Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves

1 Introduction

During the past few decades, there has been an increased amount of research on the use of fractional derivatives instead of integer derivatives and on the implementation of fractional integrals, fractional differential equations (FDEs), and fractional partial differential equations (FPDEs) in solving a series of problems arising in the pure and applied sciences [1–4]. The generalization of classical differential equations of integer order (IO) give rise to FDEs. Many complicated nonlinear phenomena that arise in the fields of heat transfer, wave propagation, and other subjects have been extensively described using fractional calculus, as is well known. Therefore, to describe physics, engineering, and other scientific subjects, FPDEs are crucial [5–8]. The fact that the response of the fractional-order system eventually converges to the response of the IO system has drawn a lot of attention to FDEs [9]. Their nonlocal characteristic is the primary benefit of employing fractional-order FDEs in various mathematical modelling. Many works have been accomplished to obtain the necessary mathematical methods for dealing with fractional problems. The goal of our investigation is to look for approximate results of time-fractional coupled Boussinesq–Burgers equation (TFCBBE) that is a nonlinear system. We investigate a time-fractional model of the coupled Boussinesq–Burgers equation (CBBE), which arises from the study of fluid flow dynamics, specifically the transmission of waves in shallow water. This model describes a wide range of physical phenomena in disciplines such as chemical physics, solid-state physics, fluid mechanics, and plasma waves. Exploring its solutions and properties is essential to understanding these phenomena, as solutions not only address specific problems but also offer deeper insights into the underlying physical processes. For instance, nonlinear wave behaviours in fluid dynamics, plasma, and optical fibers are often represented by characteristic bell-shaped and kink-shaped solutions.

As noted in the literature, Zhang et al. [10] proposed a ( 2 + 1 ) -dimensional Boussinesq–Burgers soliton (BBS) equation and its connection with the BBS hierarchy. They proved that quasi-periodic solutions of the ( 2 + 1 ) -dimensional BBE are derived by resorting to the Riemann θ functions. Zhang et al. [11] investigated the exact solutions of the generalized Boussinesq equation and the BBE using the modified mapping method. Rady et al. [12] showed that the periodic wave solutions for BBE are obtained by using the Jacobi elliptic function method and presented the properties of some periodic and multiple soliton solutions. Chen and Li [13] presented a new Darboux transformation with multiple parameters for BBE using Darboux transformation and found new soliton solutions of the BBS. Gupta and Ray [14] proposed the comparison between the homotopy perturbation method (HPM) and the optimal homotopy asymptotic method (OHAM) using approximate solution of nonlinear BBE and concluded that optimal homotopy asymptotic method is accurate with a smaller number of iterations compared to the HPM, while the OHAM introduces a simple way to control convergence region.

Wang et al. [15] investigated the approximate solution of fractal CBBE using the semi-inverse method and established the fractal variational formulation of fractal CBBE. Kumar et al. [16] represented the approximate solution of CBBE using residual power series method and compared their results with the analytical solutions as well as the solutions shown on different graphical representations. They also obtained the solution by the modified homotopy analyzis transform method. Khater and Kumar [17] constructed exact solutions for the fractional CBBE using fractional complex transform and the modified Riemann Liouville derivative sense. Shi et al. [18] investigated into a few of the time-fractional CBBE’s invariant characteristics. First, using the discovered symmetries, the fractional CBBE is converted into nonlinear time FDEs. Second, they used a power series expansion method to solve the reduced system of fractional ordinary differential equations. Third, the new conservation theorem is used to derive the conservation rules of the fractional CBBE. Ravi et al. [19] analyzed the exact solutions of CBBE using analytically as well use the Exp-function method. They also shown their result graphically. Ali et al. [20] investigated the analytical solution of CBBE through the sine-Gordon expansion method and results shown on 2 D and 3 D plotting. Heydari and Avazzadeh [21] introduced the approximate solution of CBBE by using the concept of the Caputo form. Zarin et al. [22] analyzed the fractional-order susceptible infected recovered again susceptible model using Caputo derivatives and solved numerically using the Haar collocation-Broyden method. Jitsinchayakul et al. [23] proposed the fractional-order COVID-19 epidemic model using the Atangana–Baleanu–Caputo fractional operator and analyzed the results. Chu et al. [24] presented a fractional-order SARS-CoV-2 transmission model using the Atangana–Baleanu derivative and solved numerically using Newton polynomial-based iterative scheme. Zarin et al. [25] investigated reaction–diffusion COVID-19 model using the next-generation method and solved numerically using finite difference and a one-step explicit meshless scheme. Zhang and Whang [26] presented the numerical solution of coupled Burgers equations using a space-time semi-analytical meshless scheme and radial and non-radial basis functions. Wang et al. [27] investigated the approximate solution of the Burgers equation using the multistage homotopy asymptotic method and compared their result for a large time span.

