Numerical integration of the system of governing equations which define a boundary value problem written down in the form of a coupled system of first-order ordinary differential equations is shown to be a powerful technique. After presenting the basic approach the paper critically examines the numerical schemes available for situations when the boundary value problem so defined has boundary layer characteristics. One such method which is originally due to Goldberg, S e t h and Alspaugh3 is described in detail, with documentation in the form of a flow diagram and a FORTRAN listing of a working subroutine. The method is shown to be computationally efficient and reliable for the solution of a class of problems in the field of solid mechanics. Potential use of the method for the solution of magnetostatic problems is indicated.
INTRODUCTIONA large number of problems ranging from beams to shells in solid mechanics can be formulated as a two-point boundary value problem governed by a set of linear first-order ordinary differential equations in the interval s1 s s s sz. y(s) is an n-dimensional vector of dependent variables, A(s) is an n x n coefficient matrix, and p(s) is an n-dimensional vector of nonhomogeneous (loading) terms. The boundary conditions at the two termini along s may be written as: at s = sl, any n/2 elements of y(s) are specified, and at s = s2, any n/2 elements of y(s) are also specified. Thus the formulation indicates a method for solving problems with mixed nonhomogeneous boundary conditions. Further it may be noted that although two-point boundary value problems are discussed in this paper, multipoint boundary value problems may be treated by the same technique.Boundary value problems in ordinary differential equations are meant not only for formulations for one-dimensional problems but also for formulations which approximate solutions of certain two-and three-dimensional problems in solid mechanics whose behaviour is governed by a system of partial differential equations. The use of the well-known beam functions in the case of thin plates and shells in one of the surface co-ordinate directions, double Fourier series for thick plates and shells, Harmonic analysis for shells of revolution, etc., lists a few of such formulations. It may be noted that any boundary value problem for higher order differential i-Assistant Professor, presently a Visitor in the Department of Civil Engineering, University of Wales, Swansea, U.K. equations may always be written as a system of a coupled system of first-order equations as defined by equation (1). From the available literature on the subject one finds that the state-of-the-art, both from the analytical and numerical viewpoints, on the solution of boundary value problems is not as well established as it is for the initial value problems in ordinary differential equations where there are a number of successful and well-tested algorithms for numerical solution. This is precisely the reason why there are a large number of methods for the numerical solution of boun...
