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Kutta Martin:     more detail

21. Physics - Kinematics - Martin Baker
These approximations can be made more accurate by using Eulers Method orRungekutta Method. Position vectors. Copyright (C) martin Baker 2003.
http://www.martinb.com/physics/kinematics/
##### Physics - Kinematics
Kinematics: The study and description of motion, without regard to its causes, for example, we can calculate the end point of a robot arm from the angles of all its joints. Alternatively, given the end point of the robot arm, we could calculate the angles and settings of all its joints required to put it there (inverse kinematics - IK). Kinematics can be studied without regard to mass or physical quantities that depend on mass. We will talk about dynamics later. One way to think about the difference between kinematics and dynamics is that dynamics is the cause of motion and kinematics is the effect. Kinematics involves position, velocity and acceleration (and their rotational equivalents).
• Position is the point in space that an object occupies, this needs to be defined in some coordinate system Velocity is the rate of change of position with respect to time. Acceleration is the rate of change of velocity with respect to time.
Although I am leaving the dynamics to later it is worth mentioning here that, if there are no net forces acting on an object, then it will have a constant velocity. Also if there is a constant net force acting on an object, like gravity for instance, then it will have constant acceleration. So these special cases of constant velocity and of constant acceleration are worth considering in more detail.
##### Movement in one dimension
If an object is moving in a straight line, and if we measure its position along that line, then its position, velocity and acceleration can all be represented by scalar quantities. This makes the analysis much easier, so lets start there.

22. Hollis: Differential Equations
Google search) Hodgkin, Alan (Nature) Hooke, Robert Huxley, Andrew (sfn.org) Jacobi,Carl Jordan, Camille Kirchhoff, Gustav kutta, martin Wilhelm L'Hôpital
http://www.math.armstrong.edu/faculty/hollis/dewbvp/
 Differential Equations with Boundary Value Problems by Selwyn Hollis Contents and Preface Marketing Blurb Book Site @ Prentice Hall ... Solutions Manual Technology Mathematica Maple Java M ... ATLAB Sundry Items Problem graphics and extra graphical problems for Section 3.1. Please send bug reports here Professors: Please send me an email Some Biographical References The following are links to information on most of the mathematicians/scientists whose names appear in the book. Unless otherwise noted, each of these is a link to the MacTutor History of Mathematics Archive at the University of St Andrews, Scotland. Abel, Niels Henrik Airy, George Banach, Stefan Bendixson, Ivar ... Edelstein-Keshet, Leah (U. BC) Euler, Leonhard Fourier, Joseph Frobenius, Georg Gauss, Carl Friedrich ... Hertz, Heinrich Rudolf (Google search) Hodgkin, Alan Nature Hooke, Robert Huxley, Andrew (sfn.org) Jacobi, Carl Jordan, Camille Kirchhoff, Gustav Kutta, Martin Wilhelm ... Lorenz, Edward N. (xrefer.com) Lotka, Alfred (Google search) Lyapunov, Aleksandr Maclaurin, Colin Malthus, Thomas (Google search) Menten, Maud

23. 37-021|NSR|NSR Allgemein|Wer Is Wer?
kutta, martin, deutscher Mathematiker (1867-1944),Runge-kutta-Verfahren (WR). L Lagrange, Joseph-Louis,
http://www.wr.inf.ethz.ch/education/nsr/general/who.html
Willkommen Kontakt Vorlesung Inhalt ... NSR im WWW Wer is wer?
##### NSR Celebreties
schreibt eine Mail! (Brent und Neville habe ich leider vergeblich gesucht.) A Aitken, Alexander Craig Aitken-Neville-Schema B Banach, Stefan Banach-Raum C Cauchy, Augustin Louis Cotes, Roger englischer Mathematiker (1682-1716). Newton-Cotes-Quadratur-Formeln (WR) Cramer, Gabriel Cramersche Regel D Dirichlet, Gustav Peter Lejeune E Euler, Leonhard Euler-Maclaurin'schen Summenformel (WR) wichtig. F Frobenius, Ferdinand Georg deutscher Mathematiker (1849-1917). Frobenius-Norm G Galerkin, Boris Grigorievich russischer Mathematiker (1871-1945), Galerkin-Bedingung (WR) Galois, Evariste Gander, Walter Adaptive Quadratur (WR) Gauss, Carl Friedrich Gauss-Algorithmus , die Normalengleichungen , die Gauss-Quadratur (WR) und die Gauss-Seidel-Iteration Givens, Wallace amerikanischer Mathematiker (1911-1993), QR-Zerlegung ( Givens-Rotationen Golub, Gene amerikanischer Mathematiker und Informatiker, hat u.a. den Algorithmus zur Berechnung der gefunden. Gonnet, Gaston H. Gram, Jorgen Pedersen QR-Zerlegung nach Gram-Schmidt (Orthogonalisierungsverfahren) H Hermite, Charles

