Numerical solution of linear two‐point boundary problems via the fundamental‐matrix method

A numerical method is presented for the‐solution of linear systems of differential equations with initial‐value or two‐point boundary conditions. For y ′( x ) = A ( x ) y ( x ) + f ( x ) the domain of interest [ a , b ] is divided into an appropriate number L of subintervals. The coefficient matrix A ( x ) is replaced by its value A k at a point x k within the K th subinterval, thus replacing the original system by the L discretized systems y k ( x ) = A k y k ( x ) + f k ( x ), k = 1,2,…, L . The fundamental matrix solution Φ k ( x , x k ) over each subinterval is found by computing the eigenvalues and eigenvectors of each A k . By matching the solutions y k ( x ) at the L – 1 equispaced grid points defining the limits of the subintervals and the boundary conditions, the two‐point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different y k ( x ). The values of y 1 ( a ) and y L ( b ) are used to calculate the missing boundary conditions. For initial‐value problems this method is equivalent to a one‐step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method, 1 the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.