Discrete-Continuous Methods for Solution

Abstract

Analytical-numerical methods for solving boundary-value and boundary-value eigenvalue problems for systems of ordinary differential equations and partial differential equations with variable coefficients are presented. In order to solve one-dimensional problems, the discrete-orthogonalization method is proposed. This approach is based on reducing the boundary-value problem to a number of Cauchy problems followed by their orthogonalization at some points of the integration interval which provides stability of calculations. In the case of the boundary-value eigenvalue problems, this approach is employed in combination with the incremental search method. In order to solve two-dimensional problems, the original system of partial differential system is reduced to systems of ordinary differential equations while making use of spline approximation and solving them by the discrete-orthogonalization method. Employing spline-functions is favorable, first, because of stability with respect to local disturbances, i.e., in contrast to polynomial approximation the spline behavior in the vicinity of a point does not influence the spline behavior as a whole; second, more satisfactory convergence is achieved, in contrast to the case of polynomials being applied as approximation functions; third, simplicity and convenience in calculation and implementation of spline-functions with the help of modern computers results. Besides, a nontraditional approach to solving problems of that class is proposed. It makes use of discrete Fourier series, i.e., Fourier series for functions specified on the discrete set of points. The two-dimensional boundary-value problem is solved by reducing it to a one-dimensional one after introducing auxiliary functions and separation of variables by using discrete Fourier series. Taking into account the calculation possibilities of modern computers, which make it possible to calculate a large number of series terms, the problem can be solved with high accuracy.