A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation

Normurodov, Ch.B., Shakaeva, E.E., Ziyakulova, Sh.A., Нормуродов, Ч.Б., Шакаева, Э.Э., Зиякулова, Ш.А.

Ҳисоблаш ва амалий математика муаммолари · 2025-yil

Annotatsiya

The article presents a numerical modeling approach for solving a non-homogeneous singularly perturbed fourth-order equation using by a discrete variant of the pre integration method. In the proposed method, the basis functions are Chebyshev poly nomials of the first kind. The highest derivative of the differential equation is expanded into a finite series in terms of these polynomials, with unknown expansion coefficients. All lower derivatives and the solution of the differential equation are represented in the form of the chosen expansion. The resulting series are substituted into the differential equation, leading to a system of discrete equations. For Chebyshev polynomials, there is a discrete integration formula that reduces the order of the derivative. By applying this formula, the discrete system is integrated four times, resulting in a system of algebraic equations where the number of equations is less than the number of unknown coefficients. The missing equations are obtained from the boundary conditions of the problem. The integration operation improves the smoothness of the approximating polynomials. For example, a zeroth-order polynomial, after four integrations, transforms into a fourth order polynomial. Numerical calculations demonstrate the high accuracy and efficiency of the proposed method, especially for very small values of the parameter e.

Maqola ma’lumotlari
MualliflarNormurodov, Ch.B., Shakaeva, E.E., Ziyakulova, Sh.A., Нормуродов, Ч.Б., Шакаева, Э.Э., Зиякулова, Ш.А.
JurnalҲисоблаш ва амалий математика муаммолари
NashriyotТошкент ахборот технологиялари университети қошидаги, Ахборот-коммуникация технологиялари илмий-инновацион, маркази
ISSN2181-8460, 2181-046X
Nashr sanasi2025-05-15
Son2
Betlar74-86
TilIngliz
Ilmiy sohaFizika-matematika
DOI10.71310/pcam.2_64.2025.07
Indeksga qo‘shilgan2026-08-27

DOI: 10.71310/pcam.2_64.2025.07 · Maqolaning asl sahifasi

Kalit so‘zlar

Chebyshev polynomials, discrete variant, pre-integration, high accuracy, small parameter, Полиномы Чебышева, дискретный вариант, предварительное интегрирование, высокая точность, малый параметр

Ilmiy soha

Fizika-matematika

Iqtibos keltirish

APANormurodov, Ch.B., Shakaeva, E.E., Ziyakulova, Sh.A., Нормуродов, Ч.Б., Шакаева, Э.Э., & Зиякулова, Ш.А. (2025). A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation. Ҳисоблаш ва амалий математика муаммолари, (2), 74-86. https://doi.org/10.71310/pcam.2_64.2025.07
GOST (OAK)Normurodov, Ch.B. va b. A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation // Ҳисоблаш ва амалий математика муаммолари. – 2025. – № 2. – P. 74-86. – DOI: 10.71310/pcam.2_64.2025.07.
MLANormurodov, Ch.B., Shakaeva, E.E., Ziyakulova, Sh.A., Нормуродов, Ч.Б., Шакаева, Э.Э., and Зиякулова, Ш.А. “A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation.” Ҳисоблаш ва амалий математика муаммолари, (2), 74-86, 2025. https://doi.org/10.71310/pcam.2_64.2025.07
ChicagoNormurodov, Ch.B., Shakaeva, E.E., Ziyakulova, Sh.A., Нормуродов, Ч.Б., Шакаева, Э.Э., and Зиякулова, Ш.А. “A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation.” Ҳисоблаш ва амалий математика муаммолари, (2), 74-86 (2025). https://doi.org/10.71310/pcam.2_64.2025.07
HarvardNormurodov, Ch.B., Shakaeva, E.E., Ziyakulova, Sh.A., Нормуродов, Ч.Б., Шакаева, Э.Э., and Зиякулова, Ш.А (2025) ‘A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation’, Ҳисоблаш ва амалий математика муаммолари, (2), 74-86. https://doi.org/10.71310/pcam.2_64.2025.07
BibTeX
@article{normurodovchb2025,
  title = {A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation},
  author = {Normurodov, Ch.B. and Shakaeva, E.E. and Ziyakulova, Sh.A. and Нормуродов, Ч.Б. and Шакаева, Э.Э. and Зиякулова, Ш.А.},
  journal = {Ҳисоблаш ва амалий математика муаммолари},
  year = {2025},
  number = {2},
  pages = {74-86},
  doi = {10.71310/pcam.2_64.2025.07},
  url = {https://doi.org/10.71310/pcam.2_64.2025.07}
}

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