ODE notes from AMSC-CMSC 460

ODE notes from AMSC-CMSC 460 - AMSC/CMSC 460 Computational...

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AMSC/CMSC 460 Computational Methods, Fall 2001 UNIT 9: Ordinary Differential Equations Dianne P. O’Leary c ± 2001,2002 Numerical Solution of Ordinary Differential Equations Reference: Chapter 9. Jargon: ODE = ordinary differential equation Notation: y ( i ) will denote the i th component of the vector y . y i will denote our approximation to the function y evaluated at t i . The plan ODEs: manipulation and properties Standard form Solution families Stability ODEs: numerical methods Using the software Euler’s method The backward Euler method The Adams family Building a practical ode solver Runge-Kutta methods ODEs: Manipulation and properties (What we need to know from your previous MATH class) Standard form Solution families Stability Standard form of the ODE 1
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We’ll only work with problems in standard form , because that is what the software needs: y 0 = f ( t, y ) y (0) = y 0 where the function y has m components, y 0 means the derivative with respect to t , and y 0 is a given vector of numbers. Writing this component-by-component yields y 0 (1) = f 1 ( t, y (1) ,...,y ( m ) ) ... y 0 ( m ) = f m ( t, y (1) ( m ) ) with y (1) (0) ( m ) (0) given numbers. Note: It is not essential that we start at t =0 ; any value will do. Writing problems in standard form Example: Volterra’s model of rabbits (with an infinite food supply) and foxes (feeding on the rabbits): dr dt =2 r - αrf df dt = - f + αrf r (0) = r 0 f (0) = f 0 The parameter α is the encounter factor ,w ith α meaning no interaction. Let y (1) = r and y (2) = f . Then we can write this in standard form. Example: a second order equation u 00 = g ( t, u, u 0 ) u (0) = u 0 u 0 (0) = v 0 where u 0 and v 0 are given numbers. Let y (1) = u and y (2) = u 0 . Then we can write this in standard form. 2
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Two notes: We won’t consider boundary value problems in this course. Every integration problem can be written as an ODE, but the converse is not true. Example: Z b a f ( t ) dt is equal to y ( b ) , where y 0 = f ( t ) y ( a )=0 Solution families Given y 0 = f ( t, y ) ,the family of solutions is the set of all functions y that satisfy this equation. Three examples of solution families Example 1: parallel solutions y 0 = e - t has the solutions y ( t )= c - e - t where c is an arbitrary constant. Example 2: solutions that converge y 0 = - y has the solution y ( t ce - t where c is an arbitrary constant. Example 3: solutions that diverge y 0 = y has the solution y ( t ce t where c is an arbitrary constant. 3
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Stability of ODEs Loosely speaking, An ODE is stable if family members move toward each other as t increases. See example 2. An ODE is unstable if family members move apart as t increases. See example 3. An ODE is on the stability boundary if family members stay parallel. See example 1. More precisely, let f y = ∂f/∂y. Then a single ODE is stable at a point ˆ t, ˆ y if f y ( ˆ t, ˆ y ) < 0 .
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ODE notes from AMSC-CMSC 460 - AMSC/CMSC 460 Computational...

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