24. Www.wr.inf.ethz.ch/education/nsr00/general/who.prehtml
listentry name= kutta, martin desc= deutscher
http://www.wr.inf.ethz.ch/education/nsr00/general/who.prehtml
 NSR allgemein Wer is wer? Wer is wer? NSR Celebreties schreibt eine Mail! (Brent und Neville habe ich leider vergeblich gesucht.) A Aitken-Neville-Schema ." url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aitken.html"> B Banach-Raum ." url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Banach.html"> C ." url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html"> <#list::entry name="Cotes, Roger" desc="englischer Mathematiker (1682-1716). Newton-Cotes-Quadratur-Formeln (WR) ." url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cotes.html"> Cramersche Regel D " url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html"> E Euler-Maclaurin'schen Summenformel (WR) wichtig." url="http://physics.hallym.ac.kr/reference/physicist/Euler.html"> F <#list::entry name="Frobenius, Ferdinand Georg" desc="deutscher Mathematiker (1849-1917). Frobenius-Norm ." url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html"> G <#list::entry name="Galerkin, Boris Grigorievich" desc="russischer Mathematiker (1871-1945), Galerkin-Bedingung (WR) ." url="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Galerkin.html"> Adaptive Quadratur (WR) ." url="http://www.inf.ethz.ch/personal/gander/"> Gauss-Algorithmus , die Normalengleichungen , die Gauss-Quadratur (WR) und die Gauss-Seidel-Iteration ." url="http://www.geocities.com/RainForest/Vines/2977/gauss/gauss.html"> <#list::entry name="Givens, Wallace" desc="amerikanischer Mathematiker (1911-1993), QR-Zerlegung (

 25. Www.iper1.com - Martin Wilhelm Kutta Translate this page Cerca la rima. martin Wilhelm kutta. Premio Bagutta Borutta Calcuttagommagutta martin Wilhelm kutta Resiutta Teresa di Calcutta ahttp://www.iper1.com/rime/index.asp?cerca=Martin Wilhelm Kutta

