ode219week2 - Math 219 Lecture 3 Ali Devin Sezer October 5...

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Math 219, Lecture 3 Ali Devin Sezer October 5, 2010 1 Numerical approximation of differential equa- tions, Euler’s method Consider the first order initial value problem: dy dt = f ( t, y ) , y (0) = y 0 . (1) The variable t is denoted with the letter “t” because many times this variable denotes time and y ( t ) represents the evolution of a quantity in time. Suppose that f and ∂f ∂y are continuous functions. The existence and uniqueness theorem tells us that there is a unique differentiable function y satisfying (1). The theorem doesn’t say anything about how to find the solution; it only says that there is exactly one solution. The theorem doesn’t say what the solution is. Let us now review a method that actually constructs an approximate solution. We would like to find a function y : [0 , ) R satisfying (1). Fix a small time step δ and consider the points t 0 = 0, t 1 = δ , t 2 = 2 δ , t 3 = 3 δ . . . , t n = . A very simple method to approximate y ( t n ) is as follows: 1. Start with y
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