Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method

Cover Page

Cite item

Full Text

Abstract

Earlier we developed a stable fast numerical algorithm for solving ordinary differential equations of the first order. The method based on the Chebyshev collocation allows solving both initial value problems and problems with a fixed condition at an arbitrary point of the interval with equal success. The algorithm for solving the boundary value problem practically implements a single-pass analogue of the shooting method traditionally used in such cases. In this paper, we extend the developed algorithm to the class of linear ODEs of the second order. Active use of the method of integrating factors and the d’Alembert method allows us to reduce the method for solving second-order equations to a sequence of solutions of a pair of first-order equations. The general solution of the initial or boundary value problem for an inhomogeneous equation of the second order is represented as a sum of basic solutions with unknown constant coefficients. This approach ensures numerical stability, clarity, and simplicity of the algorithm.

Full Text

1. Introduction The paper studies a method for solving linear ordinary differential equations (ODEs) of the second order using integrating factors [1-3]. The method of integrating factors in combination with the Chebyshev collocation method [4] was previously applied by the authors to solve first-order ODEs (of general form) [5]. Moreover, the Chebyshev collocation method was successfully applied by the authors to solve second-order linear ODEs (LODEs) using both differentiation matrices [6] and integration matrices [7]. K.P. Lovetskiy et al. developed and applied a modified Chebyshev collocation method, which turned out to be not only more reliable, but also significantly more efficient compared to previous versions of the collocation method and other Runge-Kutta-type methods (see [5-9]) or shooting method [10]. At the first stage of the two-stage modified method proposed by the authors, when expanding the approximate solution in Chebyshev polynomials (of the first or second kind), the corresponding special Gauss-Chebyshev-Lobatto grids are used, on which the search for a part of the coefficients of the approximate general solution of the ODE is reduced to solving non-degenerate and wellconditioned (with diagonal matrices) System of Linear Algebraic Equations (SLAE). At the second stage, the solution is refined by using correctly formulated initial (or boundary) conditions. In this case, the SLAE with a positive definite diagonal matrix is solved first, and then the low-dimensional (one- or two-dimensional) SLAE is solved with respect to the first coefficients of the expansion of the solution in Chebyshev polynomials. The method allows solving with equal efficiency both initial problems and problems with conditions at arbitrary points, previously solved, e.g., by the shooting method, which thus loses its relevance. Thus, we propose a constructive algorithm for approximate numerical solution of a wide class of LODEs. At the same time, the stage of the algorithm, consisting of solving the SLAE with a diagonal matrix, actually does not require computational costs, because it is reduced to a set of a small number of the simplest computational procedures. And only at the final stage, comprising the calculation of the first pair of coefficients of the expansion of the final particular solution, it is necessary to solve two-dimensional linear algebraic systems of equations determined by the initial or boundary conditions. The method has proven itself to be perfect in solving one-point problems for first-order ODEs (see [5, 8, 9]). The application of the modified Chebyshev collocation method to solving second-order ODEs has also demonstrated high efficiency. We solve two-point problems for second-order linear ODEs using [11] the two-stage Chebyshev collocation method. The first stage is devoted to finding an approximate solution to the ODE in the form of a Chebyshev polynomial [12] with undetermined first coefficients. At the second stage, the first coefficients (if they exist) are found by solving a 2 × 2 SLAE [5-7, 13]. The first stage can be implemented in several not entirely equivalent ways [14]. Ref. [6] presents the Chebyshev collocation method for obtaining a solution to a second order LODE using the Chebyshev differentiation matrix [15]. The paper [7] implements the Chebyshev collocation method for obtaining a solution to a second order LODE using the Chebyshev antidifferentiation matrix. The authors noted that constructing a general (complete) solution from the individual partial solutions of the LODE obtained in this way seems to be a computationally complex task. At the same time, using an intermediate method that makes use of integrating factors to reduce the LODE to the form of a total derivative allows one to obtain general (complete) solutions of the second-order LODE more efficiently. In the present article we seek approximate solutions of linear second-order ODEs of a rather general form
×

About the authors

Konstantin P. Lovetskiy

RUDN University

Email: lovetskiy-kp@rudn.ru
ORCID iD: 0000-0002-3645-1060

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Mikhail D. Malykh

RUDN University; Joint Institute for Nuclear Research

Email: malykh-md@rudn.ru
ORCID iD: 0000-0001-6541-6603
Scopus Author ID: 6602318510
ResearcherId: P-8123-2016

Doctor of Physical and Mathematical Sciences, Head of the Department of Computational Mathematics and Artificial Intelligence of RUDN University