26. Hostnames
Knoll, Max, 1935, theory of the scanning electron microscope. kutta, martin,1901, Rungekutta method (differentail equations) and Zhukovsky-kuttay aerofoil.
http://www-mae.engr.ucf.edu/names.html
##### Potential hostnames
Welcome to the hostname contest page! The following is the list of potential hostnames that might be used for any new UNIX machines that the department gets. alembert ardenne babcock baekeland barlow benz bessemer biot borries boyle braun burke carothers carpenter chilton clariaut clausius cochran colburn coriolis crosthwait daimler darby darcy diesel draper dunlop eiffel euler faber gaetano gelb goodyear gustave hancock hillier hooke howe huygens kaplan kelvin knoll lagrange lamb lanza lerond lighthill mach maudslay moody nusselt oatley ohain otis otto pelton plunkett poisson prebus rankine reynolds rolla ruska savart schumann sikorsky stanton venturi wankel weisbach whittle wilcox zeppelin fourier If you'd like to add to this list, send me a note, or use the handy form at the bootom of this page. I'm kinda picky about the names, though... The kinds of names I'm looking for
• must not already be used by a computer in the UCF COE.
• must be releated to Materials, Mechanical, or Aerospace engineering somehow.
• preferably, should not be a name already in use by any computer in the official UCF computer name lists. (but this rule has been broken before)
• 27. Literature Martin Moessner Martin Mössner
Translate this page 1082, Uri M. Ascher and Linda R. Petzold , Projected implicit runge-kutta methodsfor differential-algebraic equations . 1272, martin Aupperle , Die Kunst
http://sport1.uibk.ac.at/mm/Bib/bib.html
 xx xx . xx , xx Peter Kaps and Werner Nachbauer and Snow Friction and Drag in Alpine Skiing . ? Proceedings of the 5th Annual Congress of the European Colledge of Sport Science (ECSS) , ? Janne Avela and Paavo V. Komi and Jyrki Komulainen and Werner Nachbauer Reaktionskraft und Abscherfestigkeit von Pistenschnee II . Department of Sport Science , University of Innsbruck, Austria , Report 2001 and Werner Nachbauer and Kurt Schindelwig and Fritz Brunner and Gerhard Innerhofer and Franz Bruck Versuche mit dem Skischlitten in Stuben . Department of Sport Science , University of Innsbruck, Austria , Report 2001 Peter Kaps and and Werner Nachbauer and Rolf Stenberg Pressure distribution under a ski during carved turns . Sience and Skiing , and Hermann Schwameder and Christian Raschner and Stefan Lindiger and Elmar Kornexl , Verlag Dr. Kovac , Hamburg , pp. 180-202 , 2001 and Werner Nachbauer Reaktionskraft und Abscherfestigkeit von Pistenschnee . Oral Presentation at the University of Technology, Department of Mechanics, Vienna , 23 Mai 2001

28. 8 FSM Intrm. 218-224.
JIM kutta, HALVERSON NIMEISA, RESAUO martin, ERADIO WILLIAM, FRANCIS RUBEN,JOHNSON SILANDER and the STATE OF CHUUK, Defendants. CIVIL ACTION NO.
http://www.fsmlaw.org/fsm/decisions/vol8/8fsm218_224.htm
 THE SUPREME COURT OF THE FEDERATED STATES OF MICRONESIA TRIAL DIVISION Cite as Davis v. Kutta 8 FSM Intrm. 218 (Chk. 1997) [8 FSM Intrm. 218] MENRY DAVIS, Plaintiff, vs. JIM KUTTA, HALVERSON NIMEISA, RESAUO MARTIN, ERADIO WILLIAM, FRANCIS RUBEN, JOHNSON SILANDER and the STATE OF CHUUK, Defendants. CIVIL ACTION NO. 1992-1039 ORDER AND MEMORANDUM OF DECISION Martin Yinug Associate Justice Decided: December 5, 1997 APPEARANCES: For the Plaintiff: R. Barrie Michelsen, Esq. Law Offices of R. Barrie Michelsen P.O. Box 1450 Kolonia, Pohnpei FM 96941 For the Defendants: Wesley Simina, Esq. Attorney General Office of the Chuuk Attorney General P.O. Box 189 Weno, Chuuk FM 96942 [8 FSM Intrm. 219] HEADNOTES Attorney, Trial Counselor and Client Fees; Civil Rights A successful plaintiff under the civil rights statute, 11 F.S.M.C. 701(3), is entitled to an award for costs and reasonable attorney's fees. Davis v. Kutta, 8 FSM Intrm. 218, 220 (Chk. 1997). Attorney, Trial Counselor and Client

