Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter–Saxton Equation

Abstract

In this article, a novel and efficient collocation method based on Fibonacci wavelets is proposed for the numerical solution of the non-linear Hunter–Saxton equation. Firstly, the operational matrices of integration associated with the Fibonacci wavelets are constructed by following the strategy of Chen and Hsiao. The operational matrices merged with the collocation method are used to convert the given problem into a system of algebraic equations that can be solved by any classical method, such as Newton’s method. Moreover, the non-linearity arising in the Hunter–Saxton equation is handled by invoking the quasi-linearization technique. To show the efficiency and accuracy of the Fibonacci-wavelet-based numerical technique, the approximate solutions of the non-linear Hunter–Saxton equation with other numerical methods including the Haar wavelet, trigonometric B-spline, and Laguerre wavelet methods are compared. The numerical outcomes demonstrate that the proposed method yields a much more stable solution and a better approximation than the existing ones.