1. Introduction
Non-linear partial differential equations play a vital role in modeling real-life phenomena, especially in the field of applied sciences such as meteorology, aerospace engineering, oceanography, geology, astronomy, and many other disciplines. For the realistic treatment of these non-linear models, several numerical techniques such as the barycentric interpolation collocation method [
1], Fourier spectral method [
2], and reproducing kernel method [
3] have been adopted for the numerical treatment of these non-linear partial differential equations, such as the Klein–Gordon–Zakharov equations, Schrödinger equation, Duffing systems, and many more. The Hunter–Saxton equation is one among those non-linear partial differential equations that describe waves of a nematic liquid crystal in an enormous director field and arise at the short-wave limit of the Camassa–Holm equation—an integrable model of shallow-water waves propagating uni-directionally across a flat bottom [
4]. This equation has both the bi-Hamiltonian and bi-variational structure of integrable partial differential equations, often referred to as the Hunter–Saxton equation, which re-expresses the geodesic flow in the diffeomorphism group of a circle. Formally, the non-linear Hunter–Saxton equation is given by
Equation (
1) can also be redrafted as
where
and
denote the space and time coordinates, respectively, with initial and boundary conditions:
In practice, it is more appropriate to impose a boundary condition of the form:
or even the initial and boundary conditions:
where
, and
are sufficiently smooth functions.
Keeping in view the prolificacy of the Hunter–Saxton equation governed by the differential Equation (
2), a substantial amount of research has been carried out in the quest to obtain numerical solutions via several analytical and numerical techniques. For instance, Behzadi [
5] employed the homotopy analysis method (HAM) for the approximate solution of the non-linear Hunter–Saxton equation and showed that the HAM is more rapidly convergent than certain other iterative methods such as the variational iteration method, Adomian decomposition method, modified variational iteration method, and modified Adomian decomposition method, whereas Wei et al. [
6] established the global solutions of the periodic Hunter–Saxton equation with weak dissipation. Aratyn et al. [
7] obtained exact rational solutions of the generalized Hunter–Saxton equation by using the Padé approximation. Besides, Rouhparvar [
8] proposed the reduced differential transform method for obtaining the solution of the non-linear Hunter–Saxton equation, which does not require linearization, discretization, nor perturbation of the underlying non-linear differential equation. Hashmi et al. [
9] examined the efficiency of the cubic trigonometric
B-spline method for obtaining the numerical solution of the non-linear Hunter–Saxton equation. Karaagac et al. [
10] applied the finite element method by reducing the governing non-linear partial differential equation to a system of ordinary differential equations in terms of shape functions. Sato [
11] formulated a stable and convergent finite difference method for the modified Hunter–Saxton equation on a periodic domain. Nonetheless, very recently, Ahmad et al. [
12] demonstrated a highly applicable aspect of the Hunter–Saxton equation while undertaking the numerical treatment of the liquid crystal model described through the Hunter–Saxton equation by employing the Sinc collocation method via the theta weighted scheme.
Over the last couple of decades, wavelet theory has seen unprecedented progress in the form of innovative and efficient numerical algorithms. Wavelet-based numerical schemes have dethroned the usual numerical methods and have received much attention from researchers working in various disciplines of science and engineering [
13,
14]. In particular, the wavelet-based matrix methods have acquired a respectable status in numerical analysis due to their easy implementation, simple procedure, and rapid convergence. Moreover, they are computer-oriented and do not require calculating the inverse of the wavelet matrix, and less CPU time is needed as the major blocks of the wavelet operational matrix are computed once and reused in subsequent computations. As of now, numerous linear and non-linear differential equations have been solved by employing various wavelet families, including the Haar wavelets, Euler wavelets, Bernoulli wavelets, Chebyshev wavelets, Legendre wavelets, Hermite wavelets, Gegenbauer wavelets, and ultraspherical wavelets [
15,
16,
17,
18,
19,
20,
21,
22]. To harness the advantages of wavelet-based numerical methods in the context of the non-linear Hunter–Saxton equation, many wavelet-based methods have been employed for obtaining the numerical solution. To mention a few, Arbabi et al. [
23] utilized the Haar wavelet quasi-linearization approach to obtain the approximate solution of the Hunter–Saxton equation. Recently, Srinivasa et al. [
24] developed a collocation technique based on the Laguerre wavelet for the numerical solution of the Hunter–Saxton equation and exhibited the uniform convergence of the employed method.
The origin of the Fibonacci numbers dates back to the early 1200s with the seminal work of Italian mathematician Fibonacci studying the famous rabbit problem. The Fibonacci numbers
with initial values
are the terms of the sequence
. Using the Fibonacci-like recurrence relations, Catalan and Jacobsthal introduced a large class of polynomials commonly known as Fibonacci polynomials. The polynomials
introduced by Catalan are defined by the recurrence relation
with
, whereas the polynomials introduced by Jacobsthal have the general form
,
with
. Later on, Byrd defined another class of Fibonacci polynomials of the form
, where
. Over a couple of decades, some prominent offshoots of the Fibonacci polynomials have appeared in the open literature. For instance, Falcon and Plaza [
25] introduced the notion of
h-Fibonacci polynomials and studied the derivatives of these polynomials in terms of the convolution of
h-Fibonacci polynomials. Mathematically, the
-Fibonacci polynomials
are defined by the recurrence relation
, where
,
, and
is a polynomial with real coefficients. It is pertinent to mention that for
and
, one can obtain the Fibonacci polynomials of the Catalan and Byrd type, respectively [
26]. Lee and Asci [
27] define another novel generalization of Fibonacci polynomials coined as
-Fibonacci polynomials and obtained several results regarding the factorization of the Pascal matrix involving these polynomials. Such polynomials are defined by the recurrence relation
, where
, and
and
are polynomials with real coefficients. Subsequently, Catarino [
28] defined a quaternion analogue of the
h-Fibonacci polynomials of the form
where
is the
-Fibonacci polynomial. Recently, Strzałka 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 Chen and Hsiao [
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.