Boundary Value Techniques for the Numerical Solution of Certain Initial Value Problems in Ordinary Differential Equations

Linear initial value problems, particularly involving first order differential equations, can be transformed into systems of higher order and treated as boundary value problems. The type of difference equations used to replace the associated second order boundary value problem are y n - 2 y n +1 + y n +2 = h 2 ∑ β i y″ n + i + h 3 ; ∑ δ i y ‴ n + i + · · ·, n = 1, 2, · · ·, N - 1 and - y N + y N +1 = h ( b 0 y′ N + b 1 y′ N +1 ) + h 2 ;( c 0 y″ N + c 1 y″ N +1 ) + · · ·. Numerical techniques referred to as M 1, M 2, and M 3 have been developed in which error is O ( h 4 ), O ( h 6 ) and O ( h 8 ), respectively. Experimental results have been given to demonstrate the usefulness of method M 3 over M 1 or M 2.