# Numerical Methods for Differential Equations

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This Demonstration shows the exact and the numerical solutions using a variety of simple numerical methods for ordinary differential equations. Use the sliders to vary the initial value or to change the number of steps or the method.

Contributed by: Edda Eich-Soellner (February 2008)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The Demonstration shows various methods for ODEs:

* Euler's method is the simplest method for the numerical solution of an ordinary differential equation . Starting from an initial point , ) and dividing the interval [, ] that is under consideration into steps results in a step size ; the solution value at point is recursively computed using , .

* Implicit Euler method

* Heun's method

* classical Runge-Kutta method of order 4

The last right-hand side given belongs to a stiff equation, such that the behavior of the method for this type of equation can be studied. See M. Heath, Scientific Computing: An Introductory Survey, New York: McGraw-Hill, 2002.

Note that *Mathematica* provides all of the methods outlined here and many others as part of the NDSolve framework. In contrast to the simple implementations used here, *Mathematica* uses more advanced methods which are e.g. equipped with error estimation and step size selection strategies as well as a stiffness switching; see *Mathematica*'s advanced documentation for NDSolve.

## Permanent Citation

"Numerical Methods for Differential Equations"

http://demonstrations.wolfram.com/NumericalMethodsForDifferentialEquations/

Wolfram Demonstrations Project

Published: February 26 2008