This paper presents a highly accurate and efficient spectral-grid method for the numerical solution of a boundary value problem for a fourth-order ordinary differential equation with a small parameter at the highest derivative. The method is based on Chebyshev polynomials of the first kind and involves partitioning the integration interval into multiple subintervals, ensuring the continuity of the solution and its derivatives up to the third order across the subintervals. This is particularly important in the pres ence of boundary layers that arise due to the small parameter at the highest derivative, rendering the problem singularly perturbed. An analysis of the method’s accuracy and stability is carried out at various levels of grid subdivision and numbers of approximating polynomials. Numerical experiments are presented, demonstrating high precision in the computation of both the solution and its derivatives up to the fourth order, even for very small values of the parameter (down to 10−9). The results show that increasing the number of grid elements and Chebyshev polynomials leads to a geometric reduc tion in absolute errors, confirming the convergence of the proposed approach. Thus, the proposed spectral-grid method is not only computationally efficient but also sufficiently robust and flexible for solving a wide range of problems involving singularly perturbed high-order differential equations.
| Mualliflar | Normurodov, Ch.B., Murodov, S.K., Shakaeva, E.E., Нормуродов, Ч.Б., Муродов, С.К., Шакаева, Э.Э. |
|---|---|
| Jurnal | Ҳисоблаш ва амалий математика муаммолари |
| Nashriyot | Тошкент ахборот технологиялари университети қошидаги, Ахборот-коммуникация технологиялари илмий-инновацион, маркази |
| ISSN | 2181-8460, 2181-046X |
| Nashr sanasi | 2025-05-15 |
| Son | 2 |
| Betlar | 54-63 |
| Til | Ingliz |
| Ilmiy soha | Fizika-matematika |
| DOI | 10.71310/pcam.2_64.2025.05 |
| Indeksga qo‘shilgan | 2026-08-27 |
DOI: 10.71310/pcam.2_64.2025.05 · Maqolaning asl sahifasi
spectral-grid method, number of elements and polynomials, small parameter, high accuracy, efficiency, maximum absolute errors, спектрально-сеточный метод, число элементов и полиномов, малый параметр, высокая точность, эффективность, максимальные абсолютные по грешности
| APA | Normurodov, Ch.B., Murodov, S.K., Shakaeva, E.E., Нормуродов, Ч.Б., Муродов, С.К., & Шакаева, Э.Э. (2025). Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative. Ҳисоблаш ва амалий математика муаммолари, (2), 54-63. https://doi.org/10.71310/pcam.2_64.2025.05 |
|---|---|
| GOST (OAK) | Normurodov, Ch.B. va b. Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative // Ҳисоблаш ва амалий математика муаммолари. – 2025. – № 2. – P. 54-63. – DOI: 10.71310/pcam.2_64.2025.05. |
| MLA | Normurodov, Ch.B., Murodov, S.K., Shakaeva, E.E., Нормуродов, Ч.Б., Муродов, С.К., and Шакаева, Э.Э. “Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative.” Ҳисоблаш ва амалий математика муаммолари, (2), 54-63, 2025. https://doi.org/10.71310/pcam.2_64.2025.05 |
| Chicago | Normurodov, Ch.B., Murodov, S.K., Shakaeva, E.E., Нормуродов, Ч.Б., Муродов, С.К., and Шакаева, Э.Э. “Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative.” Ҳисоблаш ва амалий математика муаммолари, (2), 54-63 (2025). https://doi.org/10.71310/pcam.2_64.2025.05 |
| Harvard | Normurodov, Ch.B., Murodov, S.K., Shakaeva, E.E., Нормуродов, Ч.Б., Муродов, С.К., and Шакаева, Э.Э (2025) ‘Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative’, Ҳисоблаш ва амалий математика муаммолари, (2), 54-63. https://doi.org/10.71310/pcam.2_64.2025.05 |
| BibTeX | @article{normurodovchb2025,
title = {Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative},
author = {Normurodov, Ch.B. and Murodov, S.K. and Shakaeva, E.E. and Нормуродов, Ч.Б. and Муродов, С.К. and Шакаева, Э.Э.},
journal = {Ҳисоблаш ва амалий математика муаммолари},
year = {2025},
number = {2},
pages = {54-63},
doi = {10.71310/pcam.2_64.2025.05},
url = {https://doi.org/10.71310/pcam.2_64.2025.05}
} |
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