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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] |