# Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation
## 1 Introduction
[PERSON] and [PERSON] [27] define another novel generalization of Fibonacci polynomials coined as \((p,q)\)-Fibonacci polynomials and obtained several results regarding the factorization of the Pascal matrix involving these polynomials. Such polynomials are defined by the recurrence relation \(F_{p,q,m+1}(\eta)=p(\eta)F_{p,q,m}(\eta)+q(\eta)F_{p,q,m-1},m\geq 1\), where \(F_{p,q,0}=0\), \(F_{p,q,1}=1\), and \(p(\eta)\) and \(q(\eta)\) are polynomials with real coefficients. Subsequently, [PERSON] [28] defined a quaternion analogue of the \(h\)-Fibonacci polynomials of the form \(Q_{h,m}(\eta)=\sum_{l=0}^{3}F_{h,m+l}(\eta)e_{l}\), where \(F_{h,m}(\eta)\) is the \(h(\eta)\)-Fibonacci polynomial. Recently, [PERSON] et al. [29] introduced a new class of Fibonacci polynomials coined as distance Fibonacci polynomials, which embodies the classical Fibonacci, Jacobsthal, and Narayana polynomials simultaneously. Owing to the nice characteristics of the Fibonacci polynomials, they have been extensively employed for solving differential equations [30], integro-differential equations [31], fractional delay differential equations [32], fractional differential equations [33], and many more. Fibonacci wavelets are a novel addition to the class of compactly supported wavelets, having the main advantage of being polynomial-based and, as such, enjoying smoothness at the expense of having compact support. In fact, these wavelets are not generated by the orthogonal polynomials; however, the Fibonacci polynomials can be represented in terms of orthogonal Chebychev orthogonal polynomials of the second kind. Unlike the shifted Legendre polynomials, the Fibonacci polynomials possess fewer terms, resulting in faster computation, thus minimizing the computational errors. The Fibonacci wavelets are dependent on two parameters \(k\) and \(M\), which increases the rate convergence of the Fibonacci wavelet series. Indeed, the convergence of the Fibonacci wavelet method is faster than the orthogonal wavelet methods and many other numerical methods. Motivated and inspired by the nice characteristics of the Fibonacci wavelets, our goal is to introduce a new numerical technique based on Fibonacci wavelets for obtaining the solution of the non-linear Hunter-Saxton Equation (2). At the outset, the operational matrices of integration associated with Fibonacci wavelets are constructed by following the strategy of [PERSON] and [PERSON] [34], then the quasi-linearization technique is implemented to convert the underlying non-linear equation into the linearized form. Finally, the operational matrices of integration are employed to convert the problem under consideration into a system of algebraic equations, which are then solved with any conventional method, such as the Newton method. The remainder of the article is organized as follows: Section 2 is completely devoted to an elucidation of the Fibonacci wavelet method, including the construction of operational matrices. In Section 3, the proposed method is implemented for solving the non-linear Hunter-Saxton equation. The error and convergence of the proposed method are briefly illustrated in Section 4. In Section 5, the validity and efficiency of the proposed wavelet scheme are demonstrated on certain test problems. The article ends with an epilogue in Section 6.