The reduction of a multipoint problem for a system of differential equations to cauchy problems by the method of orthogonal transformations

Abstract

IN the numerical solution of boundary and multipoint problems for a set of first-order ordinary linear differential equations the general solution is usually found in the form of a linear combination of solutions with arbitrary constant coefficients, forming a fundamental set, and of the particular solution of a non-homogeneous set of equations. In those cases where the solution of the boundary or multipoint problem is not a rapidly increasing function, but the solution of the corresponding homogeneous set, on the other hand, increases rapidly and the interval of values of the argument is large, such a linear combination may turn out to be unadvantageous from the point of view of calculation, since it involves the loss of significant digits. To avoid these difficulties other methods of reducing boundary and multipoint problems to initial problems have been put forward [1–4]. In this paper we shall consider one such method. In its general idea the method put forward here is close to those which are considered in [1, 2], and differs from them in a new principle for constructing the equations for the transfer of boundary conditions. This paper is also a generalization of the results of [5] to the case of an arbitrary number of first-order linear differential equations. We note that in the theory of differential equations orthogonal transformations are known, which reduce the system of equations to triangular form (with which we shall be essentially concerned). In these constructions, however, the fundamental system of solutions is used (see, for instance, [6], pp. 145–148), which, as has already been remarked, creates difficulties in the numerical solution of the problem. It appears to us that the method out forward here. of reducing multipoint problems to Cauchy problems, may find an application in practical calculations, and so in many cases we have reduced the argument to the description of concrete numerical algorithms and have given a fairly full description of all the necessary calculations.