Convergence of the matrix method of numerical integration of the boundary value problems for linear nonhomogeneous ordinary differential second order equations with variable coefficients

The problems of stability and convergence of previously proposed matrix method of numerical integration of boundary value problems with boundary conditions of the first, second and third kinds of nonhomogeneous linear ordinary differential second order equations with variable coeffcients are considered. Using of the Taylor polynomials of arbitrary degrees allowed to increase the approximation order of the method to an arbitrary natural number and to refuse from the approximation of derivatives by finite differences. When choosing the second degree Taylor polynomials the equation of the method coincided with the known equations of the traditional method of numerical integration of the boundary value problems where the derivatives are approximated by finite differences. It was shown that a suffcient criterion of stability when used in the method of Taylor polynomials of the third degree and more coincides with the suffcient criterion of stability of the traditional grid method for the numerical integration of boundary value problems with boundary conditions of the first, second and third kind. Theoretically, it is established that the degree of convergence of the matrix method for integration of boundary value problems with boundary conditions of the first kind is proportional to the degree of the used Taylor polynomials in the case, when the degree is even, and is proportional to the number that is one less than the degree if it is odd; when integrating the boundary value problems with boundary conditions of the second and third kind the degree of convergence of the method is proportional to the degree of the used Taylor polynomials regardless of its parity and one less than it. The obtained theoretical results are confirmed by numerical experiments.

ordinary differential equations, boundary value problems, approximation order, stability, convergence, numerical methods, Taylor polynomials