NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD

Ali Sercan Karakaş; Murat Yağmurlu

doi:10.51477/mejs.1870269

NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD

Abstract

The present manuscript aims to give the details of the application of the finite elements method which is based upon the cubic Hermite Least Squares (CHLSM) method to a widely used prototype test problem. The fundamental objective of the present study is to establish a foundation for applying the Least Squares finite element method which is based upon cubic Hermite basis functions to different types of differential equations, including fractional, partial, and ordinary order equations. The explanation of the method on the differential equation under consideration, its application, and the simulation of the program written with the obtained scheme are detailed in the present study. The main contribution of the present study is to show the application of Cubic Hermite Least Squares Method on a typical model problem. The handled problem is selected as an ordinary differential equation for easy presentation of the method, but researchers can easily apply the presented method for other ordinary, partial even fractional order differential equations.

Keywords

Ethical Statement

No ethical approval was required for this study as it did not involve human or animal subjects. The research utilized numerical data and was exempt from ethics approval

References

S. Kutluay, N. M. Yağmurlu, and A. S. Karakaş, “An effective numerical approach based on cubic Hermite B-spline collocation method for solving the 1D heat conduction equation,” New Trends in Mathematical Sciences, vol. 10, no. 4, pp. 20–31, 2022.
I. A. Ganaie and V. K. Kukreja, “Numerical solution of Burgers' equation by cubic Hermite collocation method,” Applied Mathematics and Computation, vol. 237, no 2014 , pp. 571–581, 2014.
I. A. Ganaie, S. Arora, and V. K. Kukreja, “Cubic Hermite collocation method for solving boundary value problems with Dirichlet, Neumann, and Robin conditions,” International Journal of Engineering Mathematics, vol. 2014, Art. no. 65209, pp.1-8 , 2014.
R. Mohammadzadeh, M. Lakestani, and M. Dehghan, “Collocation method for the numerical solutions of Lane–Emden type equations using cubic Hermite spline functions,” Mathematical Methods in the Applied Sciences, vol. 37, no. 9, pp. 1303–1317, 2014.
A. A. Soliman, “A Galerkin solution for Burgers' equation using cubic B-spline finite elements,” Abstract and Applied Analysis, vol. 2012, Art. no. 817635, pp.1-15 , 2012.
N. M. Yağmurlu and A. S. Karakaş, “Numerical solutions of the equal width equation by trigonometric cubic B-spline collocation method based on Rubin–Graves type linearization,” Numerical Methods for Partial Differential Equations, vol. 36, no. 5, pp. 1170–1183, 2020.
R. K. Nagaich and H. Kumar, “Hermite collocation method for numerical solution of second order parabolic partial differential equations,” International Journal of Applied Mathematics and Statistical Sciences, vol. 3, no. 3, pp. 45–52, 2014.
I. A. Ganie, S. Arora, and V. K. Kukreja, “Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation,” International Journal of Computer Mathematics, vol. 93, no. 1, pp. 223–235, 2016.

