41. Holistic Finite Differences Accurately Model The Dynamics Of The {Kuramoto-Sivas ANZIAM J. 42 (E) ppC918C935, 2000. Holistic finite differences accurately modelthe dynamics of the KuramotoSivashinsky equation. T. MacKenzie and AJ Roberts. http://anziamj.austms.org.au/V42/CTAC99/Mack/

ANZIAM J. 42 (E) ppC918C935, 2000. Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation T. MacKenzie and A. J. Roberts (Received 7 August 2000) Abstract We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing insulating internal boundaries which are later removed. The Kuramoto-Sivashinsky equation is used as an example to show how holistic finite differences may be applied to fourth order, nonlinear, spatio-temporal dynamical systems. This novel centre manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms. Download copy Adobe PDF (145K) Postscript (212K) DVI (43K) Browse the article Browse the DVI file on your computer. You must be running X-Windows locally so our server can open a window for you.

42. Finite Differences In Acoustics A very simple example of a finite difference solution to the Helmholtzequation Hermann von Helmholtz 1821 1894. The Helmholtz http://www.tele.ntnu.no/akustikk/person/kristiansen/findif.html

A very simple example of a finite difference solution to the Helmholtz equation Hermann von Helmholtz 1821 - 1894 The Helmholtz equation is the harmonic time variant of the acoustic wave equation. To calculate the resonant frequencies of a one-dimensional duct which has rigid walls at both ends is a classic excersise.The analytical result for the first eigenvalue (the square of the wavenumber) of a unity length duct is pi squared. In this example we first did a direct application of the finite difference technique. Afterwards we saw how the results could be improved upon by Richardson extrapolation. This extrapolation uses the originally calculated values. The results of the investigation are presented in the table below. All calculations were performed in MATLAB. Subdivisions direct fin.dif. solution Richardson extrapolation Analytical solution Note that: The Richardson extrapolated values are correct to 12 decimal places manipulating the 7 first 2nd column values. Using even more values are detremental to the results as we get into conflict with the machine accuracy.

43. 4.4 Methods For European Options: Finite Differences (FD) 4.4.1 Naive implementation with. 4.4 Methods for European optionsfinite differences (FD). The difficulty of extending analytical http://www.lifelong-learners.com/opt/SYL/s4node11.php3

Previous: 4.3.2 A Black-Scholes formula Up: 4 EUROPEAN PAYOFF DYNAMICS Next: 4.4.1 Naive implementation with 4.4 Methods for European options: finite differences (FD) The difficulty of extending analytical derivations much further motivates the development of numerical tools to calculate the numerical value of options directly with a computer. Finite differences are often used because they are relatively simple to formulate and propose an algorithm that converges to the solution; they are however quite tricky to use. In this section, we will consider applications based only on the explicit 2-level scheme, showing later in the next chapters how the implicit Crank-Nicholson method with finite differences is in fact a special case of a more general and more robust formulation based on finite elements Subsections 4.4.1 Naive implementation with a regular sampling of the underlying 4.4.2 Improved scheme using log-normal variables www.lifelong-learners.com

44. Finite Differences Of 1/(B^i-A) By Noam D. Elkies finite differences of 1/(B^iA) by Noam D. Elkies. reply to this messagepost a message on a new topic Back to sci.math.research Subject http://mathforum.org/epigone/sci.math.research/dwongcrermnang

Finite differences of 1/(B^i-A) by Noam D. Elkies reply to this message post a message on a new topic Back to sci.math.research Subject: Finite differences of 1/(B^i-A) Author: elkies_AT_m@h.harvard.edu Organization: University of Illinois at Urbana-Champaign Date: The Math Forum

45. Math Forum - Ask Dr. Math Drexel dragon Donate to the Math Forum Associated Topics Dr. MathHome Search Dr. Math Method of finite differences. Date 10 http://mathforum.org/library/drmath/view/53223.html

