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Topic: Numerical ordinary differential equations

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  Learn more about Differential equation in the online encyclopedia.   (Site not responding. Last check: 2007-10-11)
Ordinary differential equations are to be distinguished from partial differential equations where is a function of several variables, and the differential equation involves partial derivatives.
Differential equations are used to construct mathematical models of physical phenomena such as fluid dynamics or celestial mechanics.
Therefore, the study of differential equations is a wide field in both pure and applied mathematics.
www.onlineencyclopedia.org /d/di/differential_equation.html   (443 words)

 Differential Equations
On the other hand, linear differential equations are of such importance in terms of applications, theory, and solution techniques that they warrant a strong and separate emphasis.
Finally, the dif­ferential equations course is one of the few undergraduate courses where it is possible to give students a glimpse of the nature of contemporary mathematical research.
We expect students to understand the meaning of the variables and parameters in a differential equation and to be able to interpret this meaning in terms of a particular model.
www.wordtrade.com /science/mathematics/differentialequations.htm   (4849 words)

 PlanetMath: differential equation
A differential equation is an equation involving an unknown function of one or more variables, its derivatives and the independent variables.
An ordinary differential equation (ODE) is a differential equation where the unknown function depends on a single variable.
This is version 8 of differential equation, born on 2002-05-30, modified 2005-05-09.
planetmath.org /encyclopedia/PDE.html   (432 words)

 Ordinary differential equation   (Site not responding. Last check: 2007-10-11)
Ordinary differential equations are to be distinguished from partial differential equations where there are several independent variables involving partial derivatives.
The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention.
Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.
www.tocatch.info /en/ODE.htm   (1618 words)

 Numerical ordinary differential equations - Wikipedia, the free encyclopedia
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs).
Ordinary differential equations occur in many scientific disciplines, for instance in mechanics, chemistry, biology, and economics.
Numerical analysis is not only the design of numerical methods, but also their analysis.
en.wikipedia.org /wiki/Numerical_ordinary_differential_equations   (1515 words)

 34: Ordinary differential equations
Ordinary differential equations are equations to be solved in which the unknown element is a function, rather than a number, and in which the known information relates that function to its derivatives.
The solutions to many classic differential equations, particularly linear second-order differential equations, cannot be expressed in terms of the elementary functions but are themselves studied in 33: Special Functions.
Numerical solutions of differential equations is a branch of Numerical Analysis.
www.math.niu.edu /~rusin/known-math/index/34-XX.html   (771 words)

 An Essay on the Numerical Solution of Differential Equations - for students
In more realistic models, solutions of differential equations cannot be found explicitly in terms of known functions, and the alternative is to determine an approximate solution for given data through numerical computations on a computer.
There are three sources of errors affecting the reliability of a numerical weather forecast: (i) measurement errors in data (or lack of data) (ii) approximation errors in modeling and (iii) approximation errors in computation.
The initial data at the start of the computer simulation are always measured with some error; the set of differential equations in the computer model only approximately describes the evolution of the atmosphere; and finally the numerical solution of the differential equations is only an approximation of the true solution.
www.math.colostate.edu /~estep/education/essays/intro.html   (1744 words)

 Web Directory » Web Directory » Science » Math » Numerical Analysis » People
Numerical analysis, partial differential equations, mechanics; the numerical solution of the equations of general relativity.
Numerical analysis of methods for partial differential equations; numerical linear algebra.
Numerical methods for PDEs and in particular finite element methods; multigrid methods for theoretical analysis, algorithmic developments and practical applications.
www.dcpages.com /DC_ODP/?c=Science/Math/Numerical_Analysis/People   (625 words)

 Math 211: Ordinary Differential Equations and Linear Algebra   (Site not responding. Last check: 2007-10-11)
This course focuses on ordinary differential equations and some of their many applications.
Differential equations are widely used to model phenomena that arise in the sciences and engineering.
The textbook is Ordinary Differential Equations by John C. Polking, Albert Boggess, and David Arnold.
www.owlnet.rice.edu /~math211   (405 words)

 35: Partial differential equations
Like ordinary differential equations, partial differential equations are equations to be solved in which the unknown element is a function, but in PDEs the function is one of several variables, and so of course the known information relates the function and its partial derivatives with respect to the several variables.
Linear differential equations occur perhaps most frequently in applications (in settings in which a superposition principle is appropriate.) When these differential equations are first-order, they share many features with ordinary differential equations.
For example, integral techniques (solving a differential equation by computing a convolution, say) lead to integral operators (transforms on functions spaces); these and differential operators lead in turn to general pseudodifferential operators on function spaces.
www.math.niu.edu /~rusin/known-math/index/35-XX.html   (1097 words)

