Email this article 
SPECTRAL METHODS FOR SOLVING DIFFERENTIAL AND FUNCTIONAL EQUATIONS
- Authors: Varin V.P1
-
Affiliations:
- Keldysh Institute of Applied Mathematics RAS
- Issue: Vol 64, No 5 (2024)
- Pages: 713-728
- Section: General numerical methods
- URL: https://ruspoj.com/0044-4669/article/view/665074
- DOI: https://doi.org/10.31857/S0044466924050022
- EDN: https://elibrary.ru/YDNGIE
- ID: 665074
Cite item
Open Access
Access granted
Subscription or Fee Access
Abstract
The operator approach previously developed for the spectral method using Legendre polynomials is generalized here to any systems of basis functions (not necessarily orthogonal) that satisfy two conditions: the result of the operation of multiplication by x or differentiation with respect to x is expressed in the same functions. All systems of classical orthogonal polynomials meet these conditions. In particular, a spectral method utilizing Chebyshev polynomials is constructed, which is most efficient for numerical calculations. This method is applied for the numerical solution of linear functional equations that arise in generalized series summation problems, aswell as in problems of analytic continuation of discrete mappings. It is also shown how these methods solve nonstandard and nonlinear boundary value problems for which conventional algorithms are not applicable.
About the authors
V. P Varin
Keldysh Institute of Applied Mathematics RAS
Email: varin@keldysh.ru
Moscow, Russia
References
- Варин В.П. Аппроксимация дифференциальных операторов с учетом граничных условий //Ж. вычисл. матем. и матем. физ. 2023. Т 63. №8. С. 1251-1271.
- Варин В.П. Аппроксимация дифференциальных операторов с учетом граничных условий // Препринты ИПМ им. М.В. Келдыша. 2022. № 77.
- Wilf H.S. Mathematics for the physical sciences. New-York. Wiley. 1962.
- Gantmacher F.R. Application of the Theory of Matrices. New-York. Chelsea Press. 1960.
- Boyd J.P., Petschek R. The Relationships Between Chebyshev, Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions // J. of Scientific Computing. 2014. V. 59. P. 1-27.
- Варин В.П. Факториальное преобразование некоторых классических комбинаторных последовательностей //Ж. вычисл. матем. и матем. физ. 2018. Т. 59. № 6. С. 1747-1770.
- Pashkovskii S. Computational Application of Chebyshev Polynomials and Series Moscow. Nauka. 1983. [in Russsian].
- Варин В.П. Инвариантные кривые некоторых дискретных динамических систем // Ж. вычисл. матем. и матем. физ. 2022. Т. 62. № 2. С. 199-216.
- Варин В.П. Функциональное суммирование рядов // Ж. вычисл. матем. и матем. физ. 2023. Т. 63. № 1. С. 3-17.
Supplementary files
