tailieunhanh - Two Point Boundary Value Problems part 3

The shooting method described in § tacitly assumed that the “shots” would be able to traverse the entire domain of integration, even at the early stages of convergence to a correct solution. In some problems it can happen | 760 Chapter 17. Two Point Boundary Value Problems Shooting to a Fitting Point The shooting method described in tacitly assumed that the shots would be able to traverse the entire domain of integration even at the early stages of convergence to a correct solution. In some problems it can happen that for very wrong starting conditions an initial solution can t even get from xi to x2 without s 5 j p encountering some incalculable or catastrophic result. For example the argument of a square root might go negative causing the numerical code to crash. Simple g er z shooting would be stymied. S A different but related case is where the endpoints are both singular points of the set of ODEs. One frequently needs to use special methods to integrate near the singular points analytic asymptotic expansions for example. In such cases it is feasible to integrate in the direction away from a singular point using the special 8 method to get through the first little bit and then reading off initial values for o . I g further numerical integration. However it is usually not feasible to integrate into 8 g a singular point if only because one has not usually expended the same analytic effort to obtain expansions of wrong solutions near the singular point those not satisfying the desired boundary condition . The solution to the above mentioned difficulties is shooting to a fitting point. 0 Instead of integrating from xi to x2 we integrate first from xi to some point xf that ji is between xi and x2 and second from x2 in the opposite direction to xf. 155 o a If as before the number of boundary conditions imposed at xi is n1 and the - number imposed at x2 is n2 then there are n2 freely specifiable starting values at x1 and n1 freely specifiable starting values at x2. If you are confused by this go g- .g back to . We can therefore define an n2-vector V i of starting parameters S at xi and a prescription load1 x1 v1 y for mapping V i into a y that satisfies g 5 the boundary .