N. M. Yağmurlu and A. S. Karakaş, “A novel perspective for simulations of the MEW equation by trigonometric cubic B-spline collocation method based on Rubin–Graves type linearization,” Computational Methods for Differential Equations, vol. 10, no. 4, pp. 1046–1058, 2022.
I. A. Ganie, B. Gupta, N. Parumasur, P. Singh, and V. K. Kukreja, “Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equations,” Applied Mathematics and Computation, vol. 220, no 2013, pp. 560–567, 2013.
T. Zhao and Y. Wu, “Hermite cubic spline collocation method for nonlinear fractional differential equations with variable-order,” Symmetry, vol. 13, no. 11, Art. no. 2142, pp.1-29, 2021.
S. Kutluay, N. M. Yağmurlu, and A. S. Karakaş, “A robust quintic Hermite collocation method for one-dimensional heat conduction equation,” Journal of Mathematical Sciences and Modelling, vol. 7, no. 2, pp. 82–89, 2024.
H. Kumar, “Numerical study of differential equations via cubic Hermite collocation technique,” Naturalista Campano, vol. 28, no. 1, pp. 519–537, 2024.
S. Kutluay, N. M. Yağmurlu, and A. S. Karakaş, “A novel perspective for simulations of the modified equal-width wave equation by cubic Hermite B-spline collocation method,” Wave Motion, vol. 129, Art. no. 103448, pp.1-16, 2024.
A. M. Ayebire, I. Kaur, D. A. Alemar, M. S. Manshahia, and S. Arora, “A robust technique of cubic Hermite splines to study the non-linear reaction-diffusion equation with variable coefficients,” AIMS Mathematics, vol. 9, no. 4, pp. 8192–8213, 2024.
S. Kutluay, N. M. Yağmurlu, and A. S. Karakaş, “A robust septic Hermite collocation technique for Dirichlet boundary condition heat conduction equation,” International Journal of Mathematics and Computer in Engineering, vol. 3, no. 2, pp. 253–266, 2025.
A. M. Ayebire, S. Sahani, P. Priyanka, and S. Arora, “Numerical study of soliton behavior of generalised Kuramoto–Sivashinsky type equations with Hermite splines,” AIMS Mathematics, vol. 10, no. 2, pp. 2098–2130, 2025.
V. K. Kukreja, “Highly accurate spline collocation technique for the numerical solution of generalized Burgers–Fisher's problem,” Computational Methods for Differential Equations, vol. 13, no. 2, pp.578-591, 2025.
A. Devi, A. Kumari, N. Parumasur, P. Singh, and V. K. Kukreja, “Numerical study of fractional-order Burgers–Huxley equation using modified cubic splines approximation,” Fractal and Fractional, vol. 9, no. 12, Art. no. 780, 2025.
M. Arı, “Quintic Hermite spline collocation approach to the time fractional RLW and MRLW equations,” Ain Shams Engineering Journal, vol. 17, no. 3, Art. no. 104011, pp. 1-12, 2026.
M. Arı, “A cubic Hermite collocation approach for solving the Korteweg–De Vries–Burgers equation,” Eskişehir Technical University Journal of Science and Technology A – Applied Sciences and Engineering, vol. 26, no. 4, pp. 445–459, 2025.
J. N. Reddy, An Introduction to the Finite Element Method, 2nd ed. New York, NY, USA: McGraw-Hill, 1993.

Details

Primary Language

English

Subjects

Numerical Analysis, Finite Element Analysis

Journal Section

Research Article

Authors

Ali Sercan Karakaş ^*
0000-0001-8622-1127
Türkiye

Murat Yağmurlu
0000-0003-1593-0254
Türkiye

Publication Date

June 29, 2026

Submission Date

January 23, 2026

Acceptance Date

June 23, 2026

Published in Issue

Year 2026 Volume: 12 Number: 1

DOI

https://doi.org/10.51477/mejs.1870269

IZ

https://izlik.org/JA73TN68CF

Cite

RIS / Bibtex

APA

Karakaş, A. S., & Yağmurlu, M. (2026). NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD. Middle East Journal of Science, 12(1), 74-85. https://doi.org/10.51477/mejs.1870269

AMA

1.Karakaş AS, Yağmurlu M. NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD. MEJS. 2026;12(1):74-85. doi:10.51477/mejs.1870269

Chicago

Karakaş, Ali Sercan, and Murat Yağmurlu. 2026. “NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD”. Middle East Journal of Science 12 (1): 74-85. https://doi.org/10.51477/mejs.1870269.

EndNote

Karakaş AS, Yağmurlu M (June 1, 2026) NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD. Middle East Journal of Science 12 1 74–85.

IEEE

[1]A. S. Karakaş and M. Yağmurlu, “NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD”, MEJS, vol. 12, no. 1, pp. 74–85, June 2026, doi: 10.51477/mejs.1870269.

ISNAD

Karakaş, Ali Sercan - Yağmurlu, Murat. “NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD”. Middle East Journal of Science 12/1 (June 1, 2026): 74-85. https://doi.org/10.51477/mejs.1870269.

JAMA

1.Karakaş AS, Yağmurlu M. NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD. MEJS. 2026;12:74–85.

MLA

Karakaş, Ali Sercan, and Murat Yağmurlu. “NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD”. Middle East Journal of Science, vol. 12, no. 1, June 2026, pp. 74-85, doi:10.51477/mejs.1870269.

Vancouver

1.Ali Sercan Karakaş, Murat Yağmurlu. NUMERICAL SOLUTION DIFFERENTIAL EQUATION VIA CUBIC HERMITE LEAST SQUARE METHOD. MEJS. 2026 Jun. 1;12(1):74-85. doi:10.51477/mejs.1870269