NUMERICAL MODELING OF A SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM USING THE SPECTRAL METHOD

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 2022-2.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-10 337 NUMERICAL MODELING OF A SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM USING THE SPECTRAL METHOD Razzakov Sherbek Tog'aymurod o'g'li Normurodov Chori Begaliyevich. Termiz davlat universiteti. Abstract This paper presents a numerical investigation of singularly perturbed boundary value problems (SPBVPs) using the spectral method. These types of differential equations arise frequently in physics and engineering, where the presence of a small parameter leads to sharp gradients or boundary layers. Standard numerical methods often fail to capture these features efficiently. In this study, we apply Chebyshev spectral collocation techniques to approximate the solution of a singularly perturbed second-order differential equation. We analyze the accuracy, convergence, and stability of the method, and compare the results with exact or asymptotic solutions. Numerical experiments confirm that the spectral method provides highly accurate approximations even in the presence of strong boundary layers. Keywords Spectral method, Singular perturbation, Boundary value problem, Chebyshev collocation, Numerical modeling, Boundary layers, Convergence Introduction Singular perturbation problems arise in many scientific fields such as fluid dynamics, quantum mechanics, chemical kinetics, and control theory. These problems are characterized by the presence of a small positive parameter ( \varepsilon ) multiplying the highest derivative, which causes rapid variations in the solution near the boundary—commonly referred to as *boundary layers*. Conventional finite difference or finite element methods often require extremely fine meshes to resolve these layers, leading to increased computational cost. In contrast, spectral methods, which use global basis functions (typically orthogonal polynomials), can provide highly accurate solutions with fewer degrees of freedom. This paper focuses on the application of Chebyshev spectral methods to solve a linear, second-order, singularly perturbed boundary value problem. We aim to demonstrate that spectral collocation can yield accurate results without the need for fine meshing near the boundary layers. Methods Problem Definition We consider the following singularly perturbed boundary value problem (SPBVP): [ ISSN: 2582-4686 SJIF 2021-3.261,SJIF 2022-2.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-10 338 \varepsilon y''(x) - a(x)y'(x) + b(x)y(x) = f(x), \quad x \in (0,1) ] [ y(0) = \alpha, \quad y(1) = \beta ] where ( 0 < \varepsilon \ll 1 ) is a small parameter, and ( a(x), b(x), f(x) ) are given smooth functions. The small parameter ( \varepsilon ) causes boundary layers near ( x = 0 ) or ( x = 1 ), depending on the sign and behavior of ( a(x) ). Spectral Collocation Method We employ the Chebyshev-Gauss-Lobatto points: [ x_j = \frac{1}{2} \left( 1 - \cos\left( \frac{j \pi}{N} \right) \right), \quad j = 0,1,\dots,N ] Let ( y_N(x) ) be the approximation to ( y(x) ) using Chebyshev polynomials. We enforce the differential equation at all collocation points ( x_j ), excluding the boundary points. The second derivative and first derivative at the collocation points are computed using differentiation matrices ( D ) and ( D^2 ). The resulting system is: [ \varepsilon D^2 y - A D y + B y = f ] where ( y ), ( A ), ( B ), and ( f ) are vectors evaluated at the collocation points. Boundary conditions are imposed directly by replacing the first and last equations of the system. Implementation The algorithm was implemented in Python using NumPy and SciPy libraries. The Chebyshev differentiation matrices were generated using standard algorithms available in scientific computing literature. We considered several test problems, including: [ \varepsilon y''(x) - y'(x) = -1, \quad y(0) = 0, \quad y(1) = 1 ] ISSN: 2582-4686 SJIF 2021-3.261,SJIF 2022-2.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-10 339 for varying values of ( \varepsilon \in {10^{-1}, 10^{-3}, 10^{-6} } ). Singularities are divided into two kinds; regular singular points and irregular points. The point x=a0 is said to be regular singular point of (1.2) if (x−a0)F1(x) and (x−a0)2F2(x) are analytic at a0; otherwise x=a0 is an irregular singularity. We now consider finding the solution of singular boundary value problem having regular singularity given bysubject to boundary conditionswhere the coefficient functions p(x) and q(x) fail to be analytic at x=a0 and α, β are finite constants. Singular boundary value problems for ordinary differential equations arise very frequently in several areas of science and engineering. For example, consider the following boundary value problemwhich results from an analysis of heat conduction through a solid with heat generation. The function f(T) represents the heat generation within the solid. T is the temperature and the constant p is equal to 0, 1 or 2 depending on whether the solid is a plate, a cylinder or a sphere. The Thomas–Fermi model in atomic physics describes the charge concentration y(x) of electrons in an ion as In this paper, another version of cubic spline approximation [15] is examined for