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

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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.

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

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