INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 431
NUMERICAL MODELING OF THE BOUNDARY PROBLEM FOR A SINGULAR
EXCITED EQUATION WITH THE SPECTRAL METHOD
Razzakov Sherbek Tagaymurodvich
Termez state universette master of Applied Mathematics
Normurodov Chori Begalievich
scientific leader PhD
Annotation:
in this work, the issue of numerical solution of a boundary issue for a singular
excited differential equation is considered. In this case, a solution is built on the basis of
spectral methods and their accuracy, stability and efficiency in calculation are analyzed. The
results of the study show that spectral methods provide high-resolution results for singular
excited issues and can be useful tools in representing complex physical models.
Keywords:
singular excited equation, spectral method, boundary issue, numerical modeling,
high precision, finite layers
Singular excited equations are widely used in the fields of mathematical physics and
engineering, being particularly important in modeling problems with boundary layers and
very small parameters. Since it is difficult to solve these equations with a conventional
number of methods, special methods are used in their solution — in particular, spectral
methods. Spectral methods are characterized by providing high accuracy as well as being
efficient in computing. In this work, using spectral methods, the boundary issue of the
singular excited equation is modeled numerically, the results are analyzed and their accuracy
assessed. Singular excited differential equations (SQDT) are found in the modeling of many
physical, engineering and biological processes. These equations are defined in the presence of
small parameters, and their solutions usually have boundary layers. Traditional numerical
methods in determining these layers2. Literature Review Spectral methods, particularly
Chebyshev and Fourier-based approaches, are widely used in solving Sqdts. These methods
provide high-resolution solutions, but require special lattices or transformations if boundary
layers are present. Miller, O'riordan, & Shishkin (1996): provides a detailed account of
adapted nets and spectral methods for Sqdts. Roos, Stynes, & Tobiska (1996): provides
important results on numerical methods for Sqdts and their convergence. With the modern
hybrid approach, in recent years, spectral methods have been combined with other numerical
methods to develop hybrid approaches. These approaches provide more effective results in
solving Sqdts. Ravi Kanth & Kumar (2017): offers customized spline techniques for two-
parameter Sqdts. Rakmaiah & Phaneendra (2022): develops exponentially adapted schemes
for two boundary layer Sqdts.Khan &
Research Methodology The classical spectral approach is the issue of the problem of
singularly excited boundary value:\varepsilon y "(x) + A(x)y'(x) + b(x)y(x) = f (x), \quad x
\in(0,1), \quad y(0) = \alpha, \quad y (1) = \beta is a small parameter here and the solution has
boundary layers. Chebyshev spectral method Lattice selection: a lattice is constructed on the
basis of Chebyshev nodes. Representation of functions: the solution is expressed by
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 432
Chebyshev polynomials. Differential operators: are derivatives using Chebyshev differential
matrices. Boundary conditions: boundary conditions are added to Matrix equations. Solution:
the resulting system of linear equations is solved. A modern hybrid approach, which
considers the following singular excited boundary value issue:\varepsilon y "(x) + p(x)y'(x) +
q(x)y(x) = r (x), \quad x \in(0,1), \quad y(0) = a, \quad y (1) = B is a small execution here,
and the solution ha4.1. Classical Spectral Approach Precision: the Chebyshev spectral
method provides high-resolution solutions. Layers: in the presence of boundary layers,
special nets are required. Calculation cost: due to matrix operations, the calculation cost can
be high. The modern hybrid approach is precision: the Shishkin net and B-spline approach
more accurately represent layers.Flexibility: flexible for a variety of boundary conditions and
functions.Computational efficiency: the computational cost is relatively low and solutions are
obtained faster. In solving inference singular excited differential equations, spectral methods
are characterized by high accuracy and fast convergence properties. Classical spectral
methods, especially on the basis of Chebyshev polynomials, require special lattices or
transformations if there are boundary layers, although they are effective for simple cases.
Modern hybrid approaches, such as the basis of the Shishkin net and B-spline functions
This work addressed the issue of numerical modeling of the boundary issue for singular
excited equations using spectral methods. Singular excited equations are a special case of
ordinary differential equations, in the solutions of which, depending on a small parameter,
drastic changes occur, namely boundary layers. When solving such equations by
conventional numerical methods, problems of decreased accuracy and stability in calculation
arise. For this reason, spectral methods with high resolution and able to accurately represent
boundary properties give an advantage.In the course of the study, the theoretical foundations
of spectral methods were analyzed, the main elements of which — the system of orthogonal
functions, the Galerkin approach and the properties of complete smoothness-were studied in
detail. An adapted algorithm for the singular excited problem was developed and modeling
work was carried out in the MATLAB software environment.Results from Anan.Literature
used
List of bibliography:
1. Ismailov, A., Jalil, M. A., Abdullah, Z., & Abd Rahim, N. H. (2016, Aug. A comparative
study of stemming algorithms for use with the Uzbek language. In 2016 3rd International
conference on computer and information sciences (ICCOINS) (pp. 712). IEEE. 2. Jalil, M.
M., Ismailov, A., Abd Rahim, N. H., & Abdullah, Z. (2017). The Development of the Uzbek
Stemming Algorithm. Advanced Science Letters, 23 (5), 4171-4174. 3. Abdurakhmonova, N.,
Alisher, I., & Sayfulleyeva, R. (2022, September). MorphUz: Morphological Analyzer for the
Uzbek Language. In 2022 7th International Conference on Computer Science and
Engineering (UBMK) (pp. 61-66). IEEE. 4. Ismailov, A. S., & Zhurayev, Z. B. Study of
arduino microcontroller board. 5. Ismailov, A. S., Alijanov, D. D., Zhurayev, Z. B., &
Kurbanov, M. U. Research on renewable energy sources in Uzbekistan. 6. Ismailov, A. S.,
Shamsiyeva, G., Abdurakhmonova, N., & Navoi, A. Statistical machin
