SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES

Allaberen Ashyralyev; Özgür Yıldırım

SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES

Abstract

In this paper, we investigate asymptotic formulas and $\varepsilon$-uniform difference schemes for the numerical solution of perturbation problems. Unfortunately, both of these numerical methods involving sine and cosine fitting operator functions, which depend on $\rho$ ($\rho ={\tau}/{\varepsilon }$, $\varepsilon$ small parameter and $\tau$ stepsize in $t$), are highly challenging with the realization to use symbolic differential algorithms. We present a symbolic differentiation algorithm developed for the numerical solution of hyperbolic perturbation problems with Neumann boundary conditions, employing asymptotic formulas and $\varepsilon$-uniform difference schemes. The symbolic differential algorithms are implemented by Matlab, and the results of numerical experiments are presented.

Keywords

Thanks

We would like to thank the following institutions for their support. This work was supported by the ”RUDN University Program 5-100”. This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No.AP05131225). We want to thank to the anonymous referees.

References

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[6] Shishkin, G. I., Kolmogorov, V. L., (1997), Numerical methods for singularly perturbed boundary value problems modeling diffusion processes, In: Miller, J. J. H., (ed.), Singular Perturbation Problems in Chemical Physics, Advances in Chemical Physics Series, Vol. 97, Wiley, New York, pp. 181-462.
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Details

Primary Language

English

Subjects

Numerical Analysis, Ordinary Differential Equations, Difference Equations and Dynamical Systems, Partial Differential Equations, Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

Allaberen Ashyralyev
0000-0002-4153-6624
Türkiye

Özgür Yıldırım ^*
0000-0003-1375-2503
Türkiye

Publication Date

September 7, 2026

Submission Date

May 13, 2025

Acceptance Date

December 10, 2025

Published in Issue

Year 2026 Volume: 16 Number: 9

IZ

https://izlik.org/JA64BF84RK

Cite

RIS / Bibtex

APA

Ashyralyev, A., & Yıldırım, Ö. (2026). SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES. TWMS Journal of Applied and Engineering Mathematics, 16(9), 1172-1180. https://izlik.org/JA64BF84RK

AMA

1.Ashyralyev A, Yıldırım Ö. SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES. JAEM. 2026;16(9):1172-1180. https://izlik.org/JA64BF84RK

Chicago

Ashyralyev, Allaberen, and Özgür Yıldırım. 2026. “SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES”. TWMS Journal of Applied and Engineering Mathematics 16 (9): 1172-80. https://izlik.org/JA64BF84RK.

EndNote

Ashyralyev A, Yıldırım Ö (September 1, 2026) SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES. TWMS Journal of Applied and Engineering Mathematics 16 9 1172–1180.

IEEE

[1]A. Ashyralyev and Ö. Yıldırım, “SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES”, JAEM, vol. 16, no. 9, pp. 1172–1180, Sept. 2026, [Online]. Available: https://izlik.org/JA64BF84RK

ISNAD

Ashyralyev, Allaberen - Yıldırım, Özgür. “SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES”. TWMS Journal of Applied and Engineering Mathematics 16/9 (September 1, 2026): 1172-1180. https://izlik.org/JA64BF84RK.

JAMA

1.Ashyralyev A, Yıldırım Ö. SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES. JAEM. 2026;16:1172–1180.

MLA

Ashyralyev, Allaberen, and Özgür Yıldırım. “SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 9, Sept. 2026, pp. 1172-80, https://izlik.org/JA64BF84RK.

Vancouver

1.Allaberen Ashyralyev, Özgür Yıldırım. SYMBOLIC DIFFERENTIATION ALGORITHMS FOR HYPERBOLIC PERTURBATION PROBLEMS WITH NEUMANN CONDITIONS USING ASYMPTOTIC FORMULAS AND UNIFORM DIFFERENCE SCHEMES. JAEM [Internet]. 2026 Sep. 1;16(9):1172-80. Available from: https://izlik.org/JA64BF84RK