply called ''the sigmoid''. It is also sometimes called the ''expit'', being the inverse of the logit. History The logistic function was introduced in a series of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Population Growth
Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to 7.9 billion in 2020. The UN projected population to keep growing, and estimates have put the total population at 8.6 billion by mid-2030, 9.8 billion by mid-2050 and 11.2 billion by 2100. However, some academics outside the UN have increasingly developed human population models that account for additional downward pressures on population growth; in such a scenario population would peak before 2100. World human population has been growing since the end of the Black Death, around the year 1350. A mix of technological advancement that improved agricultural productivity and sanitation and medical advancement that reduced mortality increased population growth. In some geographies, this has slowed through the process called the demographic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Carrying Capacity
The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the environment's maximal load, which in population ecology corresponds to the population equilibrium, when the number of deaths in a population equals the number of births (as well as immigration and emigration). The effect of carrying capacity on population dynamics is modelled with a logistic function. Carrying capacity is applied to the maximum population an environment can support in ecology, agriculture and fisheries. The term carrying capacity has been applied to a few different processes in the past before finally being applied to population limits in the 1950s. The notion of carrying capacity for humans is covered by the notion of sustainable population. At the global scale, scientific data indicates that humans are living beyond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Flip-flop (electronics)
In electronics, a flip-flop or latch is a circuit that has two stable states and can be used to store state information – a bistable multivibrator. The circuit can be made to change state by signals applied to one or more control inputs and will have one or two outputs. It is the basic storage element in sequential logic. Flip-flops and latches are fundamental building blocks of digital electronics systems used in computers, communications, and many other types of systems. Flip-flops and latches are used as data storage elements. A flip-flop is a device which stores a single ''bit'' (binary digit) of data; one of its two states represents a "one" and the other represents a "zero". Such data storage can be used for storage of ''state'', and such a circuit is described as sequential logic in electronics. When used in a finite-state machine, the output and next state depend not only on its current input, but also on its current state (and hence, previous inputs). It can als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Volterra Integral Equation
In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is : f(t) = \int_a^t K(t,s)\,x(s)\,ds where ''f'' is a given function and ''x'' is an unknown function to be solved for. A linear Volterra equation of the second kind is : x(t) = f(t) + \int_a^t K(t,s)x(s)\,ds. In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. A linear Volterra integral equation is a convolution equation if : x(t) = f(t) + \int_^t K(t-s)x(s)\,ds. The function K in the integral is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel of the form K(t,s) = (t^2-s^2)^ with 0Defining x_ = x(s_), f_ = f(t_), and K_ = K( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lotka–Volterra Equations
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations: \begin \frac &= \alpha x - \beta x y, \\ \frac &= \delta x y - \gamma y, \end where * is the number of prey (for example, rabbits); * is the number of some predator (for example, foxes); *\tfrac and \tfrac represent the instantaneous growth rates of the two populations; * represents time; *, , , are positive real parameters describing the interaction of the two species. The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism. History The Lotka–Volterra predat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bifurcation Theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them . Bifurcation types It is useful to divide bifurcations into two principal classes: * Local bifurcations, which can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lyapunov Function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Volterra Series
The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of this system depends strictly on the input at that particular time. In the Volterra series the output of the nonlinear system depends on the input to the system at ''all'' other times. This provides the ability to capture the "memory" effect of devices like capacitors and inductors. It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also used in electrical engineering to model intermodulation distortion in many devices, including power amplifiers and frequency mixers. Its main advantage lies in its generality: it can represent a wide range of systems. Thus it is sometimes considered a non-parametric model. In mathematics, a Volterra series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |