Initial Value Techniques for Boundary Value Problems

The first method is one which utilizes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundary conditions. Suppose we have the nonlinear boundary value problem

Convert this second-order equation into two first-order equations along with the boundary conditions written to include a parameter, s.

The parameters c₀ and c₁ are specified by the analyst such that

and this insures that the first boundary condition is satisfied for any value of the parameter s. If we knew the proper value for s we could evaluate u(0) and u'(0) and integrate the equation as an initial value problem. We would like to choose the parameter s so that the last boundary condition is satisfied. Define the function

and we would like this to be zero. Note that the solution depends on s.

We must iterate on s to find the solution to the problem. Note that the condition at x = 0 is satisfied for any s, the differential equation is satisfied by our integration routine, and we only have to insure that the last boundary condition is satisfied. A successive substitution method would use

Keller [1972] showed that if

for some N and 0 < m < 2 G, where G increases as N increases, then the iteration scheme converges as k approaches infinity.

Newton's method of iteration would use

The function dc/ds is determined by integrating two more equations obtained by differentiating the original equations with respect to s. Let

Then the additional differential equations are obtained by differentiating the equations using

giving

We integrate these equations along with the initial conditions

Then the derivative is

See also:

Multiple conditions

Turn the problem inside out

May work in only one direction

Sensitivity