Tuesday, February 2, 2010

Shooting Methods for ODE's

Finding the solution of a linear boundary problem is assisted by the linear structure of the equation and the use of two special initial value problems. Suppose that u(t) is the unique solution to the I.V.P.

[Graphics:Images/ShootingMod_gr_34.gif] with [Graphics:Images/ShootingMod_gr_35.gif].

Furthermore, suppose that v(t) is the unique solution to the I.V.P.

[Graphics:Images/ShootingMod_gr_36.gif] with [Graphics:Images/ShootingMod_gr_37.gif].

Then the linear combination

[Graphics:Images/ShootingMod_gr_38.gif].

is a solution to [Graphics:Images/ShootingMod_gr_39.gif] with [Graphics:Images/ShootingMod_gr_40.gif].

Program (Linear Shooting Method):

To approximate the solution of the boundary value problem

[Graphics:Images/ShootingMod_gr_41.gif] with [Graphics:Images/ShootingMod_gr_42.gif]

over the interval [a,b] by using the Runge-Kutta method of order n=4.

The method involves solving a two systems of equations over [Graphics:Images/ShootingMod_gr_43.gif]. First solve

[Graphics:Images/ShootingMod_gr_44.gif] with [Graphics:Images/ShootingMod_gr_45.gif],
[Graphics:Images/ShootingMod_gr_46.gif] and [Graphics:Images/ShootingMod_gr_47.gif].

Then solve

[Graphics:Images/ShootingMod_gr_48.gif] with [Graphics:Images/ShootingMod_gr_49.gif],
[Graphics:Images/ShootingMod_gr_50.gif] and [Graphics:Images/ShootingMod_gr_51.gif].

Finally, the desired solution x(t) is the linear combination

[Graphics:Images/ShootingMod_gr_52.gif].

The subroutine Runge2D will be used to construct the two solutions [Graphics:Images/ShootingMod_gr_53.gif], and [Graphics:Images/ShootingMod_gr_54.gif].

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Introductory Methods of Numerical Analysis