Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order

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Abstract

The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, ‘inverse’ with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.

About the authors

Konstantin P. Lovetskiy

Peoples’ Friendship University of Russia (RUDN University)

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

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

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

Dmitry S. Kulyabov

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research

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

Docent, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Applied Probability and Informatics

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

Ali Weddeye Hissein

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: 1032209306@rudn.ru
ORCID iD: 0000-0003-1100-4966

student of Department of Applied Probability and Informatics

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

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