A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS

Kübra Erdem Biçer; Gökçe Yıldız

A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS

Abstract

This article examines the solutions of high-order nonlinear ordinary differential equations with cubic terms under initial conditions using Bell polynomials, their derivatives, and collocation points. The nonlinear differential equation and the corresponding conditions are transformed into matrix form by means of Bell polynomials and reduced to an algebraic system. From the solution of this system, the unknown Bell coefficients are determined. By substituting these coefficients, the approximate solution of the problem is expressed in terms of Bell polynomials. To illustrate the method, some numerical examples are presented. For these examples, the Bell solutions and the absolute error functions are calculated, and the results are shown in tables and figures for comparison with the exact solutions.

Keywords

References

Abbasbandy, S., (2006), Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Applied Mathematics and Computation, 172 (1), pp. 459-464.
Abbasbandy, S., (2007), A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, Journal of Computational and Applied Mathematics, 207 (1), pp. 90-96.
Abbasbandy, S., (2006), Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Applied Mathematics and Computation, 175 (1), pp. 18-23.
Akyüz Daşcığlu, A., Yaslan, H. C., (2006), An approximation method for solution of nonlinear integral equations, Applied Mathematics and Computation, 174 (2), pp. 1050-1058.
Bayin, S. S., (1978), Solutions of Einstein’s field equations for static fluid spheres, Physical Review D, 18 (8), pp. 2745-2751.
Belbachir, H., Mihoubi, M., (2009), A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould-Quaintance formulas, European Journal of Combinatorics, 30 (5), pp. 1254-1256.
Bell, ET, (1934), Exponential polynomials, Annals of Mathematics, 35 (2), pp. 258-277.
Borghero, F., Melis, A., (1990), On Szebehely’s problem for holonomic systems involving generalized potential functions, Celestial Mechanics and Dynamical Astronomy, 49 (3), pp. 273-284.

Çağlar, H., Çağlar, N., Özer, M., Valaristos, A., Anagnostopoulos, A. N., (2010), B-spline method for solving Bratu’s problem, International Journal of Computer Mathematics, 87 (8), pp. 1753-1765.
Çam, Ş., (2005), Stirling Sayıları, Matematik Dünyası, (31), pp. 30-34.
Darvishi, M. T., Kheybari, S., (2011), An approximate solution of the classical Van der Pol oscillator coupled gyroscopically to a linear oscillator using parameter-expansion method, International Journal of Engineering and Natural Sciences, 5 (1), pp. 208-210.
Deeba, E., Khuri, S. A., Xie, S., (2000), An algorithm for solving boundary value problems, Journal of Computational Physics, 159 (1), pp. 264-283.
El-Tawil, MA, Bahnasawi, A. A., Abdel-Naby, A., (2004), Solving Riccati differential equation using Adomian’s decomposition method, Applied Mathematics and Computation, 157 (2), pp. 477-486.
Erdem Bicer, K., Sezer, M., (2019), A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials, Thermal Science, 23 (Suppl. 1), pp. S345-S352.
Garcia Macias, A., Mielke, A., (1997), E.W: Stewart-Lyth second order approach as an Abel equation for reconstructing inflationary dynamics, Physics Letters A, 229 (1-2), pp. 32-36.
Gavrilov, VR, Ivashchuk, V. D., Melnikov, V. N., (1996), Multidimensional integrable vacuum cosmology with two curvatures, Classical and Quantum Gravity, 13 (11), pp. 3039-3056.
Geng, FZ, Lin, Y. Z., Cui, M. G., (2009), A piecewise variational iteration method for Riccati differential equations, Computers and Mathematics with Applications, 58 (11-12), pp. 2379-2387.
Güler, C., (2007), A new numerical algorithm for the Abel equation of the second kind, International Journal of Computer Mathematics, 84 (6), pp. 803-813.
Gümgüm, S., Baykuş, N., Kürkçü, Ö., Sezer, M., (2020), Lucas polynomial approach for second order nonlinear differential equations, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 24 (1), pp. 230-236.
Haager, G., Mars, M., (1998), A self-similar inhomogeneous dust cosmology, Classical and Quantum Gravity, 15 (5), pp. 1567-1580.
Kazemi, NA, Pashazadeh, A. Z., Kılı¸cman, A., (2013), An efficient approach for solving nonlinear Troesch’s and Bratu’s problems by wavelet analysis method, Mathematical Problems in Engineering, 2013, Article ID 763920, 10 pages.
Kim, T., Kim, D., WooJang, G., (2019), On central complete and incomplete Bell polynomials, Symmetry, 11 (3), Article 288.
Kimiaeifar, A., Saidi, A. R., Sohouli, A. R., Ganji, D., (2010), Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods, Current Applied Physics, 10 (1), pp. 121-128.
Lakestani, M., Dehghan, M., (2010), Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions, Computer Physics Communications, 181 (5), pp. 957-966.
Lebrun, JPM, (1990), On two coupled Abel-type differential equations arising in a magnetostatic problem, Il Nuovo Cimento A, 103 (11), pp. 1369-1379.
Mak, MK, Harko, T., (1998), Full causal bulk-viscous cosmological models, Journal of Mathematical Physics, 39 (9), pp. 5458-5476.
Mak, MK, Harko, T., (1999), Addendum to “Exact causal viscous cosmologies”, General Relativity and Gravitation, 31 (2), pp. 273-274.
Maleknejad, K., Mahmoudi, Y., (2003), Taylor polynomial solutions of high–order nonlinear Volterra-Fredholm integro-differential equation, Applied Mathematics and Computation, 145 (2), pp. 641-653.
Mandelzweig, VB, Tabakin, F., (2001), Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications, 141 (2), pp. 268-281.
Mohsen, A., (2014), A simple solution of the Bratu problem, Computers and Mathematics with Applications, 67 (5), pp. 896-902.
Öztürk, Y., Gülsu, M., (2016), The approximate solution of high order nonlinear ordinary differential equations by improved collocation method with terms of shifted Chebyshev polynomials, International Journal of Applied and Computational Mathematics, 2 (4), pp. 529-531.
Roman, S., (1984), The exponential polynomials and the Bell polynomials, The Umbral Calculus, Academic Press, New York, pp. 63-67, 82-87.
Supriya, M., Banamali, R., Sourav, D., (2010), Solution of the Duffing–van der Pol oscillator equation by a differential transform method, The Royal Swedish Academy of Sciences, 83 (1), pp. 1-10.
Tang, BQ, Li, X. F., (2007), A new method for determining the solution of Riccati differential equations, Applied Mathematics and Computation, 194 (2), pp. 431-440.
Van Gorder, RA, (2010), Recursive relations for Bell polynomials of arbitrary positive non-integer order, International Mathematical Forum, 5 (37), pp. 1819-1821.
Venkatesh, SG, Ayyaswamy, S. R., Balachandar, S. R., (2012), The Legendre wavelet method for solving initial value problems of Bratu-type, Computers and Mathematics with Applications, 63 (7), pp. 1287-1295.
Yalçınbaş, S., (2002), Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation, 127 (2), pp. 196-206.
Yüzbaşı, Ş., (2017), A numerical scheme for solutions of a class of nonlinear differential equations, Journal of Taibah University for Science, 11 (4), pp. 1165-1181.
Yüzbaşı, Ş., Şahin, N., (2012), On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials, Journal of Numerical Mathematics, 20 (1), pp. 55-79.
Yüzbaşı, Ş¸., (2015), A collocation method based on the Bessel functions of the first kind for singularly perturbed differential equations and residual correction, Mathematical Methods in the Applied Sciences, 38 (14), pp. 3033-3042.
Yüzbaşı, S¸., Şahin, N., Sezer, M., (2011), Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases, Computers and Mathematics with Applications, 61 (9), pp. 3079-3096.
YÜzbaşı, Ş., Şahin, N., Sezer, M., (2024), Fractional Bell collocation method for solving linear fractional integro-differential equations, Mathematical Sciences, 18 (1), pp. 29-40.
Wazwaz, AM, (2005), Adomian decomposition method for a reliable treatment of the Bratu-type equations, Applied Mathematics and Computation, 166 (3), pp. 652-663.

