Two-Point Boundary Problems

Initial value problems are relatively simple to solve. All of the dependent variables are assigned values at the start of the range of integration. The solution then marches out from the starting point to whatever independent variable value is desired. The only decision is which integration algorithm to use.

With two-point boundary problems, boundary conditions are specified at both ends of the integration range. Some of the variables at each end of the integration range will be unspecified by boundary conditions. These are called free variables. Not only do you have to integrate the ODEs, but you also must assign values to the free variables at the beginning of the integration range such that the boundary conditions at the end of the range of integration are satisfied. Unless you make a very good initial guess, it is likely that the solution will have to be iterated. In addition to selecting an integration algorithm you must also develop an iteration scheme that will efficiently converge to the proper solution.

As you probably guessed, there are several techniques to solve two-point boundary problems. The method that we will implement in this chapter is called shooting .