Mathematics and Statistics Vol. 12(3), pp. 270 - 282
DOI: 10.13189/ms.2024.120307
Reprint (PDF) (1137Kb)

A Numerical Study of Newell-Whitehead-Segel Type Equations Using Fourth Order Cubic B-spline Collocation Method

Maheshwar Pathak ¹, Rachna Bhatia ^2,*, Pratibha Joshi ³, Ramesh Chand Mittal ^4
1 Department of Mathematics, Sharda University, Greater Noida, India
² Department of Mathematics, School of Advanced Science, Vellore Institute of Technology, Vellore, India
³ Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida, India
⁴ Department of Mathematics, Jaypee Institute of Information Technology, Noida, India

ABSTRACT

Newell-Whitehead-Segel (NWS) type equations arise in solid-state physics, optics, dispersion, convection system, mathematical biology, quantum mechanics, plasma physics and oil pollution in ocean environment. Extensive applications of such type of equations draw attention of scientists toward their numerical solutions. In this work, we propose fourth order numerical method based on cubic B-spline functions for the numerical solutions of nonlinear NWS type equations. The Crank Nicolson finite difference scheme is used to discretize the equation and quasi-linearization is use to linearize the nonlinear term. As a result, we get a system of linear equation, which we solve using Gauss elimination method. Stability analysis has been carried out by a thorough Fourier series analysis and stability conditions have been obtained. The scheme has been applied to five numerical problems having quadratic, cubic and forth order nonlinear terms. The effectiveness and robustness of the proposed technique have been demonstrated by comparing the obtained numerical results with the exact solutions and numerical results obtained by other existing methods. A comparison of the numerical results obtained using the proposed technique with exact solutions shows excellent agreement. Graphs of numerical solutions have been drawn at different times and also compared with the graphs of the exact solutions. The comparative analysis shows that the proposed scheme outperformed other methods in terms of accuracy and produced good results.

KEYWORDS
Non-linear Partial Differential Equation, Allen-Cahn Equation, Newell-Whitehead-Segel Equation, Cubic B-spline Method, Collocation Method

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Maheshwar Pathak , Rachna Bhatia , Pratibha Joshi , Ramesh Chand Mittal , "A Numerical Study of Newell-Whitehead-Segel Type Equations Using Fourth Order Cubic B-spline Collocation Method," Mathematics and Statistics, Vol. 12, No. 3, pp. 270 - 282, 2024. DOI: 10.13189/ms.2024.120307.

(b). APA Format:
Maheshwar Pathak , Rachna Bhatia , Pratibha Joshi , Ramesh Chand Mittal (2024). A Numerical Study of Newell-Whitehead-Segel Type Equations Using Fourth Order Cubic B-spline Collocation Method. Mathematics and Statistics, 12(3), 270 - 282. DOI: 10.13189/ms.2024.120307.