Thus far we have only discussed initial value problems. In the projectile problems, for example, we fixed the initial position and velocity and integrated over the variables up to a final position and velocity.

But there are many ODE problems in which two or more points are fixed and solutions of the ODE must be found that satisfy these constraints. Such boundary value problems can be solved with two approaches:

1. Shooting Method
2. Finite difference method

The finite difference method works only with linear ODEs. A grid of points is defined over the intervals between the initial and final values of the independent variable. Then derivatives are substituted with difference equations between the grid points, resulting in a set of algebraic linear equations. The end points are known from the boundary conditions while the values at the inner points must be solved . The equations can be solved with relaxation methods. (See the references for more about the finite difference methods.)

Here we will focus on the shooting method since it works with either linear or non-linear equations.

References & Web Resources

• Gould & Tobochnik, An Introduction to Computer Simulation Methods...
• Tao Pang, An Introduction to Computational Physics
• Press et al, Numerical Recipes: The Art of Scientific Computing
• Bishop & Bishop, Java Gently for Scientists & Engineers