Details

Primary Language

English

Subjects

Numerical Solution of Differential and Integral Equations, Ordinary Differential Equations, Difference Equations and Dynamical Systems, Algebra and Number Theory

Journal Section

Research Article

Authors

Kübra Erdem Biçer
0000-0002-4998-6531
Türkiye

Gökçe Yıldız ^*
0000-0001-9896-6580
Türkiye

Publication Date

January 8, 2026

Submission Date

June 12, 2024

Acceptance Date

August 23, 2025

Published in Issue

Year 2026 Volume: 16 Number: 1

IZ

https://izlik.org/JA39FP92XY

Cite

RIS / Bibtex

APA

Erdem Biçer, K., & Yıldız, G. (2026). A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS. TWMS Journal of Applied and Engineering Mathematics, 16(1), 134-146. https://izlik.org/JA39FP92XY

AMA

1.Erdem Biçer K, Yıldız G. A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS. JAEM. 2026;16(1):134-146. https://izlik.org/JA39FP92XY

Chicago

Erdem Biçer, Kübra, and Gökçe Yıldız. 2026. “A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS”. TWMS Journal of Applied and Engineering Mathematics 16 (1): 134-46. https://izlik.org/JA39FP92XY.

EndNote

Erdem Biçer K, Yıldız G (January 1, 2026) A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS. TWMS Journal of Applied and Engineering Mathematics 16 1 134–146.

IEEE

[1]K. Erdem Biçer and G. Yıldız, “A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS”, JAEM, vol. 16, no. 1, pp. 134–146, Jan. 2026, [Online]. Available: https://izlik.org/JA39FP92XY

ISNAD

Erdem Biçer, Kübra - Yıldız, Gökçe. “A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS”. TWMS Journal of Applied and Engineering Mathematics 16/1 (January 1, 2026): 134-146. https://izlik.org/JA39FP92XY.

JAMA

1.Erdem Biçer K, Yıldız G. A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS. JAEM. 2026;16:134–146.

MLA

Erdem Biçer, Kübra, and Gökçe Yıldız. “A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 1, Jan. 2026, pp. 134-46, https://izlik.org/JA39FP92XY.

Vancouver

1.Kübra Erdem Biçer, Gökçe Yıldız. A NUMERICAL SOLUTION TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON BELL POLYNOMIALS. JAEM [Internet]. 2026 Jan. 1;16(1):134-46. Available from: https://izlik.org/JA39FP92XY