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The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross [1] and Lev Petrovich Pitaevskii [2]) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model. A Bose–Einstein condensate (BEC) is a gas of bosons that are in the same quantum state, and ...
Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9). Numerical solution of the KdV equation u t + uu x + δ 2 u xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx). Time evolution was done by the Zabusky–Kruskal scheme. [1]
The following is an example of an image sequence zooming to a selected c value. The magnification of the last image relative to the first one is about 10 10 to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers.
Second-order cone programming. A second-order cone program (SOCP) is a convex optimization problem of the form. where the problem parameters are , and . is the optimization variable. is the Euclidean norm and indicates transpose. 1 The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function ...
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
Geometric programming is closely related to convex optimization: any GP can be made convex by means of a change of variables. [2] GPs have numerous applications, including component sizing in IC design, [3][4] aircraft design, [5] maximum likelihood estimation for logistic regression in statistics, and parameter tuning of positive linear ...
The term "Bellman equation" usually refers to the dynamic programming equation (DPE) associated with discrete-time optimization problems. [5] In continuous-time optimization problems, the analogous equation is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation. [6][7] In discrete time any multi-stage ...
Pseudo-spectral method. Pseudo-spectral methods, [1] also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely related to spectral methods, but complement the basis by an additional ...