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Unformatted text preview: Chapter 08.03 RungeKutta 2nd Order Method for Ordinary Differential Equations After reading this chapter, you should be able to : 1. understand the RungeKutta 2nd order method for ordinary differential equations and how to use it to solve problems. What is the RungeKutta 2nd order method? The RungeKutta 2nd order method is a numerical technique used to solve an ordinary differential equation of the form ( 29 ( 29 , , y y y x f dx dy = = Only first order ordinary differential equations can be solved by using the RungeKutta 2nd order method. In other sections, we will discuss how the Euler and RungeKutta methods are used to solve higher order ordinary differential equations or coupled (simultaneous) differential equations. How does one write a first order differential equation in the above form? Example 1 Rewrite ( 29 5 , 3 . 1 2 = = + y e y dx dy x in ) ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 , 3 . 1 2 = = + y e y dx dy x ( 29 5 , 2 3 . 1 = = y y e dx dy x In this case 08.03.1 08.03.2 Chapter 08.03 ( 29 y e y x f x 2 3 . 1 , = Example 2 Rewrite ( 29 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y in ) ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y ( 29 5 , ) 3 sin( 2 2 2 = = y e y x x dx dy y In this case ( 29 y e y x x y x f 2 2 ) 3 sin( 2 , = RungeKutta 2 nd order method Eulers method is given by ( 29 h y x f y y i i i i , 1 + = + (1) where = x ) ( x y y = i i x x h = + 1 To understand the RungeKutta 2nd order method, we need to derive Eulers method from the Taylor series. ( 29 ( 29 ( 29 ... ! 3 1 ! 2 1 3 1 , 3 3 2 1 , 2 2 1 , 1 + + + + = + + + + i i y x i i y x i i y x i i x x dx y d x x dx y d x x dx dy y y i i i i i i ( 29 ( 29 ( 29 ... ) , ( ' ' ! 3 1 ) , ( ' ! 2 1 ) , ( 3 1 2 1 1 + + + + = + + + i i i i i i i i i i i i i x x y x f x x y x f x x y x f y (2) As you can see the first two terms of the Taylor series ( 29 h y x f y y i i i i , 1 + = + are Eulers method and hence can be considered to be the RungeKutta 1st order method. The true error in the approximation is given by ( 29 ( 29 ... ! 3 , ! 2 , 3 2 + + = h y x f h y x f E i i i i t (3) So what would a 2nd order method formula look like. It would include one more term of the Taylor series as follows. RungeKutta 2nd Order Method 08.03.3 ( 29 ( 29 2 1 , ! 2 1 , h y x f h y x f y y i i i i i i + + = + (4) Let us take a generic example of a first order ordinary differential equation ( 29 5 , 3 2 = = y y e dx dy x ( 29 y e y x f x 3 , 2 = Now since y is a function of x , ( 29 ( 29...
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This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.
 Spring '07
 Gallivan

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