29. 8 FSM Intrm. 228-230
JIM kutta, HALVERSON NIMEISA, RESAUO martin, JOHNSON SILANDER, the STATE OF CHUUKand the FEDERATED STATES OF MICRONESIA, Defendants. CIVIL ACTION NO.
http://www.fsmlaw.org/fsm/decisions/vol8/8fsm228_230.htm
 THE SUPREME COURT OF THE FEDERATED STATES OF MICRONESIA TRIAL DIVISION Cite as Louis v. Kutta 8 FSM Intrm. 228 (Chk. 1998) [8 FSM Intrm. 228] SIKBERT LOUIS, as personal representative of the estates of his sons Jeffrey and Jimmy Louis, deceased, Plaintiff, vs. JIM KUTTA, HALVERSON NIMEISA, RESAUO MARTIN, JOHNSON SILANDER, the STATE OF CHUUK and the FEDERATED STATES OF MICRONESIA, Defendants. CIVIL ACTION NO. 1994-1023 ORDER AND MEMORANDUM Martin Yinug Associate Justice Decided: January 15, 1998 APPEARANCES: For the Plaintiff: Frank Casiano, trial counselor Charles Greenfield, Esq. (supervising attorney) Micronesian Legal Services Corporation P.O. Box D Weno, Chuuk FM 96942 For the Movant: Terence M. Brown, Esq. (FSM) Assistant Attorney General Office of the FSM Attorney General P.O. Box PS-105 Palikir, Pohnpei FM 96941 HEADNOTES Constitutional Law A court should avoid unnecessary constitutional adjudication. Louis v. Kutta, 8 FSM Intrm. 228, 229 (Chk. 1998). Treaties Although the FSM Supreme Court has the power to interpret treaties, it should not do so if the issue may be decided on other grounds. Louis v. Kutta, 8 FSM Intrm. 228, 229-30 (Chk. 1998).

following topics Introduction to MAPLE. Eigenvalues. Fourier Series.Laplace Transforms. Rungekutta Methods. Supplementary Material.
http://www.soc.staffs.ac.uk/~cmtmfp/engmaths2a/emaths2a.html
##### Engineering Mathematics 2A
Welcome to the Web Page for the module Engineering Mathematics 2A MAPLE worksheets are available on the following topics: Introduction to MAPLE Eigenvalues Fourier Series Laplace Transforms ... Supplementary Material

31. Fachhochschule Wedel - Martin "Herbert" Dietze
h Opts Options are -r Use Runge-kutta integration -p Use
http://www.fh-wedel.de/~di/prakt-alt/cueb/c-ss98/ueb9/
Thema der Aufgabe [Seitenanfang] [weiter]
##### Aufgabenstellung
[Seitenanfang] [weiter]
##### Eine Bibliothek zur numerischen Integration
Es soll eine Bibliothek erstellt werden, die Funktionen der Form double func (double) (der Name func exp oder cos ) in den Grenzen bis mit der Tafelschrittweite h numerisch integriert. typedef double (*t_func) (double); typedef double (*t_method) (t_func, double, double, double); Es sollen folgende Funktionen exportiert werden:
• double integrate_rk (t_func func, double x1, double x2, double h) integriert die Funktion in den Grenzen von nach mit der Tafelschrittweite h stdout oder stderr
• double integrate_pg (t_func func, double x1, double x2, double h) macht dasselbe wie
• double integrate (t_method method, t_func func, double x1, double x2, double h) ruft die Integrationsfunktion
Das Testprogramm soll unter Benutzung der Funktion aus der Bibliothek die Funktionen und und h
• double und h angegeben.
• Es wurde ein nicht definierter Schalter gesetzt.
LCLint
##### Hinweise
[Seitenanfang] [weiter]
##### Hantieren mit Funktionszeigern
/* definiert einen Funktionszeigertypen * mit Rueckgabewert double und einem * double-Argument */ typedef double (*t_foo) (double);

