chap09_8up

# chap09_8up - Ordinary Differential Equations Numerical...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Scientiﬁc Computing: An Introductory Survey Chapter 9 – Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientiﬁc Computing 1 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Outline 1 Ordinary Differential Equations 2 Numerical Solution of ODEs 3 Additional Numerical Methods Michael T. Heath Scientiﬁc Computing 2 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Differential Equations Initial Value Problems Stability Differential Equations Differential equations involve derivatives of unknown solution function Ordinary differential equation (ODE): all derivatives are with respect to single independent variable, often representing time Solution of differential equation is function in inﬁnite-dimensional space of functions Numerical solution of differential equations is based on ﬁnite-dimensional approximation Differential equation is replaced by algebraic equation whose solution approximates that of given differential equation Michael T. Heath Scientiﬁc Computing 3 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Differential Equations Initial Value Problems Stability Order of ODE Order of ODE is determined by highest-order derivative of solution function appearing in ODE ODE with higher-order derivatives can be transformed into equivalent ﬁrst-order system We will discuss numerical solution methods only for ﬁrst-order ODEs Most ODE software is designed to solve only ﬁrst-order equations Michael T. Heath Scientiﬁc Computing 4 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Differential Equations Initial Value Problems Stability Higher-Order ODEs, continued For k-th order ODE y ( k ) ( t ) = f ( t,y,y ,...,y ( k-1) ) deﬁne k new unknown functions u 1 ( t ) = y ( t ) , u 2 ( t ) = y ( t ) , ..., u k ( t ) = y ( k-1) ( t ) Then original ODE is equivalent to ﬁrst-order system u 1 ( t ) u 2 ( t ) . . . u k-1 ( t ) u k ( t ) = u 2 ( t ) u 3 ( t ) . . . u k ( t ) f ( t,u 1 ,u 2 ,...,u k ) Michael T. Heath Scientiﬁc Computing 5 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Differential Equations Initial Value Problems Stability Example: Newton’s Second Law Newton’s Second Law of Motion, F = ma , is second-order ODE, since acceleration a is second derivative of position coordinate, which we denote by y Thus, ODE has form y 00 = F/m where F and m are force and mass, respectively Deﬁning u 1 = y and u 2 = y...
View Full Document

{[ snackBarMessage ]}

### Page1 / 11

chap09_8up - Ordinary Differential Equations Numerical...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online