 Numerical Solution of Differential Equations
Partial differential equations involve two or more independent variables.
Unknown functions in differential equations do not necessarily have to be represented by single symbols.
This solves a differential equation in which the derivative has a discontinuity.
documents.wolfram.com /v4/MainBook/3.9.7.html   (1032 words)

 Open Directory - Science: Math: Differential Equations
Differential Equations in Banach Algebras - Fuchsian Singularities of Linear Ordinary Differential Equations in Banach Algebras.
Differential Equations in Industry and Commerce - European TMR network coordinated at the Oxford Centre for Industrial and Applied Mathematics.
Nonlinear Differential Equations at Glasgow - The site describes research activities of the differential equations group in the mathematics department at the university of Glasgow, UK, and provides some resources of a general nature.
dmoz.org /Science/Math/Differential_Equations   (672 words)

 The Math Forum - Math Library - ODE   (Site not responding. Last check: 2007-10-11)
A short article designed to provide an introduction to ordinary differential equations, equations to be solved in which the unknown element is a function rather than a number, and in which the known information relates that function to its derivatives.
A set of interactive texts for studying single-variable and multi-variable calculus and ordinary differential equations using Maple, by two members of the Dept. of Mathematics, North Carolina State University.
Fuchsian Singularities of Linear Ordinary Differential Equations in Banach Algebras - Gerald Albrecht
mathforum.org /library/topics/ordinary_diffeq   (2278 words)

 Computational and Applied Mathematics Group (CAM)
Error analysis of the numerical solution of linear equations and least squares problems for the full rank and rank deficient cases.
Error analysis of numerical quadrature and of the numerical solution of ordinary differential equations.
Numerical quadrature: interpolature quadrature, Richardson extrapolation, Romberg Integration, Gaussian quadrature, singular integrals, adaptive quadrature.
cam.ucsd.edu /syllabi/math270.html   (204 words)

 BU Differential Equations Project
Qualitative, analytic, and numerical approaches are stressed throughout the course.
Selected animations used in the BU Differential Equations course.
Rick Wicklin at the University of Minnesota has developed some differential equations labs that are particularly well suited for use in a course that takes a dynamical systems approach.
math.bu.edu /odes   (345 words)

 Recurrences for Ordinary Differential Equations
We chose a homogeneous equation with simple characteristic roots for the preceding example so that the recurrence values could be easily compared to the analytical solution.
The main benefit of the method is that it eliminates the need to determine the characteristic roots of the differential equation, which can be difficult for high order equations.
Also, the algorithm requires no special handling of characteristic equations with repeated roots, so it is applicable to any equation of the form (1).
www.mathpages.com /home/kmath361/kmath361.htm   (1704 words)

 Numerical Solutions of Ordinary Differential Equations
The numerical methods for solving ordinary differential equations are methods of integrating a system of first order differential equations, since higher order ordinary differential equations can be reduced to a set of first order ODE's.
Common numerical methods for solving initial value problems of ordinary differential equations are summarized:
The Euler method is important in concept for it points the way of solving ODE by marching a small step at a time on the right-hand-side to approximate the "derivative" on the left-hand-side.
www.efunda.com /math/num_ode/num_ode.cfm   (142 words)

 Ordinary Differential Equations - Resources
Brook/Cole Online Source for Differential Equations Our aim is to develop the DiffEQ Resource Center to become the place where teachers and students of differential equations look first for the latest information.
The Boston University Ordinary Differential Equations Project Project by Paul Blanchard, Robert L. Devaney, and Glen R. Hall, e-mail: odes@math.bu.edu This National Science Foundation Project is designed to produce a text and related materials for the first college course in Ordinary Differential Equations.
Qualitative, analytic, and numerical approaches are stressed throughout in our approach to understanding solutions of differential equations.
www.math.tamu.edu /~Don.Allen/ODE_resources.htm   (639 words)

Matlab can numerically solve Ordinary Differential equations using 2 methods.
ODE23 uses 2nd and 3rd order Runge-Kutta formulas and ODE45 uses 4th and 5th order Runge-Kutta formulas What you first need to do is to break your ODE into a system of 1st order equations.
Your coefficients to your equations do not have to be constant.
web.mit.edu /answers/matlab/matlab_ode.html   (390 words)

