(Difference between revisions)
 Revision as of 09:11, 10 December 2005 (view source)Praveen (Talk | contribs) (corrected equation)← Older edit Latest revision as of 18:53, 13 August 2007 (view source) (What are BDF Methods?) Line 1: Line 1: - Adams' methods are a subset of the family of multistep methods used for the numerical integration of initial value problems in ODEs. Multistep methods benefit from the fact that the computations have been going on for some time, and use previously computed values of the solution (BDF methods), or the right hand side (Adams' methods), to approximate the solution at the next step. + == Introduction == + + Adams' methods are a subset of the family of [[multi-step methods]] used for the [[numerical integration]] of initial value problems in [[Ordinary Differential Equations]] (ODE's). [[Multi-step methods]] benefit from the fact that the computations have been going on for some time, and use previously computed values of the solution (BDF methods), or the right hand side (Adams' methods), to approximate the solution at the next step. Adams' methods begin by the integral approach, Adams' methods begin by the integral approach, Line 12: Line 14: [/itex] [/itex] - Since $f$ is unknown in the interval $t_n$ to $t_{n+1}$ it is approximated by an interpolating [[polynomial]] $p(t)$ using the previously computed steps $t_{n},t_{n-1},t_{n-2} ...$ and the current step at $t_{n+1}$ if an implicit method is desired. + Since $f$ is unknown in the interval $t_n$ to $t_{n+1}$ it is approximated by an [[interpolating]] [[polynomial]] $p(t)$ using the previously computed steps $t_{n},t_{n-1},t_{n-2} ...$ and the current step at $t_{n+1}$ if an implicit method is desired.{{fact}} + + == References == + + ? + + {{stub}}

## Introduction

Adams' methods are a subset of the family of multi-step methods used for the numerical integration of initial value problems in Ordinary Differential Equations (ODE's). Multi-step methods benefit from the fact that the computations have been going on for some time, and use previously computed values of the solution (BDF methods), or the right hand side (Adams' methods), to approximate the solution at the next step.

Adams' methods begin by the integral approach,

$y^\prime = f(t,y)$

$y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = y(t_{n}) + \int_{t_n}^{t_{n+1}} f(t,y(t)) dt$

Since $f$ is unknown in the interval $t_n$ to $t_{n+1}$ it is approximated by an interpolating polynomial $p(t)$ using the previously computed steps $t_{n},t_{n-1},t_{n-2} ...$ and the current step at $t_{n+1}$ if an implicit method is desired.

?