Estimation of the Order of the Matrix Method Approximation of Numerical Integration of Boundary-Value Problems for Inhomogeneous Linear Ordinary Differential Equations of the Second Order

V. N. Maklakov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 3 (2014), pp. 143-160

Résumé

Using the first three terms of Taylor expansion of the required function in the approximate derivative by finite differences leads to the second order approximation of the traditional numerical quadrature method of boundary value problems for linear ordinary second order differential equations with variable coefficients. The paper shows previously proposed numerical quadrature method using tools of matrix calculus where the approximate derivative by finite differences was not used. Agreeing to above method the arbitrary number of terms of Taylor expansion for the required solution may be used when compiling the difference equation system. When using the three first terms of expansion the difference equation system coincided with the traditional system. The estimation of residuals and the order of approximation depending on the number of the used terms of Taylor expansion is given. It is theoretically shown that for the boundary value problem with boundary conditions of the first kind the approximation method order increases in direct proportion with the increasing in the number of members used in Taylor series expansion only for odd values of this number. For even values of this number the order of approximation coincides with the order of approximation for the number less by unit of the odd values. For boundary value problems with boundary conditions of the second and third kinds the order of approximation was directly proportional to the number of used terms in the Taylor series expansion of the required solution of the problem regardless of evenness. In these cases the order of approximation of the boundary points and therefore the whole problem turned out to be one unit less than the order for the inner points of the grid for the interval of integration. The method of approximation order increase at the boundary points up to the approximation order in the inner points of the grid is presented. The theoretical conclusions are confirmed by a numerical experiment for a boundary value problem with boundary conditions of the first and third kinds.

Détail

Keywords: ordinary differential equations of second order, boundary value problems, boundary conditions of the first, second and third types, order of approximation, numerical methods, Taylor polynomials.