Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, 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 Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, 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. Among ... [...More Info...] [...Related Items...] 

Heat
In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by ..., heat is energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ... in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is govern ... or transfer of matter. The various mechanisms of energy transfer that define heat are stated in the nex ... [...More Info...] [...Related Items...] 

General Relativity
General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo'' "earth", '' metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ... theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, G ... of gravitation Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is th ... published by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a Germanborn , widely ac ... [...More Info...] [...Related Items...] 

Fluid Mechanics
Fluid mechanics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ... concerned with the mechanics Mechanics (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ... of fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...s (liquid A liquid is a nearly incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Me ... [...More Info...] [...Related Items...] 

Parabolic Partial Differential Equation
A parabolic partial differential equation is a type of partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... (PDE). Parabolic PDEs are used to describe a wide variety of timedependent phenomena, including heat conduction Thermal conduction is the transfer of internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes u ..., particle diffusion, and pricing of derivative investment instruments. Definition To define the simplest kind of parabolic PDE, consider a realvalued function u(x, y) of two independent real variables, x and y. A secondorder, linear, constantcoefficient PDE for u takes the form ... [...More Info...] [...Related Items...] 

Elliptic Partial Differential Equation
Secondorder linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ... partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...s (PDEs) are classified as either elliptic, hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ..., or parabolic. Any secondorder linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= ... [...More Info...] [...Related Items...] 

Fractional Calculus
Fractional calculus is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ... that studies the several different possibilities of defining real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ... powers or complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... powers of the differentiation Differentiation may refer to: Business * Differe ... [...More Info...] [...Related Items...] 

Stochastic Partial Differential Equation
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have relevance to quantum field theory, statistical mechanics, and Spatial analysis, spatial modeling. Examples One of the most studied SPDEs is the stochastic heat equation, which may formally be written as : \partial_t u = \Delta u + \xi\;, where \Delta is the Laplacian and \xi denotes spacetime white noise. Other examples also include stochastic versions of famous linear equations, such as wave equation and Schrödinger equation. Discussion One difficulty is their lack of regularity. In one dimensional space, solutions to the stochastic heat equation are only almost 1/2Hölder continuous in space and 1/4Hölder continuous in time. For dimensions two and higher, solutions are not even functionvalued, but can be made sense of as ra ... [...More Info...] [...Related Items...] 

Ordinary Differential Equation
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an ordinary differential equation (ODE) is a differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ... containing one or more functions of one independent variable Dependent and Independent variables are variables in mathematical modeling A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to f ... and the derivative In mathematics, the derivative of a function of a real variable m ... [...More Info...] [...Related Items...] 

Geometric Topology
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., geometric topology is the study of manifold The real projective plane is a twodimensional manifold that cannot be realized in three dimensions without selfintersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...s and maps A map is a symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meani ... between them, particularly embedding In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and relate ... [...More Info...] [...Related Items...] 

Poincaré Conjecture
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the Poincaré conjecture (, , ) is a theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... about the characterization Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative A narrative, story or tale is any account of a series of related events or experiences, whether nonfiction Nonfiction (also spelle ... of the 3sphere In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory ... [...More Info...] [...Related Items...] 

Calculus Of Variations
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shorte ... [...More Info...] [...Related Items...] 

Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity, as it relates to astronomy and the geodesy of the Earth, and later in the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the threedimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometri ... [...More Info...] [...Related Items...] 