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Unformatted text preview: Lecture 30 Euler Methods Numerical Solution of an IVP Suppose we wish to numerically solve the initial value problem ˙ y = f ( t, y ) , y ( a ) = y , (30.1) on an interval of time [ a, b ]. By a numerical solution, we must mean an approximation of the solution at a finite number of points, i.e. ( t , y ) , ( t 1 , y 1 ) , ( t 2 , y 2 ) , . . . , ( t n , y n ) , where t = a and t n = b . The first of these points is exactly the initial value. If we take n steps as above, and the steps are evenly spaced, then the time change in each step is h = b a n , (30.2) and the times t i are given simply by t i = a + ih . This leaves the most important part of finding a numerical solution: determining y 1 , y 2 , . . . , y n in a way that is as consistent as possible with (30.1). To do this, first write the differential equation in the indexed notation ˙ y i ≈ f ( t i , y i ) , (30.3) and will then replace the derivative ˙ y by a difference. There are many ways we might carry this out and in the next section we study the simplest. The Euler Method The most straight forward approach is to replace ˙ y i in (30.3) by its forward difference approximation. This gives y i +1 y i h = f ( t i , y i ) . Rearranging this gives us a way to obtain y i +1 from y i known as Euler’s method: y i +1 = y i + h f ( t i , y i ) . (30.4) With this formula, we can start from ( t , y ) and compute all the subsequent approximations ( t i , y i ). This is very easy to implement, as you can see from the following program (which can be downloaded from the class web site): 109 110 LECTURE 30. EULER METHODS function [T , Y] = myeuler(f,tspan,y0,n) % function [T , Y] = myeuler(f,tspan,y0,n)...
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 Fall '08
 Young,T
 Numerical Analysis, Approximation, Boundary value problem

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