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Unformatted text preview: 08.04.1 Chapter 08.04 Runge-Kutta 4th Order Method for Ordinary Differential Equations After reading this chapter, you should be able to 1. develop Runge-Kutta 4 th order method for solving ordinary differential equations, 2. find the effect size of step size has on the solution, 3. know the formulas for other versions of the Runge-Kutta 4 th order method What is the Runge-Kutta 4th order method? Runge-Kutta 4 th order method is a numerical technique used to solve ordinary differential equation of the form ( ) ( ) , , y y y x f dx dy = = So only first order ordinary differential equations can be solved by using the Runge-Kutta 4 th order method. In other sections, we have discussed how Euler and Runge-Kutta 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 ( ) 5 , 3 . 1 2 = = + − y e y dx dy x in ) ( ), , ( y y y x f dx dy = = form. 08.04.2 Chapter 08.04 Solution ( ) 5 , 3 . 1 2 = = + − y e y dx dy x ( ) 5 , 2 3 . 1 = − = − y y e dx dy x In this case ( ) y e y x f x 2 3 . 1 , − = − Example 2 Rewrite ( ) 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y in ) ( ), , ( y y y x f dx dy = = form. Solution ( ) 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y ( ) 5 , ) 3 sin( 2 2 2 = − = y e y x x dx dy y In this case ( ) y e y x x y x f 2 2 ) 3 sin( 2 , − = The Runge-Kutta 4 th order method is based on the following ( ) h k a k a k a k a y y i i 4 4 3 3 2 2 1 1 1 + + + + = + (1) where knowing the value of i y y = at i x , we can find the value of 1 + = i y y at 1 + i x , and i i x x h − = + 1 Equation (1) is equated to the first five terms of Taylor series...
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.
- Spring '08