mws_gen_ode_spe_shootingmethod

mws_gen_ode_spe_shootingmethod - 08.06 Shooting Method for...

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08.06.1 08.06 Shooting Method for Ordinary Differential Equations After reading this chapter, you should be able to 1. learn the shooting method algorithm to solve boundary value problems, and 2. apply shooting method to solve boundary value problems. What is the shooting method? Ordinary differential equations are given either with initial conditions or with boundary conditions. Look at the problem below. Figure 1 A cantilevered uniformly loaded beam. To find the deflection as a function of location x , due to a uniform load q , the ordinary differential equation that needs to be solved is  2 2 2 2 x L EI q dx d (1) where L is the length of the beam, E is the Young’s modulus of the beam, and I is the second moment of area of the cross-section of the beam. Two conditions are needed to solve the problem, and those are 0 0 q υ L x
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08.06.2 Chapter 08.06  0 0 dx d (2a,b) as it is a cantilevered beam at 0 x . These conditions are initial conditions as they are given at an initial point, 0 x , so that we can find the deflection along the length of the beam. Now consider a similar beam problem, where the beam is simply supported on the two ends Figure 2 A simply supported uniformly loaded beam. To find the deflection as a function of x due to the uniform load q , the ordinary differential equation that needs to be solved is L x EI qx dx d 2 2 2 (3) Two conditions are needed to solve the problem, and those are 0 0 0 L (4a,b) as it is a simply supported beam at 0 x and L x . These conditions are boundary conditions as they are given at the two boundaries, 0 x and L x . The shooting method The shooting method uses the same methods that were used in solving initial value problems. This is done by assuming initial values that would have been given if the ordinary differential equation were an initial value problem. The boundary value obtained is then compared with the actual boundary value. Using trial and error or some scientific approach, one tries to get as close to the boundary value as possible. This method is best explained by an example.
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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.

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mws_gen_ode_spe_shootingmethod - 08.06 Shooting Method for...

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