FirstEqn2 - Math 246 Fall 2009 Professor Levermore 6...

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Unformatted text preview: Math 246, Fall 2009, Professor Levermore 6. First-Order Equations: Numerical Methods For many first-order differential equations analytic methods are either difficult or impossible to apply. If one is interested in understanding how a particular solution behaves it is often easiest to use numerical methods to construct accurate approximations to the solution. Suppose we are interested in the solution Y ( t ) of the initial-value problem dy dt = f ( t, y ) , y ( t I ) = y I , (6.1) over the time interval [ t I , t F ] — i.e. for t I ≤ t ≤ t F . Here t I is called the initial time and t F is called the final time . A numerical method selects times { t n } N n =0 such that t I = t < t 1 < t 2 < ··· < t N- 1 < t N = t F , and computes values { y n } N n =0 such that y = Y ( t ) = y I and y n approximates Y ( t n ) for n = 1 , 2 , ··· , N . For good numerical methods, these approximations will improve as N increases. So for sufficiently large N you can plot the points { ( t n , y n ) } N n =0 in the ( t, y )- plane and “connect the dots” to get an accurate picture of how Y ( t ) behaves over the time interval [ t I , t F ]. Here we will introduce a few basic numerical methods in simple settings. The numer- ical methods used in software packages such as MATLAB are generally far more sophisti- cated than those we will study here. They are however built upon the same fundamental ideas as the simpler methods we will study. Throughout this section we will make the following two basic simplifications. • We will employ uniform time steps . This means that given N we set h = t F- t I N , and t n = t I + nh for n = 0 , 1 , ··· , N , (6.2) where h is called the time step . • We will employ one-step methods . This means that given f ( t, y ) and h the value of y n +1 for n = 0 , 1 , ··· , N- 1 will depend only on y n and h . Sophisticated software packages use methods in which the time step is chosen adaptively. In other words, the choice of t n +1 will depend on the behavior of recent approximations — for example, on ( t n , y n ) and ( t n- 1 , y n- 1 ). Employing uniform time steps will greatly simplify the algorithms, and thereby simplify the programming you will have to do. If you do not like the way a run looks, you will simply try again with a larger N . Similarly, sophisticated software packages will sometimes use so-called multi-step methods for which the value of y n +1 for n = m, m + 1 , ··· , N- 1 will depend on y n , y n- 1 , ··· , and y n- m for some positive integer m . Employing one-step methods will again simplify the algorithms, and thereby simplify the programming you will have to do. 1 2 6.1. Euler Methods. The simplest (and the least accurate) numerical methods are the Euler methods. These can be derived many ways. Here we give a simple approach based on the definition of the derivative through difference quotients....
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This note was uploaded on 01/24/2011 for the course MATH math246 taught by Professor Levermore during the Spring '10 term at Maryland.

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FirstEqn2 - Math 246 Fall 2009 Professor Levermore 6...

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