Study of the Dynamics of Derivatives of a Fourth-Order Differential Equation with a Small Parameter at the Highest Derivative
DOI:
https://doi.org/10.71310/pcam.4_68.2025.03Keywords:
Chebyshev polynomials, small parameter, high accuracyAbstract
The article studies the dynamics of derivatives up to the highest derivative of an ordinary differential equation of the fourth order with a small parameter at the highest derivative. The solution of the differential problem is sought in the form of a series in Chebyshev polynomials of the first kind with unknown expansion coefficients. Substituting these series into the differential problem, we obtain a system of algebraic equations for finding the unknown expansion coefficients. Using these coefficients, we calculate the solution to the problem and its derivatives. To illustrate the high accuracy of the applied method, we use the method of trial functions. The essence of the method of trial functions is as follows. A certain function is selected, it can be chosen arbitrarily. Substituting it into the main differential equation, we find the right-hand side and satisfy the corresponding boundary conditions. The resulting problem is solved by the spectral method and the approximate solution is compared with the known trial function and its derivatives for different values of the small parameter and the approximating Chebyshev polynomials.
References
Il'in A.M. Raznostnaya skhema dlya differentsial'nogo uravneniya s malym parametrom pri starshey proizvodnoy Matem. Zametka. – 1969. №6(2). – S. 237–248. https://doi.org/10.1007/BF01093706
Bakhvalov N.S. K optimizatsii metodov resheniya krayevykh zadach pri nalichii pogranichnogo sloya. ZH.vychisl. matem. i matem.fiz., – 1969. – №9(4). – S. 841–859. https://doi.org/10.1016/0041-5553(69)90038-X
Liseykin V.D., Yanenko N.N. O ravnomerno skhodyashchemsya algoritme chislennogo resheniya obyknovennogo differentsial'nogo uravneniya vtorogo poryadka s malym parametrom pri starshey proizvodnoy. Chislennyye metody mekhaniki sploshnoy sredy. Novosibirsk. – 1981. – №12(2). – S. 45–56. https://www.researchgate.net/publication/336890257_1981
Li J. Convergence analysis of finite element methods for singularly perturbed problems. Computers and Mathematics with Applications. – 2000. – №40(6). – P. 735–745. https://doi.org/10.1016/S0898-1221(00)00192-9
Duan J.S, Jing L.X., Li M. The mixed boundary value problems and Chebyshev Collocation Method for Caputo-Type fractional ordinary differential equations. Fractal fract. – 2022. –№6(3). – P. 148–156. https://doi.org/10.3390/fractalfract6030148
Bouakkaz M., Arar N. Meflah M. Enhanced numerical resolution of the Duffing and Van der Pol equations via the spectral homotopy analysis method employing Chebyshev polynomials of the first kind. J. Appl. Math. Comput. – 2025. – Vol. 71. – P. 1159–1187. https://doi.org/10.1007/s12190-024-02271-5
Benzahi A., Arar N., Abada N., Rhaima M., Mchiri L., Ben Makhlouf A. Numerical investigation of Fredholm fractional integro-differential equations by least squares method and compact combination of shifted Chebyshev polynomials.J. Nonlinear Math.Physi. – 2023. – Vol. 30. – P. 1392–1408. https://doi.org/10.1007/s44198-023-00128-2
Van T., Lyan' KH., TSzi L. Metod razdeleniya singulyarnostey Chebysheva dlya slabo singulyarnykh integral'nykh uravneniy Vol'terry vtorogo roda. Numer Algor. – 2024. – Vol. 95. – S. 1829–1854. https://doi.org/10.1007/s11075-023-01629-3
Mishchuk A.Y., Shutovskyi A.M. Optimization Properties of Generalized Chebyshev-Poisson Integrals CybernSystAnal. – 2024. – Vol. 260. – P. 613–620. https://doi.org/10.1007/s10559-024-00700-8
Atta A.G., Youssri Y.H. Avanced shifted first – kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel.Comp.Appl.Math. – 2022. – №41(381). – P. 1–19. https://doi.org/10.1007/s40314-022-02096-7
Khubedyy S.S. Priblizhennoye resheniye singulyarnogo integral'nogo uravneniya pervogo roda s ispol'zovaniyem mnogochlenov Chebysheva s nulevymi znacheniyami na obeikh konechnykh tochkakh intervala integrirovaniya. Vychisl. tekhn.Mat.i.Mat.Fiz. – 2021. – Vol. 61. – S. 1269–1275. https://doi.org/10.1134/S0965542521080030
Yadav A., Setia A., Agarval R. Analiz oshibok chislennogo metoda, osnovannogo na polinomakh Chebysheva, dlya sistemy gipersingulyarnykh integral'nykh uravneniy. Comp. Appl. Math. – 2023. – Vol. 42. – 213 p. https://doi.org/10.1007/s40314-023-02352-4
Rocha D., Brauer Z., Gershgorin Raspolozheniye nuley vozmushchennykh mnogochlenov Chebysheva vtorogo roda putem rasshireniya Math.Comput.Sci. – 2024. – №18(13). https://doi.org/10.1007/s11786-024-00582-1
Castillo K., Mbouna D., Petronilho J. A characterization of continuous q-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials, Journal of Mathematical Analysis and Applications, – 2022. – №514(2). – P. 1–16. https://doi.org/10.1016/j.jmaa.2022.126358 .