 Stochastic Differential Equations
Non-stochastic differential equations are models of dynamical systems where the state evolves continuously in time.
Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfied by the error process"]
In this paper, we develop a theory for the stochastic embedding of ordinary differential equations.
cscs.umich.edu /~crshalizi/notebooks/stoch-diff-eqs.html   (1300 words)

 MATH 432: Numerical Differential Equations - Spring 2006
Textbook: A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles (Cambridge Texts in Applied Mathematics, 1996).
Solution of ordinary differential equations by multistep and single step methods.
The diffusion equation: numerical schemes and stability analysis.
www.case.edu /artsci/math/geuzaine/432   (427 words)

 Ordinary Differential Equations
In many real life applications where we wish to evaluate a function, we are only able to define the relationship that the derivative of that function must satisfy.
One example is the dynamics of a vehicle in which we wish to determine the velocity or acceleration, however the unknowns are specified only in terms of ordinary differential equations (ODEs) and initial conditions.
We see that 20 degrees is about the limit of the small angle period (2.006 s), however at 80 degrees the period is seen to be much larger (2.293 s).
www.ent.ohiou.edu /~urieli/odes/odes.html   (1076 words)

 Numerical Differential Equations
This generates a numerical solution to the equation
For a differential equation, however, the solution is a
This solves a system of two coupled differential equations.
documents.wolfram.com /v4/MainBook/1.6.4.html   (101 words)

 CAAM 435/MATH 322   (Site not responding. Last check: 2007-10-11)
For this year only CAAM 435, Ordinary Differential Equations, and MATH 322, Introduction to Analysis II will be cross listed.
Since numerical methods are covered in CAAM 452, they will not be taught in this course.
The text for this course is Differential Equations, Dynamical Systems, and Linear Algebra, By Morris W. Hirsch and Stephen Smale.
www.owlnet.rice.edu /~math322   (423 words)

 Differential Equations
solves boundary-value or initial-value problems involving nonlinear or linear ordinary differential equations of any order, or systems of such.
The conditions may also be linear or nonlinear equations involving the unknown functions and their derivatives.
The solution produced is a continuous function in the form of finite power or trigonometric series, depending on the program, of user-specified number of terms valid in the entire defined interval and expanded about a user-chosen center of expansion.
www.numericalmathematics.com /ordinary_differential_equations.htm   (226 words)

 Wiley::Ordinary Differential Equations, 4th Edition
Numerical Solution of Ordinary Differential Equations: for Classical, Relativistic and Nano Systems
First chapters present a rigorous treatment of background material; middle chapters deal in detail with systems of nonlinear differential equations; final chapters are devoted to the study of second-order linear differential equations.
The power of the theory of ODE is illustrated throughout by deriving the properties of important special functions, such as Bessel functions, hypergeometric functions, and the more common orthogonal polynomials, from their defining differential equations and boundary conditions.
eu.wiley.com /WileyCDA/WileyTitle/productCd-0471860034.html   (209 words)

 Numerical Solution of Ordinary Differential Equations
The first two labs concern elementary numerical methods for finding approximate solutions to ordinary differential equations.
The results of some numerical experiments are presented and discussed in the first set of notes below.
(postscript file) These notes give the results of some numerical experiments designed to determine how the error generated by Euler's method, the improved Euler method and the Runge-Kutta method depend on the step size used.
www.math.ubc.ca /~feldman/math/ode.html   (572 words)

 CS 367 Numerical Methods - Syllabus
Topics include systems of linear equations, numerical integration, ordinary differential equations, and nonlinear equations.
Miscellaneous Points are assigned at the discrestion of the instructor throughout the course for class attendance, class participation such as answering questions, in-class work at the board, possible pop quizzes and/or the overall performance of each student throughout the course.
Other books on numerical methods, scientific computing, or numerical analysis available in the library or in local bookstores.
www.cs.utexas.edu /~kincaid/cs367-info.html   (420 words)

The book collects original articles on numerical analysis of ordinary differential equations and its applications.
Some of the topics covered in this volume are: discrete variable methods, Runge-Kutta methods, linear multistep methods, stability analysis, parallel implementation, self-validating numerical methods, analysis of nonlinear oscillation by numerical means, differential-algebraic and delay-differential equations, and stochastic initial value problems.
Numerical Validation for Ordinary Differential Equations Using Power Series Arithmetic (M Kashiwagi)
www.worldscibooks.com /mathematics/2719.html   (236 words)

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