Associated Topics Dr. Math Home Search Dr. Math Method of Finite Differences Date: 10/12/2000 at 14:36:30 From: Victoria Gillett Subject: Turning a series into an equation I was given the equation 3n^2 - 4n - 2. By plugging in 1, 2, 3, 4, 5 for n, I worked out the series: -3, 2, 13, 30, 53. If I were given only the series, how would I work through the series in order to find it the equation? Date: 10/12/2000 at 17:56:01 From: Doctor Greenie Subject: Re: Turning a series into an equation Hi, Victoria. Thanks for sending your question to us here at Dr. Math. The most elementary method I know of to find the equation from the series is called the method of "finite differences." The key to this method is the fact that the equation is a polynomial of degree k if and only if the k-th row of differences generated by the series is constant. That probably doesn't mean anything to you if you haven't seen the method before; I'll show you the method for your series in a few moments. I found an in-depth demonstration of WHY the method of finite differences works in "Finite Differences and Polynomials," from the Dr. Math archives, at the following URL: http://mathforum.org/dr.math/problems/harris.6.17.99.html

46. An Application Of Finite Differences An Application of finite differences. Ponder This finite differences commutewith derivatives D(D f) = D D(f). Let f R + ®R be k times differentiable. http://www.gotmath.com/diff.html

An Application of Finite Differences Ponder This... is a monthly puzzle column run by IBM Research. The problems featured in this column are often quite challenging, and they usually receive many correct solutions. The February 2000 problem was as follows: Suppose C is a positive number such that for all positive numbers N it is true that (N raised to the C power) is an integer. Does C have to be an integer? Why or why not? Surprisingly, only one correct solutions was submitted, although the problem yields quickly to the method of finite differences . In this note I will describe a solution using this method. The forward difference of a function f, denoted D f, is defined by ( D f)(n) = f(n+1) f(n). The forward difference operator can be iterated. D f)(n) D f)(n+1) D f)(n) f(n+2) 2 f(n+1) + f(n) D f)(n) D f)(n+1) D f)(n) f(n+3) 3 f(n+2) + 3 f(n+1) f(n) D f)(n) D f)(n+1) D f)(n) f(n+4) 4 f(n+3) + 6 f(n+2) 4 f(n+1) + f(n) D k f)(n) S i=0..k n i (k choose i) f(n+i) We record the following properties of the forward difference operator. If f is a polynomial of degree d, then

47. Pythagoras Of Samos - Activity 4 Return to MATHGYM Back Activity 4 finite differences and Inductive Proof. Findingthe formulas - finite differences. We start by placing the data in a table. http://www.mathgym.com.au/history/pythagoras/Activity4.htm

Return to MATHGYM Back Activity 4 - Finite Differences and Inductive Proof This Activity and the related Essays on the History of Mathematics are on-line at MATHGYM (http://www.mathgym.com.au) [-Figurate Numbers-] [-Finding the formula - Finite Differences-] [-Mathematical Induction-] In this activity you will learn both a method for finding the equation given a table of related numbers, and for proving that the equation is correct using the technique of Mathematical Induction. To fully appreciate this Activity you need some understanding of elementary algebra. Figurate Numbers We believe that the Pythagoreans arranged pebbles in the shape of a regular polygons (e.g. equilateral triangle, square, pentagonal etc ). We call these numbers figurate numbers . For example: Figurate Number Geometric shape triangular numbers square numbers pentagonal numbers It is trivial to find the n th term of the square numbers (the 10 th term is 10 = 100 ) but it is not so clear how to get the formula for the n th term of the triangular numbers. There is a useful method called "finite differences" which we can use and I will try to explain how it works below for you. It is somewhat complex so bear with me as it is also pretty powerful (and nifty). Finding the formulas - Finite Differences We start by placing the data in a table. Let's use the data for the triangular numbers. Notice that the two columns contain the term number (n) and the corresponding value for the n

48. Finite Differences And Relaxation. next up previous Next Geometric integration. Up Knowhow PreviousFronts. finite differences and relaxation. A variety of finite http://www.iac.rm.cnr.it/~natalini/prog03/node27.html

Next: Geometric integration. Up: Know-how Previous: Fronts. Finite differences and relaxation. A variety of finite difference schemes for the linear advection equation having two time levels and three space grid have been developed in [ ]. A critical analysis of them and a unifying theory is the main objective of such research. The proposed method is based on the idea to represent the solution in six-points stencil and to collocate the differential equation in a different suitable internal point. A stability and consistency analysis has been carried out. Numerical examples have shown the effectiveness and the performances of the different methods according to the choice of the parameters. Starting from the papers [ ], a lot of different approximations for nonlinear hyperbolic and parabolic conservation laws have been developed, by obtaining rigorous results of convergence and some new and efficient numerical schemes [ ]. Boundary conditions have been studied in [ ] and in the Ph.D thesis by V. Milisic, partially developed at IAC [ ]. A substantial progress in the field of well-balanced and asymptotic-preserving schemes has been given, in the context of discrete kinetic models in both rarefied and diffusive regimes, in the papers [

