chap09_8up - Ordinary Differential Equations Numerical...

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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...
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chap09_8up - Ordinary Differential Equations Numerical...

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