32. Numerical Analysis Groups, Recent Articles
22 (1996), 279292, special issue Runge-kutta Centennial martin H. Gutknecht, MarlisHochbruck, Optimized look-ahead recurrences for adjacent rows in the Padé
http://na.uni-tuebingen.de/na/arbeiten.shtml
##### Recent articles
• Harry Yserentant, The convergence of the finite mass method for flows in given force and velocity fields , In: Meshfree Methods for Partial Differential Equations (M. Griebel and M.A. Schweitzer, eds.), Lecture Notes in Computational Science and Engineering 26 (2002).
(238 kB gzipped Postscript file)
• Klaus Neymeyr, A geometric theory for preconditioned inverse iteration applied to a subspace , Math. Comp. 71 (2002), 197-216.
(92 kB gzipped Postscript file)
• Christian Lubich, Integrators for quantum dynamics: a numerical analyst's brief review
(39 kB gzipped Postscript file)
• Fast convolution for non-reflecting boundary conditions , SIAM J. Sci. Comput. 24 (2002), 161-182.
(558 kB gzipped Postscript file)
• , Wave Motion 35 (2002), 181-188.
(79 kB gzipped Postscript file)
• Christian Lubich, On dynamics and bifurcations of nonlinear evolution equations under numerical discretization , in B. Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001, 469-500.
(84 kB gzipped Postscript file)
• Harry Yserentant
• 33. Martin J. Gander
martin J. Gander Dept. 2002) A graduate course in numerical methods for ordinaryand partial differential equations, including Rungekutta, Linear Multistep
http://www.math.mcgill.ca/mgander/teaching.php
 Martin J. Gander Dept. of Mathematics and Statistics McGill University Montreal, QC, H3A 2K6 mgander@math.mcgill.ca Home Research Teaching ... Personal Current Address On leave at: Université de Genève Section de mathématiques 2-4 rue du Lièvre CH-1211 Geneve 24 Switzerland Seminars Genève: Analyse numérique McGill: CSE Applied Mathematics CSE at McGill The departments of AOS, CS, ECE, MATH and ME offer a new joint masters option in Computational Science and Engineering (CSE). A successful CSE seminar is currently running. I ran the seminar from :: Teaching Interests My teaching interests are both in Mathematics and Computer Science: in addition to undergraduate courses in both areas, I am interested and qualified to teach at the graduate level Scientific Computing, Numerical Differential Equations, Matrix Computations, Differential Equations, Parallel Computing, Numerical Dynamical Systems, Algorithms and Data Structures and Object Oriented Programing. :: Courses I teach this year in Geneva Introduction à l'analyse numérique I (Genève): A first undergraduate course introducing students from mathematics and computer science to numerical integration, interpolation and approximations, numerical ordinary differential equations and linear systems. (40 students, 2 hours lecture, 1 hour exercises and 2 hours Fortran exercises)

34. On The Positivity Of Low Order Explicit Runge-Kutta Schemes Applied In Splitting
order explicit Rungekutta schemes applied in splitting methods by A. Gerisch, R.Weiner Preprint series 99-35, Reports on Numerical Mathematics, martin-Luther
http://nummulit.mathematik.uni-halle.de/~gerisch/publications/99-35report.html
 On the positivity of low order explicit Runge-Kutta schemes applied in splitting methods by A. Gerisch, R. Weiner Preprint series: 99-35, Reports on Numerical Mathematics, Martin-Luther-University Halle-Wittenberg, December 1999. The paper is published: accepted for publication ( Computers and Mathematics with Applications, 2001 MSC 65L05 Initial value problems 65L06 Multistep, Runge-Kutta and extrapolation methods 65M20 Method of lines Abstract Splitting methods are a frequently used approach for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A -stage second order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard

35. Infoseiten Des MC Breitenbrunn (Aerodynamik)
Translate this page so groß, daß die Flügelhinterkante nicht umströmt wird und kann für reibungsfreieStrömungen zB aus der Abflußbedingung von kutta (martin Wilhelm kutta
http://www.toeging.lednet.de/flieger/profi/aerodyn.htm
##### Anschauliche Aerodynamik
Bernoulli - Gleichung Magnus - Effekt Wirbelsystem am Flugzeug Anfahrwirbel ... Randwirbel
##### 1) Bernoulli - Gleichung
Zum Seitenanfang
##### 2) Magnus - Effekt
(Heinrich Gustav Magnus 1802 - 1870) Zum Seitenanfang Zum Seitenanfang Zum Seitenanfang
##### 5) Wirbelsystem am Flugzeug
Wegen der endlichen Spannweite eines Tragflügels wird bei seiner Umströmung ein Wirbelsystem erzeugt, bestehend aus einem Anfahrwirbel, der stromab zurückbleibt und vergeht, aus einem „tragenden" Wirbel der fest mit dem Tragflügel verbunden bleibt und einem System freier Wirbel (Randwirbel), das ständig verlängert wird. Zum Seitenanfang
##### 6) Anfahrwirbel
Zum Seitenanfang
##### 7) Gebundener „tragender" Wirbel
Der Anfahrwirbel löst nach dem „Thomson Wirbelsatz" (William Thomson, 1824 - 1907) gleichzeitig eine Wirbelströmung um den Tragflügel mit entgegengesetzt gleicher Zirkulation aus (Bild 3b und 5). Sie ist gerade so groß, daß die Flügelhinterkante nicht umströmt wird und kann für reibungsfreie Strömungen z.B. aus der Abflußbedingung von Kutta (Martin Wilhelm Kutta, 1867 - 1944) und Jukowski (Nikolai Jegorowitsch Jukowski, 1847 - 1921) wie folgt bestimmt werden: Als Ergebnis des Zusammenwirkens von Anfahrwirbel, tragflügelfestem „tragendem" Wirbel und Anströmgeschwindigkeit stellt sich folgende gesunde Tragflügelumströmung ein und damit beginnt der dynamische Auftrieb (Bild 6).

36. Kepler3
they are closely related. It was published by Carle Runge (18561927)and martin kutta (1867-1944) in 1901. Euler's method and 4th
http://www.ams.org/new-in-math/cover/kepler3.html
Celestial Mechanics on a Graphing Calculator
##### 3. The Runge-Kutta algorithm
The Runge-Kutta algorithm (strictly speaking the fourth-order R-K algorithm; see example ) allows much better accuracy than Euler's method. Their relative efficiency is like that of Simpson's method and left-hand sums for approximating integrals, algorithms to which they are closely related. It was published by Carle Runge (1856-1927) and Martin Kutta (1867-1944) in 1901.
Euler's method and 4th order Runge-Kutta, applied to the restricted 2-body problem with the same initial conditions. The Runge-Kutta method easily accomplishes in 30 steps what Euler's method could not do in 1000. Even though every Runge-Kutta step is computationally the equivalent of 4 Euler steps, the savings are enormous. But when we decrease w to produce more eccentric elliptical orbits, even this powerful method starts to strain.
For w , step sizes of .1 and .05 lead to non-physical solutions. Comments: webmaster@ams.org

37. Papers By Martin Hairer
Papers by martin Hairer. numerically. The codes implement symmetricand symplectic Rungekutta, multistep, and composition methods.
http://mpej.unige.ch/~hairer/research.html
 Papers by Martin Hairer Home Papers Software Useful Links Non-Equilibrium Statistical Mechanics Diploma Thesis, Univ. of Geneva (1998) Under the direction of Prof. J.-P. Eckmann We investigate the existence of stationary states in mechanical systems with compact phase space coupled to linear heat baths at different temperatures. PostScript pdf gzipped ps gzipped pdf Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators Commun. Math. Phys. (2000), no 1, pp. 105-164 Published version on LINK Written in collaboration with J.-P. Eckmann We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hoermander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains. PostScript pdf gzipped ps gzipped pdf Spectral Properties of Hypoelliptic Operators Preprint , Geneva (2002) Written in collaboration with J.-P. Eckmann