49. Finite Differences (Diele, Gosse, Natalini, Notarnicola, Pontrelli). Up Expected results in 2003 Previous Fronts (Bertsch, Natalini). Finitedifferences (Diele, Gosse, Natalini, Notarnicola, Pontrelli). http://www.iac.rm.cnr.it/~natalini/prog03/node41.html

Next: Geometric integration (Diele). Up: Expected results in 2003 Previous: Fronts (Bertsch, Natalini). Finite differences (Diele, Gosse, Natalini, Notarnicola, Pontrelli). Starting from the previous results it is possible to deal with many different types of problems. Let us sketch some more promising directions: 1. Extensions to nonlinear and multidimensional cases of the collocation methods proposed by Funaro and Pontrelli will be analyzed. 2. Application of the ideas of well-balanced and asymptotic preserving numerical schemes to more realistic models with a continuous distribution of the velocity variable, like the radiative transfer equation. 3. Development of a new numerical approach for the approximation of the extended thermodynamics models of Brenier and Corrias, in the framework of WKB approximation in quantum and semi-classical mechanics equations. 4. Application of the diffusive relaxation schemes to problems of filtration in porous media. A special consideration will be devoted to improving the order in time of our schemes by using suitable Runge-Kutta methods and a mixing of implicit and explicit schemes. Roberto Natalini 2002-07-14

50. Solutions To Elliptic Equations By Finite Differences Solutions to Elliptic Equations by finite differences. http://www.npac.syr.edu/pub/by_module/presentations/cps615/talk14/normal/008.htm

Solutions to Elliptic Equations by Finite Differences

51. Finite Differences The finite differences Method. The Physics! calculates electromagneticfields by solving a series of differential equations. Its Uses http://www.npac.syr.edu/REU/reu95/home/granti/fd.html

The Finite Differences Method The Physics! calculates electromagnetic fields by solving a series of differential equations Its Uses: buildings with doors open structures where radiation occurs in an enclosed area, possibly causing harm to human beings or animals More Methods Planet of the Apes

52. Keyword Type Of Model: Finite Differences Model Translate this page GSF-Forschungszentrum für Umwelt und Gesundheit. keyword type of modelfinite differences model. List of all models related to this keyword http://www.gsf.de/UFIS/ufis/schlag_modelltyp/schlagwort320.html

keyword type of model: finite differences model List of all models related to this keyword UFIS model home page [index app...] [index typ...] [index brief...] [index quant...] [index equat...] Alex Fieger , Jan 30, 1995 Ekkehard Ernst , 04 Dec 1996

53. Salvo - Practical Aspects Of Prestack Depth Migration With Finite Differences prestack depth migration with finite differences . Selected Slides Practicalaspects of prestack depth migration with finite differences. http://www.cs.sandia.gov/~ccober/seismic/seg97_st11.6.html

Selected slides presented at Society of Exploration Geophysicists November 2-7,1997 Seismic Theory 11: Migration Theory "Practical aspects of prestack depth migration with finite differences" Selected Slides Practical aspects of prestack depth migration with finite differences Curtis C. Ober Last modified: October 9, 1998 Back to top of page Back to Seismic Imaging Home Page Sandia Home Page Questions and Comments

54. Finite Differences next up previous Next Localised discrete Fourier transform Up TheoryPrevious Theory finite differences. The most straightforward http://www.tcm.phy.cam.ac.uk/~pdh1001/papers/paper9/node3.html

55. RR-2231 : Mellin Transforms And Asymptotics : Finite Differences And Rice's Inte Translate this page logo inria. RR-2231 - Mellin transforms and asymptotics finite differences andRice's integrals. Flajolet, P. - Sedgewick, R. Les rapports de cet auteur. http://www.inria.fr/rrrt/rr-2231.html

