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List Of Runge–Kutta Methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation :\frac = f(t, y). Explicit and implicit methods, Explicit Runge–Kutta methods take the form :\begin y_ &= y_n + h \sum_^s b_i k_i \\ k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n+c_2h, y_n+h(a_k_1)), \\ k_3 &= f(t_n+c_3h, y_n+h(a_k_1+a_k_2)), \\ &\;\;\vdots \\ k_i &= f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). \end Stages for Explicit and implicit methods, implicit methods of s stages take the more general form, with the Explicit and implicit methods#Computation, solution to be found over all s :k_i = f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows: : \begin c_1 & a_ & a_& \dots & a_\\ c_2 & a_ & a_& \dots & a_\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_ & a_& \dots & a_ \\ \hline & b_1 & b_2 & \dots & b_s\\ \end For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rehuel Lobatto
Rehuel Lobatto (6 June 1797 – 9 February 1866 ) was a Dutch mathematician. The Gauss-Lobatto quadrature method is named after him, as are his variants on the Runge–Kutta methods for solving ODEs, and the Lobatto polynomials. He was the author of a great number of articles in scientific periodicals, as well as various schoolbooks. Lobatto was born in Amsterdam to a Portuguese Jewish family. As a schoolboy Lobatto already displayed remarkable talent for mathematics. Gotthard Deutsch, E. Slijper (1906)"LOBATTO, REHUEL" ''The Jewish Encyclopedia''. He studied mathematics under Jean Henri van Swinden at the Athenaeum Illustre of Amsterdam, earning his BA in 1812; and then with Adolphe Quetelet (coediting a volume o"Correspondance Mathématique et Physique". Working for the Dutch government - initially for the Ministry of the Interior - he became secretary of a statistical commission in 1831. From 1826 till 1849 he was editor of the '' "Jaarboekje van Lobatto"'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of Explicit and implicit methods, implicit and explicit iterative methods, List of Runge–Kutta methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding y value is y_0. The function f and the initial conditions t_0, y_0 are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Legendre Method
In numerical analysis and scientific computing, the Gauss–Legendre methods are a family of numerical methods for ordinary differential equations. Gauss–Legendre methods are implicit Runge–Kutta methods. More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method based on ''s'' points has order 2''s''. All Gauss–Legendre methods are A-stable. The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is: : The Gauss–Legendre method of order four has Butcher tableau: : The Gauss–Legendre method of order six has Butcher tableau: : The computational cost of higher-order Gauss–Legendre methods is usually excessive, and thus, they are rarely used. Intuition Gauss-Legendre Runge-Kutta (GLRK) methods solve an ordinary differential equation \dot = f(x) with x(0) = x_0. The distinguishing feature of GLRK is the estimation of x(h) - x_0 = \int_0^h f( x(t) ) \, dt wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Differential Equations
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Numerical may refer to: * Number * Numerical digit * Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Iowa
The University of Iowa (U of I, UIowa, or Iowa) is a public university, public research university in Iowa City, Iowa, United States. Founded in 1847, it is the oldest and largest university in the state. The University of Iowa is organized into 12 colleges offering more than 200 areas of study and 7 professional degrees. On an urban 1,880-acre campus on the banks of the Iowa River, the University of Iowa is Carnegie Classification of Institutions of Higher Education, classified among "R1: Doctoral Universities – Very high research activity". In fiscal year 2021, research expenditures at Iowa totaled $818 million. The university was the original developer of the Master of Fine Arts degree, and it operates the Iowa Writers' Workshop, whose alumni include 17 of the university's 46 Pulitzer Prize winners. Iowa is a member of the Association of American Universities and the Universities Research Association. Among public universities in the United States, UI was the first to beco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discontinuous Collocation Method
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all Function (mathematics), functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its Domain of a function, domain, one says that it has a discontinuity there. The Set theory, set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. The Oscillation (mathematics), oscillation of a function at a point quantifies these discontinuities as follows: * in a removable discontinuity, the distance that the value of the function is off by is the oscillation; * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides); * in an essential discontinuity (a.k.a. infinite discontinuity), oscillation measures the failure of a Limit of a function, limit to exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