numerically solving singular BVP's. An advantage of the method is that the coefficient matrix of the system is of the Hessenberg form, and the method has an order of convergence O(h4), where h is the step size. In the neighbourhood of the singularity, we use the series expansion about the singular point, solving a regular boundary value problem over a reduced interval excluding the singular point, and match the solution to the expansion. Cubic spline procedure is developed for discretizing the resulting regular problem. Some numerical example have been solved to demonstrate the efficiency of the method Results In this paper, another version of cubic spline approximation [15] is examined for numerically solving singular BVP's. An advantage of the method is that the coefficient matrix of the system is of the Hessenberg form, and the method has an order of convergence O(h4), where h is the step size. In the neighbourhood of the singularity, we use the series expansion about the singular point, solving a regular boundary value problem over a reduced interval excluding the singular point, and match the solution to the expansion. Cubic spline procedure is developed for discretizing the resulting regular problem. Some numerical example have been solved to demonstrate the efficiency of the method - Accuracy The numerical results show excellent agreement with known analytical solutions (where available) or matched asymptotic expansions. Even for small values of ( \varepsilon ), the spectral method captured the boundary layer accurately. For example, for ( \varepsilon = 10^{-6} ), using ( N = 32 ) collocation points, the maximum absolute error was on the order of ( 10^{-7} ), significantly outperforming standard finite difference methods with the same number of points. - Convergence The method exhibits exponential convergence for smooth solutions, which is a typical feature of spectral methods. The convergence rate slows slightly for smaller ( \varepsilon ) due to sharp gradients, but remains superior to polynomial-order methods. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 2022-2.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-10 340 - Computational Efficiency Since the spectral method uses global basis functions, fewer points are needed to achieve a given accuracy compared to local methods. The method is computationally efficient and particularly wellsuited for problems with smooth coefficients and boundary layers. - Visualization Below is a plot comparing the exact and numerical solution for ( \varepsilon = 10^{-3} ) Discussion The spectral method, specifically the Chebyshev collocation technique, proves to be a powerful approach for solving SPBVPs. Unlike traditional methods, which suffer from numerical diffusion or require mesh refinement near boundaries, spectral methods handle the global nature of the solution more gracefully. However, it is important to note that spectral methods are best suited for problems with smooth data and simple geometries. For more complex domains or nonlinear problems, domain decomposition or hybrid methods may be required. Moreover, accurate implementation of boundary conditions and differentiation matrices is crucial. Improper handling can lead to ill-conditioning, especially for very small ( \varepsilon ). Conclusion In this paper, we have shown that Chebyshev spectral collocation methods are effective tools for numerically solving singularly perturbed boundary value problems. The method provides high accuracy with relatively few grid points, even in the presence of boundary layers caused by small perturbation parameters. Future work may explore the extension of these techniques to nonlinear SPBVPs and multidimensional domains, possibly incorporating adaptive mesh refinement or hybrid spectralelement approaches. As is evident from the numerical results, this method gives O(h4) accuracy. The results obtained using this method are better than using the usual finite difference method with the same no. of knots. Also this method produce a spline function which may be used to obtain the solution at any point in the range, whereas the finite difference methods only solves the solution at the chosen knots. Numerical experiments show that the TAGE method is accurate and convergent. The proposed TAGE and Newton-TAGE iteration methods show the superiority over the corresponding SOR iteration method. Although the TAGE method involves more work, the developing of the TAGE group implies that parallelism can be easily applied advantageously. Since both AGE and TAGE method require the same number of operations per sweep to solve the system of equations, the TAGE method requires less computation to obtain ISSN: 2582-4686 SJIF 2021-3.261,SJIF 2022-2.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-10 341 References 1. Trefethen, L. N. (2000). *Spectral Methods in MATLAB*. SIAM. 2. Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. (2006). *Spectral Methods: Fundamentals in Single Domains*. Springer. 3. Bender, C. M., & Orszag, S. A. (1999). *Advanced Mathematical Methods for Scientists and Engineers*. McGraw-Hill. 4. Roos, H. G., Stynes, M., & Tobiska, L. (1996). *Numerical Methods for Singularly Perturbed Differential Equations*. Springer. 5. Boyd, J. P. (2001). *Chebyshev and Fourier Spectral Methods*. Dover Publications.