RR-2231 - Mellin transforms and asymptotics : finite differences and Rice's integrals Flajolet, P. Sedgewick, R. Les rapports de cet auteur Rapport de recherche de l'INRIA- Rocquencourt Page d'accueil de l'unité de recherche Pour obtenir la version papier (Donnez votre adresse postale et le no de rapport) Projet : ALGO - 21 pages - Mars 1994 - Document en anglais Page d'accueil du projet

56. RR-2979 : Mesh-Centered Finite Differences From Nodal Finite Elements Translate this page logo inria. RR-2979 - Mesh-Centered finite differences from Nodal Finite Elements.Hennart, Jean-Pierre - del Valle, Edmundo Les rapports de cet auteur. http://www.inria.fr/rrrt/rr-2979.html

RR-2979 - Mesh-Centered Finite Differences from Nodal Finite Elements Hennart, Jean-Pierre del Valle, Edmundo Les rapports de cet auteur Rapport de recherche de l'INRIA- Rocquencourt Page d'accueil de l'unité de recherche Fichier PostScript / PostScript file Fichier postscript du document : 184 Ko Fichier PDF / PDF file Fichier PDF du document : 362 Ko Projet : ESTIME Page d'accueil du projet KEY-WORDS : NODAL METHODS / MESH-CENTERED FINITE DIFFERENCE SCHEMES

57. Schaums Outline Of Calculus Of Finite Differences And Difference Equations Schaums Outline of Calculus of finite differences and Difference Equations, SchaumsOutline of Calculus of finite differences and Difference Equations by Authors http://www.wkonline.com/a/Schaums_Outline_of_Calculus_of_Finite_Differences_and_

Book > Schaums Outline of Calculus of Finite Differences and Difference Equations Schaums Outline of Calculus of Finite Differences and Difference Equations by Authors: Murray R. Spiegel Released: 01 December, 1971 ISBN: 0070602182 Paperback Sales Rank: List price: Our price: Schaums Outline of Calculus of Finite Differences and Difference Equations > Customer Reviews: Average Customer Rating: Schaums Outline of Calculus of Finite Differences and Difference Equations > Customer Review #1: Handy, but not perfect This book discusses difference calculus, sum calculus, and difference equations as well as discusses applications. With each chapter, there are plenty of explanations and examples. The book also has problems you can try to test your knowledge of the chapter. The problem I have with it is that not all the problems have answers to them. The explanations are great, but I want to make sure I understand it. There doesnt seem to be a pattern regarding which ones have answers in the back. You just have to try the problem, then look in the back to see if that one was answered.

58. Finite Differences finite differences. Please enter the number of data points youwish to use Back to Example Outputs of Numerical Methods http://www.cis.usouthal.edu/faculty/lynn/summer99/examples/fd.html

Finite Differences Please enter the number of data points you wish to use: Back to "Example Outputs of Numerical Methods"

59. Numerical Differentiation Using Finite Differences Numerical Differentiation using finite differences. Please enter the numberof data pairs you have Back to Example Outputs of Numerical Methods http://www.cis.usouthal.edu/faculty/lynn/summer99/examples/finite.html

Numerical Differentiation using Finite Differences Please enter the number of data pairs you have: Back to "Example Outputs of Numerical Methods"

60. PBS Teachersource - Mathline - Women In Math An Introduction to finite differences (Grade Levels 610) Hypatia Ada Lovelace Diophantine Equations I Diophantine Equations II High Fives! http://www.pbs.org/teachersource/mathline/concepts/womeninmath/activity5.shtm

March 18, 2003 Math Challenges Math Concepts Career Connections Activity 3: High-Fives! An Introduction to Finite Differences (Grade Levels: 6-10) Hypatia Ada Lovelace Diophantine Equations I Diophantine Equations II ... More Math Concepts The technique of finite differences was one of the keys to how the difference engine worked. Ada Lovelace studied this method. The technique of finite differences can be used in problem solving to find and extend terms in a pattern, as well as to develop a general algebraic statement. The following problems are ones that lend themselves to finite differences. At a women's basketball All Star Game, all the members of each team are introduced. There are 12 players on the East team. When a player is introduced, she runs out on the floor and gives a high-five to all her other teammates already on the floor. How many high-fives will take place during the East team introduction? Some questions to ask students to get them thinking about the problem: 1. How many players are on the East team?

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