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Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc

Iftikhar Ahmad1, Hira Ilyas1, Kadir Kutlu2

  • 1Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan.

Heliyon
|August 11, 2021
PubMed
Summary

This study introduces a sinc collocation technique for numerically solving the Hunter-Saxton equation (HSE). The method offers an accurate and efficient approach for analyzing liquid crystal models and their diverse applications.

Keywords:
Hunter-Saxton equation (HSE)Liquid crystals (LC)Numerical solution of PDEsSinc collocation method (SCM)Stability analysis

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Area of Science:

  • Physics
  • Materials Science
  • Computational Science

Background:

  • The Hunter-Saxton equation (HSE) models liquid crystal behavior, crucial for understanding phenomena like phase transitions and self-assembly.
  • Numerical methods are essential for solving the complex HSE, enabling detailed analysis of mesophase materials and processes.

Purpose of the Study:

  • To present a novel numerical treatment for the liquid crystal model described by the Hunter-Saxton equation (HSE).
  • To evaluate the efficacy and accuracy of the sinc collocation technique with a theta-weighted scheme for solving the HSE.

Main Methods:

  • The sinc collocation technique was employed to approximate solutions for the HSE.
  • The HSE was reduced to a system of algebraic equations, represented as matrices, for computational analysis.
  • A theta-weighted scheme was integrated to enhance the numerical treatment.

Main Results:

  • The sinc collocation technique demonstrated high computational efficiency and accuracy for solving the HSE.
  • Stability analysis confirmed the reliability and convergence of the proposed numerical method.
  • Graphical and tabular representations of solutions were generated for various theta values and collocation points, validating the method's performance.

Conclusions:

  • The sinc collocation technique is a highly effective and accurate computational tool for the numerical treatment of the Hunter-Saxton equation.
  • The method's stability and convergence properties ensure reliable solutions for liquid crystal modeling.
  • This approach provides a robust framework for investigating diverse applications of liquid crystals.