PERTURBED CHEBYSHEV-FINITE DIFFERENCE METHOD FOR SOLVINGTHIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS
DOI:
https://doi.org/10.62050/fjst2026.v10n2.775Keywords:
Third-order ordinary differential equations, Chebyshev polynomials basis, Finite difference method, Perturbation, Legendre polynomials basis, Boundary value problemsAbstract
Ordinary differential equations are crucial in modeling complex phenomena in fluid mechanics, structural engineering, and damping systems. However, obtaining analytical solutions remains challenging due to inherent nonlinearities and intricate boundary conditions. This paper proposes a Perturbed Chebyshev Finite Difference Method (PCFDM) specifically designed to solve third-order ordinary differential equations (ODEs). The proposed framework integrates Chebyshev basis polynomials with a Legendre-based perturbation terms to enhance the approximation space and minimize truncation errors. Numerical experiments on third-order boundary value problems associated with draining and coating flows reveal that the PCFDM significantly outperforms conventional finite difference method (FDM) in terms of accuracy and computational efficiency. The results confirm that the integration of orthogonal polynomial and perturbation provides a flexible and reliable approach for solving high-order differential equations.
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Abdulkadir, E. (2019). Solution of second order linear and nonlinear two point boundary value problems using legendre operational matrix of differentiation. American Scientific Research Journal for Engineering, Technology, and Sciences, 51(1), 225-234.
Agarwal, R. P. (1986). Boundary Value Problems for Higher Order Differential Equations. World Scientific.
Ahmed, J. (2017). Numerical solutions of third-order boundary value problems associated with draining and coating flows. Kyungpook Mathematical Journal, 4(57), 651-665.
Al-Humedi, H. O. and Al-Saadawi, F. A. (2021). The numerical solution of bioheat equation based on shifted legendre polynomial. International Journal of Nonlinear Analysis and Applications, 12(2), 1061–1070.
Alvarez-Ramirez, J. and Valdes-Parada, F. J. (2009). Non-standard finite-differences schemes for generalized reaction-diffusion equations. Journal of Computational and Applied Mathematics, 228(1), 334-343.
Amer, Y. A., Mahdy, A. M. S. and Namoos, H. A. R. (2018). Hermite collocation method for solving hammerstein integral equations. Journal of Advances in Mathematics, 14(1), 7413-7423.
Ascher, U. M., Mattheij, R. M. M. and Russell, R. D. (1995). Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM.
Bos, L., Narayan, A., Levenberg, N. and Piazzon, F. (2017). An orthogonality property of the Legendre polynomials. Constructive Approximation, 45(2017), 65-81.
Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill.
Dung, N. D. and Quang, V. V. (2024). Finite difference method for solving second-order boundary value problem with high-order accuracy. European Journal of Mathematical Analysis, 10(4), 1–11.
El-Danaf, T. S. (2008). Quartic nonpolynomial spline solutions for third order two-point boundary value problem. World Academy of Science, Engineering and Technology, 45, 453–456.
Endeshaw, B. (2019). Finite difference method for boundary value problems in ordinary differential equations. Mathematical Theory and Modeling.
Enemoh, E. N., Olagunju, A. S. and Ayoo, P. V. (2025). Improved finite difference methods for solving second-order boundary value problems of ordinary differential equations using chebyshev polynomials. International Journal of Mathematical Sciences and Optimisation: Theory and Applications, 1(11), 45-60.
Erjaee, G. H., Akrami, M. H. and Atabakzadeh, M. H. (2013). The operational matrix of fractional integration for shifted legendre polynomials. Iranian Journal of Science and Technology, Transaction A: Science, 37(4), 439-444.
Falkner, V. M. and Skan, S. W. (1931). Some approximate solutions of the boundary layer equations. Philosophical Magazine, 12(80), 865–896.
Ghimire, B. K., Li, X., Chen, C. S. and Lamichhane, A. R. (2020). Hybrid Chebyshev polynomial scheme for solving elliptic partial differential equations. Journal of Computational and Applied Mathematics, 364, 1-14.
Ishola, C. Y., Taiwo, O. A., Adebisi, A. F. and Peter, O. J. (2022). Numerical solution of two dimensional fredholm integro-differential equations by chebyshev integral operational matrix method. Journal of Applied Mathematics and Computational Mechanics, 21(1), 29-40.
Karjanto, N. (2002). Properties of chebyshev polynomials. Publication Scientifique de l’AEIF, 2(2002), 127–132.
LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM.
Mamadu, J. E. and Njoseh, I. N. (2016). Numerical solutions of volterra equations using galerkin method with certain orthogonal polynomials. Journal of Applied Mathematics and Physics, 4, 1-18.
Mason, J. C. and Handscomb, D. C. (2002). Chebyshev Polynomials. Chapman & Hall/CRC.
Ogata, K. K. (2010). Modern Control Engineering. Prentice Hall, 5th edition.
Rao, S. S. (2007). Vibration of Continuous Systems. Wiley.
Smith, G. D. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press.
Taiwo, O. A. and Fesojaye, M. O. (2017). Double perturbation collocation method for solving fractional riccati differential equations. International Journal of Mathematics and Statistics Studies, 5(4), 18-37.
Uwaherena, O. A., Adebisib, A. F. and Taiwo, O. A. (2020). Perturbed collocation method for solving singular multi-order fractional differential equations of lane-emden type. Journal of the Nigerian Society of Physical Sciences, 2(2020), 141-148.
Yaslan, H. C. and Mutlu, F. (2020). Numerical solution of the conformable differential equations via shifted legendre polynomials. International Journal of Computer Mathematics, 97(5), 1-22.
Yigit, G. and Bayram, M. (2019). Chebyshev differential quadrature for numerical solutions of third and fourth-order singular perturbation problems. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 1-8.
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