satisfy boundary conditions at more than one value of the independent variable, the resulting problem is called a two point boundary value problem. As the terminology indicates, the most common case by far is where boundary conditions are supposed | Chapter 17. Two Point Boundary Value Problems Introduction When ordinary differential equations are required to satisfy boundary conditions at more than one value of the independent variable the resulting problem is called a two point boundary value problem. As the terminology indicates the most common case by far is where boundary conditions are supposed to be satisfied at two points usually the starting and ending values of the integration. However the phrase two point boundary value problem is also used loosely to include more complicated cases . where some conditions are specified at endpoints others at interior usually singular points. The crucial distinction between initial value problems Chapter 16 and two point boundary value problems this chapter is that in the former case we are able to start an acceptable solution at its beginning initial values and just march it along by numerical integration to its end final values while in the present case the boundary conditions at the starting point do not determine a unique solution to start with and a random choice among the solutions that satisfy these incomplete starting boundary conditions is almost certain not to satisfy the boundary conditions at the other specified point s . It should not surprise you that iteration is in general required to meld these spatially scattered boundary conditions into a single global solution of the differential equations. For this reason two point boundary value problems require considerably more effort to solve than do initial value problems. You have to integrate your differential equations over the interval of interest or perform an analogous relaxation procedure see below at least several and sometimes very many times. Only in the special case of linear differential equations can you say in advance just how many such iterations will be required. The standard two point boundary value problem has the following form We desire the solution to a set of N coupled .
