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Unformatted text preview: Chapter 7 Ordinary Differential Equations Matlab has several different functions for the numerical solution of ordinary dif ferential equations. This chapter describes the simplest of these functions and then compares all of the functions for efficiency, accuracy, and special features. Stiffness is a subtle concept that plays an important role in these comparisons. 7.1 Integrating Differential Equations The initial value problem for an ordinary differential equation involves finding a function y ( t ) that satisfies dy ( t ) dt = f ( t,y ( t )) together with the initial condition y ( t ) = y . A numerical solution to this problem generates a sequence of values for the indepen dent variable, t ,t 1 ,... , and a corresponding sequence of values for the dependent variable, y ,y 1 ,... , so that each y n approximates the solution at t n : y n ≈ y ( t n ) , n = 0 , 1 ,.... Modern numerical methods automatically determine the step sizes h n = t n +1 t n so that the estimated error in the numerical solution is controlled by a specified tolerance. The fundamental theorem of calculus gives us an important connection be tween differential equations and integrals: y ( t + h ) = y ( t ) + Z t + h t f ( s,y ( s )) ds. February 16, 2008 1 2 Chapter 7. Ordinary Differential Equations We cannot use numerical quadrature directly to approximate the integral because we do not know the function y ( s ) and so cannot evaluate the integrand. Nevertheless, the basic idea is to choose a sequence of values of h so that this formula allows us to generate our numerical solution. One special case to keep in mind is the situation where f ( t,y ) is a function of t alone. The numerical solution of such simple differential equations is then just a sequence of quadratures: y n +1 = y n + Z t n +1 t n f ( s ) ds. Throughout this chapter, we frequently use “dot” notation for derivatives: ˙ y = dy ( t ) dt and ¨ y = d 2 y ( t ) dt 2 . 7.2 Systems of Equations Many mathematical models involve more than one unknown function, and second and higher order derivatives. These models can be handled by making y ( t ) a vector valued function of t . Each component is either one of the unknown functions or one of its derivatives. The Matlab vector notation is particularly convenient here. For example, the secondorder differential equation describing a simple har monic oscillator ¨ x ( t ) = x ( t ) becomes two firstorder equations. The vector y ( t ) has two components, x ( t ) and its first derivative ˙ x ( t ): y ( t ) = • x ( t ) ˙ x ( t ) ‚ . Using this vector, the differential equation is ˙ y ( t ) = • ˙ x ( t ) x ( t ) ‚ = • y 2 ( t ) y 1 ( t ) ‚ . The Matlab function defining the differential equation has t and y as input arguments and should return f ( t,y ) as a column vector. For the harmonic oscillator, the function could be an Mfile containing function ydot = harmonic(t,y) ydot = [y(2); y(1)] A more compact version uses matrix multiplication in an anonymous function,...
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 Spring '08
 BURKHAUSER
 matlab, Numerical Analysis, Tn, ORDINARY DIFFERENTIAL EQUATIONS

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