What are the main issues related to the numerical solution of two-point boundary value problems?

Answer 1

In the theory of ODEs (ordinary differential equations), an initial value problem (IVP)

y'(t)=f(t,y), y(a)=c specifies a unique condition at the point a.

A two-point boundary value problem (2PBVP) specifies conditions at two points (a and b):

y''(t)=f(t,y,y'), y(a)=c y(b)=d

As usual, you can transform a second order ODE into a system of two first order ODEs, by defining:

x1=y

x2=y'

so that:

x1'(t)=x2

x2'(t)=f(t,x1,x2)

The problem is that, while one of the two conditions, say x1(a)=c, remains valid, you are not able to translate the other, say x1(b)=d into a condition on x2(b). Hence, what you do is to create a dummy condition on x2(b), say x2(b)=e, and then you numerically solve the system for different values of e, until you find a solution that also satisfies the condition x1(b)=d.