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Gardner Equation
The Gardner equation is an integrable nonlinear partial differential equation introduced by the mathematician Clifford Gardner in 1968 to generalize KdV equation and modified KdV equation. The Gardner equation has applications in hydrodynamics, plasma physics and quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ... : \frac-(6 \varepsilon^2 u^2 + 6u) \frac+\frac=0, where \varepsilon is an arbitrary real parameter. See also * Korteweg–de Vries equation Notes References * Nonlinear partial differential equations Integrable systems {{theoretical-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gardner's Relation
Gardner's relation, or Gardner's equation, named after Gerald H. F. Gardner and L. W. Gardner, is an empirically derived equation that relates seismic P-wave velocity to the bulk density of the lithology in which the wave travels. The equation reads: :\rho = \alpha V_p^ where \rho is bulk density given in g/cm3, V_p is P-wave velocity given in ft/s, and \alpha and \beta are empirically derived constants that depend on the geology. Gardner et al. proposed that one can obtain a good fit by taking \alpha = 0.23 and \beta = 0.25. Assuming this, the equation is reduced to: :\rho = 0.23 V_p^, where the unit of V_p is feet/s. If V_p is measured in m/s, \alpha = 0.31 and the equation is: :\rho = 0.31 V_p^. This equation is very popular in petroleum exploration because it can provide information about the lithology from interval velocities obtained from seismic data. The constants \alpha and \beta are usually calibrated from sonic and density Density (volumetric mass density ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ''generic'' dynamical systems, which are more typically chaotic sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Partial Differential Equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear system, nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the Operator (mathematics), operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Gardner
Clifford Spear Gardner (January 14, 1924 – September 25, 2013) was an American mathematician specializing in applied mathematics. Career Gardner studied at Phillips Academy and Harvard, where he earned his baccalaureate in 1944. In 1953 he earned a PhD from New York University, under the supervision of Fritz John. Thereafter he worked at NASA in Langley Field, the Courant Institute of Mathematical Sciences of NYU, Lawrence Livermore National Laboratory and the Princeton Plasma Physics Laboratory. He was a mathematics professor at the University of Texas at Austin from 1967 to 1990, when he retired as professor emeritus.. In 1985 he won the Norbert Wiener Prize for his contributions to supersonic aerodynamics and plasma physics. In 2006 he received with Martin Kruskal, Robert M. Miura, and John M. Greene the Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unnormalized Modified KdV Equation
The modified Korteweg–de Vries (KdV) equation is an integrable nonlinear partial differential equation: : u_t+u_+\alpha u^2 u_x=0 \, where \alpha is an arbitrary (nonzero) constant. This is a special case of the Gardner equation The Gardner equation is an integrable nonlinear partial differential equation introduced by the mathematician Clifford Gardner in 1968 to generalize KdV equation and modified KdV equation. The Gardner equation has applications in hydrodynamics, .... See also * Korteweg–de Vries equation Notes References * * Nonlinear partial differential equations Integrable systems {{theoretical-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hydrodynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion) and (the study of water and other liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space, understanding large scale geophysical flows involving oceans/atmosphere and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plasma Physics
Plasma () is a state of matter characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form of ordinary matter in the universe, mostly in stars (including the Sun), but also dominating the rarefied intracluster medium and intergalactic medium. Plasma can be artificially generated, for example, by heating a neutral gas or subjecting it to a strong electromagnetic field. The presence of charged particles makes plasma electrically conductive, with the dynamics of individual particles and macroscopic plasma motion governed by collective electromagnetic fields and very sensitive to externally applied fields. The response of plasma to electromagnetic fields is used in many modern devices and technologies, such as plasma televisions or plasma etching. Depending on temperature and density, a certain number of neutral particles may also be present, in which case plasma is called partially ioni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on QFT. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Partial Differential Equations
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
