The evaluation of the order of approximation of~the~matrix method for numerical integration

V. N. Maklakov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 3, pp. 389-409.

Résumé

We present the first message of the cycle from two articles where the rearrangement of the order of approximation of the matrix method of numerical integration depending on the degree in the Taylor’s polynomial expansion of solutions of boundary value problems for systems of ordinary differential equations of the second order with variable coefficients with boundary conditions of the first kind were investigated. The Taylor polynomial of the second degree use at the approximation of derivatives by finite differences leads to the second order of approximation of the traditional method of nets. In the study of boundary value problems for systems of ordinary differential equations of the second order we offer the previously proposed method of numerical integration with the use of matrix calculus where the approximation of derivatives by finite differences was not performed. According to this method a certain degree of Taylor polynomial can be selected for the construction of the difference equations system. The disparity is calculated and the order of the method of approximation is assessed depending on the chosen degree of Taylor polynomial. It is theoretically shown that for the boundary value problem with boundary conditions of the first kind the order of approximation method increases with the degree of the Taylor polynomial and is equal to this degree only for its even values. For odd values of the degree the order of approximation is less by one. The theoretical conclusions are confirmed by a numerical experiment for boundary value problems with boundary conditions of the first kind.

Détail

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

Bibliographie

[1] Keller H. B., “Accurate Difference Methods for Nonlinear Two-point Boundary Value Problems”, SIAM J. Numer. Anal., 11:2 (1974), 305–320 | DOI | MR | Zbl

[2] Lentini M., Pereyra V., “A Variable Order Finite Difference Method for Nonlinear Multipoint Boundary Value Problems”, Mathematics of Computation, 28:128 (1974), 981–1003 | DOI | MR | Zbl

[3] Keller H. B., “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations: Survey and Some Resent Results on Difference Methods”, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Academic Press, New York, 1975, 27–88 | DOI | MR

[4] Godunov S. K., Ryaben'kii V. S., Raznostnye skhemy [Difference Scheme], Nauka, Moscow, 1977, 439 pp. (In Russian) | MR

[5] Formaleev V. F., Reviznikov D. L., Chislennye metody [Numerical methods], Fizmatlit, Moscow, 2004, 400 pp. (In Russian)

[6] Samarskii A. A., Teoriia raznostnykh skhem [The Theory of Difference Schemes], Nauka, Moscow, 1977, 656 pp. (In Russian) | MR

[7] Samarskii A. A., Gulin A. V., Chislennye metody [Numerical methods], Nauka, Moscow, 1973, 432 pp. (In Russian) | MR

[8] Samarskii A. A., Gulin A. V., Ustoichivost' raznostnykh skhem [The stability of difference schemes], Nauka, Moscow, 1973, 416 pp. (In Russian)

[9] Boutayeb A., Chetouani A., “Global extrapolations of numerical methods for solving a parabolic problem with non local boundary conditions”, International Journal of Computer Mathematics, 80:6 (2003), 789–797 | DOI | MR | Zbl

[10] Boutayeb A., Chetouani A., “A Numerical Comparison of Different Methods Applied to the Solution of Problems with Non Local Boundary Conditions”, Applied Mathematical Sciences, 1:44 (2007), 2173–2185 http://www.m-hikari.com/ams/ams-password-2007/ams-password41-44-2007/boutayebAMS41-44-2007.pdf | MR

[11] Radchenko V. P., Usov A. A., “Modified grid method for solving linear differential equation equipped with variable coefficients based on Taylor series”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2008, no. 2(17), 60–65 (In Russian) | DOI

[12] Maklakov V. N., “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”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, no. 36, 143–160 (In Russian) | DOI

[13] Samarskii A. A., Nikolaev E. S., Metody resheniia setochnykh uravnenii [Methods for solving grid equations], Nauka, Moscow, 1978, 592 pp. (In Russian) | MR

[14] Ryaben'kii V. S., “Necessary and sufficient conditions for good definition of boundary value problems for systems of ordinary difference equations”, U.S.S.R. Comput. Math. Math. Phys., 4:2 (1964), 43–61 | DOI | MR | Zbl

[15] Fikhtengol'ts G. M., Kurs differentsial'nogo i integral'nogo ischisleniia [Course of Differential and Integral Calculus], v. 1, Nauka, Moscow, 1970, 608 pp. (In Russian)

[16] Kurosh A. G., Kurs vysshei algebry [A Course of Higher Algebra], Nauka, Moscow, 1975, 431 pp. (In Russian) | MR

[17] Turchak L. I., Osnovy chislennykh metodov [Fundamentals of numerical methods], Nauka, Moscow, 1987, 320 pp. (In Russian) | MR

[18] Zaks L., Statisticheskoe otsenivanie [Statistical estimation], Statistika, Moscow, 1976, 598 pp. (In Russian) | MR