 # Chebyshev collocation method for solving second order ODEs using integration matrices

## Abstract

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

## Full Text

1. Introduction Ordinary differential equations (ODEs) and systems of ODEs of the second order describe most problems in classical mechanics. Most oscillatory processes are described by second order ODEs or systems of ODEs. Second order ODE systems describe a number of optical diffraction problems (see, for example, ). The model of adiabatic guided wave propagation of polarized light in integrated optical waveguides is also described by a system of two coupled oscillators [2-4]. There are many different methods for exact and approximate solution of initial/boundary value problems for different classes of second order ordinary differential equations. Among them, the spectral methods of expansion in Chebyshev polynomials consistently occupy a well-deserved place. In 1991, L. Greengard  formulated a method for solving a two-point boundary value problem for second order ODEs with constant coefficients, based on expanding the solution into a series of Chebyshev polynomials of the first kind. The method became stably referred to as the pseudospectral collocation method. In the same paper, mathematical constructions were introduced, which later received the names “differentiation matrix” and “integration matrix” (or “antidifferentiation matrix”). A detailed description of the properties of matrices that determine the relationship between the expansion coefficients in a series of approximated functions and the expansion coefficients of their derivatives and antiderivatives in the same set of basis functions is given in . Greengard obtained estimates for the norms of these matrices and their condition numbers - large values for differentiation matrices and small values for integration (antidifferentiation) matrices. Despite the poor conditionality of differentiation matrices, many authors used them to solve initial and boundary problems for ODEs of various orders. This is explained by the more familiar and therefore ‘convenient’ representation of physical models using the language of mathematical formulas. The instability of widely used [7, 8] algorithms has been overcome by applying methods of preconditioning to the corresponding systems of linear algebraic equations. As a result of numerous studies, methods based on integration matrices in the physical space and in the spectral representation turned out to be the most preferable . It is important to note that none of the applied methods for solving ODEs based on Chebyshev integration matrices [9, 10] allows obtaining systems of linear equations with sparse matrices . The dense filling of matrices is a consequence of attempts to introduce boundary conditions into the system of linear algebraic equations along with differential relations . The high sparseness of the matrices can be maintained by improving the algorithm by switching to the two-stage method. In this case, at the first stage, differential conditions are considered, which allow fixing the leading coefficients in the expansion of the solution into a series, thus defining the ‘general solution’. The next step uses boundary/initial conditions to determine a pair (for second order equations) of missing coefficients. This makes it possible to obtain a complete set of expansion coefficients for the desired ‘particular’ solution. The results of studies  demonstrate that the method of Chebyshev collocation that ensures the best accuracy in solving initial and boundary value problems is the method using Chebyshev integration matrices in the spectral space. This approach effectively relies on the use of operations with sparse matrices and its computational costs are quite comparable with the Fourier spectral discretization. 2. Setting of the problem We consider an approximate solution to the two-point boundary value problem for the second-order differential equation having the form 
×

### 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 Applied Probability and Informatics

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

### Dmitry S. Kulyabov

RUDN University; Joint Institute for Nuclear Research

Author for correspondence.
Email: kulyabov-ds@rudn.ru
ORCID iD: 0000-0002-0877-7063

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 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 Applied Probability and Informatics of Peoples’ Friendship University of Russia named after Patrice Lumumba (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, Moscow Region, 141980, Russian Federation

### Stepan V. Sergeev

RUDN University

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

PhD student of Department of Applied Probability and Informatics

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

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