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

封面

如何引用文章

全文:

详细

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.

作者简介

Konstantin 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 Malykh

RUDN University; Joint Institute for Nuclear Research

Email: malykh-md@rudn.ru
ORCID iD: 0000-0001-6541-6603
Scopus 作者 ID: 6602318510
Researcher ID: 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 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 Sergeev

RUDN University

编辑信件的主要联系方式.
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

参考

  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).

补充文件

附件文件
动作
1. JATS XML
下载