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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 UrbanaChampaign 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 inﬁnitedimensional space of functions Numerical solution of differential equations is based on ﬁnitedimensional 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 highestorder derivative of solution function appearing in ODE ODE with higherorder derivatives can be transformed into equivalent ﬁrstorder system We will discuss numerical solution methods only for ﬁrstorder ODEs Most ODE software is designed to solve only ﬁrstorder equations Michael T. Heath Scientiﬁc Computing 4 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Differential Equations Initial Value Problems Stability HigherOrder ODEs, continued For kth order ODE y ( k ) ( t ) = f ( t,y,y ,...,y ( k1) ) deﬁne k new unknown functions u 1 ( t ) = y ( t ) , u 2 ( t ) = y ( t ) , ..., u k ( t ) = y ( k1) ( t ) Then original ODE is equivalent to ﬁrstorder system u 1 ( t ) u 2 ( t ) . . . u k1 ( 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 secondorder 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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 Fall '11
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 Mechanical Engineering, ORDINARY DIFFERENTIAL EQUATIONS, Numerical ordinary differential equations, Michael T. Heath

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