Wang X., Hu J. An Identity Involving the Integral of the First-Kind Chebyshev Polynomials, Mathematical Problems in Engineering. – 2018. – Volum 2. – P. 1–5. https://doi.org/10.1155/2018/7186940 .
Dressler M., Foucart S., Joldes M., De Klerk E., Lasserre J.B., Xu Y. Least multivariate chebyshev polynomials on diagonally-determined domains. – 2024. https://doi.org/10.48550/arXiv.2405.19219
G‥ulsu M., ‥ Ozt‥urk Y. Approximate solution of the singular-perturbation problem on Chebyshev-Gauss grid, – 2011. – Vol. 1. – P. 209–218. http://dx.doi.org/10.4236/ajcm.2011.14024
Dolbeeva S.F., Rozhdestvenskaya E.A. Contrasting step-type structures for an ordinary second-order differential equation with a small parameter at the highest derivative,J.Math.and Math.Phys. – 2009. №49(12). – P. 2034–2046. https://doi.org/10.1134/S0965542509120045
Denisov A.M. Iterative Method for Solving an Inverse Problem for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative Diff Equat. – 2019. – Vol. 55. – P. 940–948. https://doi.org/10.1134/S0012266119070073
Tsekhan O.B. Decomposition of singularly perturbed functional-differential systems based on a nondegenerate transformation.J Math Sci. – 2025. – Vol. 288. – P. 128–141. https://doi.org/10.1007/s10958-025-07672-w
Danilin A.R. Asymptotics of the solution of a bisingular optimal distributed control problem in a convex domain with a small parameter multiplying a highest derivative Comput. Math. and Math. Phys. – 2024. – Vol. 64. – P. 941–953. https://doi.org/10.1134/S0965542524700210
Denisov A.M. The inverse problem for the heat equation in the case of a small heat capacity coefficient Comput Math Model. – 2023. – Vol. 34. – P. 208–216. https://doi.org/10.1007/s10598-025-09612-4
Normurodov Ch.B., Tursunova B.A. Numerical modeling of the boundary value problem of an ordinary differential equation with a small parameter at the highest derivative by Chebyshev polynomials of the second kind. Results in Applied mathematics. – 2023. – Vol.
https://doi.org/10.1016/j.rinam.2023.100388
Abutaliev F.B., Narmuradov Ch.B. Mathematical modeling of the problem of hydrodynamic stability. Tashkent: Fan va texnologiya: – 2011. – 188 p.
Narmuratov Ch.B. About one effective method for solving the Orr-Sommerfeld equation Math modeling. – 2005. – №9(17). – P. 35–42.
Normurodov Ch.B., Toyirov A.X., Ziyakulova Sh.A., Viswanathan K.K. Convergence of pectral-Grid Method for Byurgers equation with initial-boundary conditions. Mathematics and Statistics. – 2024. – №12(2). – P. 115–125. https://doi.org/10.13189/ms.2024.120201
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Ch.B. Normurodov
This work is licensed under a Creative Commons Attribution 4.0 International License.
