Two-Point Boundary Value Problems

Abstract

Solve y″ = f ( x, y, y ′), y ( a ) = α, y ( b ) = β Introduction In two-point boundary value problems the auxiliary conditions associated with the differential equation, called the boundary conditions , are specified at two different values of x . This seemingly small departure from initial value problems has a major repercussion – it makes boundary value problems considerably more difficult to solve. In an initial value problem, we were able to start at the point were the initial values were given and march the solution forward as far as needed. This technique does not work for boundary value problems, because there are not enough starting conditions available at either end point to produce a unique solution. One way to overcome the lack of starting conditions is to guess the missing boundary values at one end. The resulting solution is very unlikely to satisfy boundary conditions at the other end, but by inspecting the discrepancy we can estimate what changes to make to the initial conditions before integrating again. This iterative procedure is known as the shooting method . The name is derived from analogy with target shooting – take a shot and observe where it hits the target, then correct the aim and shoot again. Another means of solving two-point boundary value problems is the finite difference method , where the differential equations are approximated by finite differences at evenly spaced mesh points. As a consequence, a differential equation is transformed into set of simultaneous algebraic equations. The two methods have a common problem: they give rise to nonlinear sets of equations if the differential equation is not linear. As we noted in Chapter 4, all methods of solving nonlinear equations are iterative procedures that can consume a lot of computational resources.