HYBRID CUBIC B-SPLINE COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF THE ALLEN–CAHN EQUATION

Roohi Aaysha; Muhammad Amin; Iqra safdar; Muhammad Usama

doi:10.71146/kjmr861

Authors

Roohi Aaysha Faculty of Sciences, The Superior University Lahore, Pakistan. Author
Muhammad Amin Faculty of Sciences, The Superior University Lahore, Pakistan. Author
Iqra safdar Faculty of Sciences, The Superior University Lahore, Pakistan. Author
Muhammad Usama Faculty of Sciences, The Superior University Lahore, Pakistan. Author

DOI:

https://doi.org/10.71146/kjmr861

Keywords:

Allen–Cahn equation, Hybrid cubic B-spline Method, Numerical solution, Finite difference method, Stability and convergence analysis

Abstract

The Allen Cahn equation is a famous nonlinear partial differential equation which is applied to characterize phase separation and motion of interfaces in multi-component systems. It is used in materials science, fluid dynamics, and image processing, where numerical approximation of results is needed to be accurate. This paper suggests a numerical solution of the Allen-Cahn equation based on a hybrid cubic B-spline technique. The approach employs the smoothness and local support of cubic B-spline basis functions and includes a hybrid formulation to enhance accuracy and computational stability.

The spatial discretization in the proposed scheme is done by performing hybrid cubic B-spline interpolation and time discretization by finite difference method. Nonlinear term is tackled using an appropriate linearization approach. Numerical experiments are given to investigate the performance of the method. The findings reveal that the scheme is very accurate, converges, and has less numerical error. The hybrid cubic B-spline method is therefore a safe and effective method to use in solving nonlinear reaction-diffusion equations like the Allen-Cahn equation.

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