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

Leonid A. Sevastianov

RUDN University; Joint Institute for Nuclear Research

Email: sevastianov-la@rudn.ru
ORCID iD: 0000-0002-1856-4643

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Computational Mathematics and Artificial Intelligence of RUDN University, Leading Researcher of Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

Stepan V. Sergeev

RUDN University

Author for correspondence.
Email: 1142220124@rudn.ru
ORCID iD: 0009-0004-1159-4745

PhD student of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

References

  1. Tenenbaum, M. & Pollard, H. Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences (Dover, Mineola, New York, 1986).
  2. Yeomans, J. M. Complex Numbers and Differential Equations. Lecture Notes for the Oxford Physics course 2014.
  3. Binney, J. J. Complex Numbers and Ordinary Differential Equations. Lecture Notes for the Oxford Physics course 2002.
  4. Boyd, J. P. Chebyshev and Fourier Spectral Methods 2nd. 665 pp. (Dover, Mineola, New York, 2000).
  5. Sevastianov, L. A., Lovetskiy, K. P. & Kulyabov, D. S. Multistage collocation pseudo-spectral method for the solution of the first order linear ODE Russian. in 2022 VIII International Conference on Information Technology and Nanotechnology (ITNT) (2022), 1-6. doi: 10.1109/ITNT55410.2022.9848731.
  6. Lovetskiy, K. P., Kulyabov, D. S., Sevastianov, L. A. & Sergeev, S. V. Multi-stage numerical method of collocations for solving second-order ODEs. Russian. Tomsk State University Journal of Control and Computer Science 2023, 45-52. doi: 10.17223/19988605/63/6 (2023).
  7. Lovetskiy, K. P., Kulyabov, D. S., Sevastianov, L. A. & Sergeev, S. V. Chebyshev collocation method for solving second order ODEs using integration matrices. Discrete and Continuous Models and Applied Computational Science 31, 150-163. doi: 10.22363/2658-4670-2023-31-2-150-163 (2023).
  8. Lovetskiy, K. P., Sevastianov, L. A., Kulyabov, D. S. & Nikolaev, N. E. Regularized computation of oscillatory integrals with stationary points. Journal of Computational Science 26, 22-27. doi: 10.1016/j.jocs.2018.03.001 (2018).
  9. Lovetskiy, K. P., Sevastianov, L. A., Hnatič, M. & Kulyabov, D. S. Numerical Integration of Highly Oscillatory Functions with and without Stationary Points. Mathematics 12, 307. doi: 10.3390/math12020307 (2024).
  10. Zill, D. G. & Cullen, M. R. Differential Equations with Boundary-Value Problems. Cengage Learning 2008.
  11. Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. Spectral Methods: Fundamentals in Single Domains (Springer Verlag, 2006).
  12. Mason, J. C. & Handscomb, D. C. Chebyshev polynomials doi: 10.1201/9781420036114 (Chapman and Hall/CRC Press, 2002).
  13. Lovetskiy, K. P., Sergeev, S. V., Kulyabov, D. S. & Sevastianov, L. A. Application of the Chebyshev collocation method tosolve boundary value problems of heat conduction. Discrete and Continuous Models and Applied Computational Science 32, 74-85. doi: 10.22363/2658-4670-2024-32-1-74-85 (2024).
  14. Fornberg, B. A Practical Guide to Pseudospectral Methods (Cambridge University Press, New York, 1996).
  15. Amiraslani, A., Corless, R. M. & Gunasingam, M. Differentiation matrices for univariate polynomials. Numerical Algorithms 83, 1-31. doi: 10.1007/s11075-019-00668-z (2023).
  16. Lebl, J. Notes on Diffy Qs: Differential Equations for Engineers 2024.
  17. Trench, W. F. Elementary Differential Equations with Boundary Value Problems 2013.
  18. Polyanin, A. D. & V.F. Zaitsev, V. F. Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems (CRC Press, Boca Raton-London, 2018).
  19. Schekochihin, A. A. Lectures on ordinary differential equations. Lectures for the Oxford 1st-year undergraduate physics course, paper CP3/4 2018.
  20. Tikhonov, A. N., Vasil’eva, A. B. & Sveshnikov, A. G. Differential Equations (Springer, Berlin, 1985).
  21. Sevastianov, L. A., Lovetskiy, K. P. & Kulyabov, D. S. A new approach to the formation of systems of linear algebraic equations for solving ordinary differential equations by the collocation method. Russian. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics 23, 36-47. doi: 10.18500/1816-9791-2023-23-1-36-47 (2023).

Copyright (c) 2024 Lovetskiy K.P., Malykh M.D., Sevastianov L.A., Sergeev S.V.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies