###### Asked in DifferentialsMathematical FinanceAlgebra

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

## Answer

###### Wiki User

###### August 30, 2007 12:43PM

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**.