Boundary-Value Problems

As we have seen, initial-value problems are quite straightforward to solve. Once a complete set of initial conditions is specified, we can simply proceed forward in time, step by step, in a completely deterministic fashion, using one of the integrators (Midpoint or Runge-Kutta-4) discussed in class. Not all ordinary differential equations are so easy to handle, however. In boundary-value problems, the constraints on the system are specified at several different points (usually in space, but occasionally in time). A different strategy is required.

In general, a system of N ordinary differential equations requires N constraints to determine the solution. In principle, these constraints, or boundary conditions, may all be specified at different points. However, we will confine ourselves here to so-called two-point boundary-value problems, where the boundary conditions are specified at just two locations. Since the independent variable in boundary-value problems is usually space, as opposed to time in initial-value problems, in these pages we will denote the independent variable by x and the dependent variable(s) by y. (Naturally, this change of notation makes no difference to either the nature of the problem or the method of solution!)

We can specify a two-point boundary-value problem as follows. The system of equations is

$$\frac{dy}{dx} = f(y,x)\,,$$

with the constraints

$$\begin{eqnarray*} \gamma_i(y,x) &=& 0 \ \ (i = 1,\ldots,m) \ \ {\rm at} \ \ x ... ...) &=& 0 \ \ (i = m+1,\ldots,N) \ \ {\rm at} \ \ x = b > a\,.\\ \end{eqnarray*}$$

The constraints $\gamma$ can be arbitrary functions of y and x. However, a simple, and fairly typical, example might be that m components of y are specified at x = a (so $\gamma_i(y,a) \equiv y_i - \alpha_i$ ), and N-m components (possibly the same ones) are specified at x = b ( $\gamma_i(y,b) \equiv y_i - \beta_i$ ).

For now, we will concentrate on the case N = 2, m = 1, with $~y_1~$ specified at both x = a and x = b, and $~y_2~$ unconstrained. (As a concrete example, the system could represent the solution of a second-order differential equation with the values of the solution fixed at two points, but with the derivatives ($~y_2~$ ) freely variable.)