38. Euler Og Runge-Kutta Metoder
Biografi af Carle Runge (18561927) Biografi af martin kutta (1867-1944) Metodener også kendt som Heun's metode og er simpelthen en forbedring af Eulers
http://www.frhavn-gym.dk/matematik/mrunge.html
 Dette er et interaktivt kursus i numerisk løsning af differentialligninger ud fra Euler's metode og Runge-Kutta metoder. Klassetrin:Højt niveau, Gymnasiet. Tag allerførst en kopi af denne side Introduktion Ved praktisk løsning af differentialligninger stilles man ofte over for et Begyndelsesværdiproblem Vi har differentialligningen med et startpunkt og skal undersøge, hvordan udviklingen er som tiden går. Det viser sig nu, at der under generelle betingelser er en entydig løsning. Augustin Cauchy (1789-1857) har som den første studeret på dette eksistens- og entydighedsproblem for differentialligninger. Linieelementer Den første angrebsvinkel til vort arbejde med problemet skal være Linieelementer Betragt f.eks. differentialligningen (¤) dy/dt = -2*t*y Vælges (t,y)=(2,½) bliver ifølge (¤) dy/dt= -2. Vi tegner nu igennem (2,½) en lille liniestykke med hældning -2 og snakker om Linieelementet (2,½,-2) Hvis vi tegner linieelementer for et utal af punkter i planen, antydes der en serie af løsningskurver. På SOS-math finder du linieelementer (slope field) og gode eksempler.

39. Euler's Metode Og Runge-Kutta Metoder (interaktivt)
gennemgår 2'ordens Rungekutta s. 128-129 Du kan starte med at læselidt om Carle Runge (1856-1927) og martin kutta (1867-1944);
http://www.frhavn-gym.dk/~st/runge.html
##### Differentialligninger
Dette er et interaktivt kursus i differentialligninger ud fra Euler's metode og Runge-Kutta metoder.
(M) betyder Blomhøj/Frisdahl: Modelsnak Fag 1985.
Tag allerførst en kopi af denne side.
a. Læs (M) side 118-120 (til "Cauchy").
b. Læreren snakker om side 120-121
Som nævnt har A. Cauchy (1789-1857) som den første studeret på eksistens- og entydighedsproblemet inden for differentialligninger.
c. Læreren gennemgår nu metoden opfundet af L. Euler (1707-1783) (s. 121-123)
Figuren nederst side 123 omhandler GIER. Artikel om GIER's forgænger, DASK (brugt til folketingsvalget 1960) Billede af DASK og GIER
d. Læreren gennemgår princippet i eks. 4.1. s. 124
e. Nu til differentialligningsprogrammet
• Vælg Euler (tangent line)
• Indtast y i rubrikken for dy/dt
• Indtast t
• Indtast y
• Indtast t
• Indtast step size h =0.5
• Vælg Graph and Data points
• Klik på "Submit"
• Udskriv den fremkomne løsning. Sammenlign med tabel 4b og fig. 4c s.125.
• Tilføj en søjle med e t på udskriften
• Beregn med lommeregner den relative fejl og angiv den i endnu en søjle.
• Tabellen afleveres
• Eksperimenter med mindre værdier for step size
• Gentag 6-11 med en fornuftig h-værdi og beregn ca. 10 af de relative fejl
• 40. Prime Numbers
Rungekutta method, based on the work of martin kutta(1867-1944), and the methodof successive approximations, based on the work of Emile Picard (1856-1941).
http://hypatia.math.uri.edu/~kulenm/diffeqaturi/m381f00fp/theron/theronmp.html
##### Recursive Relations
Number theory index History Topics Index It is from these recursive equations that some mathematical wonders are created. We begin with plane filling curves or fractals, which are curves that fill planes without any holes. The first such curve was discovered by Guiseppe Peano in 1890. Other mathematicians who used difference equations in their work with plane filling curves include David Hilbert (1862-1943), and Niels Fabian Von Koch (1870-1924). The relevant work all three will be discussed in the following. As will the work of Emile Picard (1856-1941) and Martin Kutta (1867-1944), both of whom used recursive equations in solutions to differential equations. There are curves that fill a plane without holes. The first such curve was discovered by Guiseppe Peano in 1890 and the second by D. Hilbert (1862-1943). Calling them Peano Monster Curves, B. Mandelbrot collected a series of quotations in support of this terminology.
Fractal Curves and Dimension
Fractals burst into the open in early 1970s. Their breathtaking beauty captivated many a layman and a professional alike. Striking fractal images can often be

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