Numerical integration by the matrix method and evaluation of the approximation order of difference boundary value problems for non-homogeneous linear ordinary differential equations of the fourth order with variable coefficients

V. N. Maklakov ; M. A. Ilicheva

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 1, pp. 137-162

Résumé

The use of the second degree Taylor polynomial in approximation of derivatives by finite difference method leads to the second order approximation of the traditional grid method for numerical integration of boundary value problems for non-homogeneous linear ordinary differential equations of the second order with variable coefficients. The study considers a previously proposed method of numerical integration using matrix calculus which didn’t include the approximation of derivatives by finite difference method for boundary value problems of non-homogeneous fourth-order linear ordinary differential equations with variable coefficients. According to this method, when creating a system of difference equations, an arbitrary degree of the Taylor polynomial can be chosen in the expansion of the sought-for solution of the problem into a Taylor series. In this paper, the possible boundary conditions of a differential boundary value problem are written both in the form of derived degrees from zero to three, and in the form of linear combinations of these degrees. The boundary problem is called symmetric if the numbers of the boundary conditions in the left and right boundaries coincide and are equal to two, otherwise it is asymmetric. For a differential boundary value problem, an approximate difference boundary value problem in the form of two subsystems has been built. The first subsystem includes equations for which the boundary conditions of the boundary value problem were not used; the second one includes four equations in the construction of which the boundary conditions of the problem were used. Theoretically, the patterns between the order of approximation and the degree of the Taylor polynomial were identified. The results are as follows: a) the approximation order of the first and second subsystems is proportional to the degree of the Taylor polynomial used; b) the approximation order of the first subsystem is two units less than the degree Taylor polynomial with its even value and three units less with its odd value; c) the approximation order of the second subsystem is three units less than the degree Taylor polynomial regardless of both even-parity or odd-parity of this degree, and the degree of the highest derivative in the boundary conditions of the boundary value problem. The approximation order of the difference boundary value problem with all possible combinations of boundary conditions is calculated. The theoretical conclusions are confirmed by numerical experiments.

Détail

Keywords: ordinary differential equations, boundary value problems, approximation order, numerical methods, Taylor series.