Numerical integration by the matrix method of boundary value problems for linear inhomogeneous ordinary differential equations of the third order with variable coefficients

V. N. Maklakov ; Ya. G. Stelmakh

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 1, pp. 153-183

Résumé

The use of the Taylor polynomial of the second degree when approximating the derivatives by finite differences leads to the second order of approximation of the traditional method of nets in the numerical integration of second-order ordinary differential equations with variable coefficients. In the study of boundary-value problems for the third-order ordinary differential equations with variable coefficients, we offer the previously proposed method of numerical integration, using the means of the matrix calculus, in which approximation of the derivatives by finite differences was not used. According to this method, in the construction of a system of difference equations, an arbitrary power of the Taylor polynomial in the expansion of the desired solution of the problem in a Taylor series can be chosen. The disparity is calculated and an estimate of the order of approximation of the method is given depending on the chosen degree of the Taylor polynomial using the four-point pattern. The regularities between the order of approximation of the matrix method and the degree of the used Taylor polynomial are theoretically revealed. We found out that the order of approximation is proportional to the degree of the used Taylor polynomial and less by two than it. We propose a procedure for constructing a fictitious boundary condition that allows us to construct a closed system of difference equations for the matrix method of numerical integration. The system of difference equations is divided into two subsystems: the first subsystem consists of two equations, the first of which contains the given value of the derivative in the boundary conditions of the problem, the second one contains the value calculated from the fictitious boundary condition; the second subsystem consists of the remaining difference equations of the constructed closed system. The disparity is calculated and an estimate of the order of approximation of the method is given depending on the chosen degree of the Taylor polynomial using the five-point pattern. The regularities between the order of approximation of the matrix method and the degree of the used Taylor polynomial are theoretically revealed. The following is revealed: a) the order of approximation of the first subsystem, the second subsystem with an even value of the degree of the Taylor polynomial and the whole problem is proportional to this degree and less than it by two; b) the order of approximation of the second subsystem with an odd value of the degree of the Taylor polynomial is proportional to this degree and less than it by one.

Détail

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