Iqbal et al. [28] investigated the theoretical foundation of the nonlinear Schrödinger equation is connected with real-world ocean engineering applications in this article, which offers solutions that demonstrate several wave forms, including solitons and periodic waves, utilizing three different approaches. Deepening our grasp of wave dynamics and improving underwater engineering and design are the goals of these efforts. Rehman et al. [29] used two new mathematical techniques to analyze the ( 3 + 1 ) -dimensional extended quantum Zakharov–Kuznetsov equation in plasma and obtain a variety of soliton solutions. The results provide useful tools for investigating nonlinear evolution equations and provide fresh perspectives on the dynamics of the model, as demonstrated by 2 D , 3 D , and contour plots. Iqbal et al. [30] utilized novel techniques to acquire different optical soliton solutions of the Sasa–Satsuma equation, which simulates the propagation of femtosecond pulses in optical fibres. These results improve our knowledge of optical phenomena and present new methods for studying nonlinear optical systems. Zhu et al. [31] investigated special solitary wave solutions of the Heisenberg ferromagnetic spin chain equation and uncover its function in nonlinear wave dynamics. They show how these solutions improve knowledge of solitons in mathematical physics and its applications in domains such as fluid dynamics and fibre optics using straightforward but powerful analytical methods. Kai and Yin [32] presented a broad multi-wave solution framework by using the Cole–Hopf transformation to reveal the linear structure of the Sharma–Tasso–Olver–Burgers equation. Notably, it uses bifurcation techniques to expose the paths of dark-bright and kink–kink soliton molecules, as well as their fusion and fission processes. He and Kai [33] analyzed Kudryashov’s equation with third-order dispersion, periodic and soliton solutions, and exploring modulation instability and chaotic behaviour under many perturbations. The findings provide valuable insights into the equation’s wave dynamics and potential applications in complex phenomena. Xie et al. [34] proposed freedom asymmetric Duffing system of degree two with fractional damping and stability analysis using approximate methods. The study highlights the impact of fractional damping on controlling rotor vibrations and practical applications in vibration management. Kumar et al. [35] presented the exact solitary wave solutions of the strain wave equation using the generalized exponential rational function method and proved the effectiveness of the method with clear physical interpretations. Kumar et al. [36] investigated the (1 + 1)-dimensional longitudinal wave equation to derive a novel solitary wave using the unified method and the unified Riccati equation expansion method. Kumar et al. [37] analyzed the Wazwaz–Benjamin–Bona–Mahony equation using Adomian, Lie isomorphism, and unified Riccati equation expansion methods to obtain exact soliton solutions. Hussain et al. [38] solved the nonlinear Benjamin Bona Mahony Burgers equation to obtain the soliton solutions using extended exp ( − Φ ( χ ) ) expansion and Kudryashov methods and analyzis of the fluid dynamics. Hussain et al. [39] investigated the nonlinear Sharma Tasso Olver Burgers equation using the extended ( Ǵ G 2 ) expansion method to obtain soliton solutions and analyzis the stability, and sensitivity.

The present study employs the quintic B-spline (QBS) method in conjunction with the Caputo derivative for the numerical solution of the TFCBBE. The L 1 formula is applied for handling the non-IO derivative, whereas the discretization of the space derivative is carried out with the help of B-spline functions. The benefits of utilizing the spline collocation technique in comparison to the currently available methods lie in its ability to effectively provide a piecewise continuous, closed-form solution. In addition, it offers a more straightforward approach and can be applied to a diverse set of problems related to partial and FPDEs. The superiority of this approach compared to alternative techniques lies in the fact that upon obtaining the solution, the requisite data for interpolation at various points within the range becomes accessible. The primary advantage of this technique is its capability to closely approximate the analytical curve up to a specified level of continuity. Consequently, the spline methodology offers more precise outcomes when estimating values at any given location in the field, in contrast to its predecessors.

Several researchers have used numerical methods to solve FPDEs. The presented method is innovative in that it uses a Crank–Nicolson scheme to discretize the fractional derivative. This method is chosen for its suitability in capturing complex nonlinear wave behaviours, such as bell-shaped and kink-shaped waves in fluid dynamics and optics. Spatial derivatives for TECBBE are discretized using QBS functions. The proposed scheme is novel, and as far as we are aware, the fractional derivative in the Caputo sense has never been employed for the numerical solution of TFCBBE using QBS functions before. The article commences by presenting the TFCBBE as a means for determining numerical solutions to the TFCBBE with the help of the Crank–Nicolson approach. This innovative technique depends on the θ -weighted scheme and the Caputo operator. Initially, we constructed the QBS function across the entire domain and employed the Caputo time fractional derivative for temporal discretization. In addition, a numerical method is introduced that utilizes a θ -weighted scheme and quasi-linearization. Subsequently, a stability analyzis is conducted. Finally, two numerical test problems are provided to demonstrate the suggested technique, together with their corresponding graphical representation. Concluding remarks are provided at the conclusion of the article.

In this work, the QBS functions are used to solve the TFCBBE numerically. They are piecewise polynomial functions, which are smooth and offer flexibility in approximating solutions. The QBS, which are fifth-degree polynomials, offer C 4 continuity, ensuring smooth transitions at each joint between piecewise segments, which is valuable for accurately approximating solutions to differential equations. The CBBE model has been previously solved using the QBS approach combined with the Crank–Nicolson formulation. Moreover, the time-fractional version of this model has been addressed using the time-spectrum function method, the power series expansion method and the homotopy perturbation method. However, no research has been conducted for solving the TFCBBE model using the QBS approach alongside the Crank–Nicolson scheme. This gap highlights the unique contribution of this study, offering a novel approach for solving the time-fractional version of the model. The novelty of this work lies in the numerical solution of the fractional CBBE by employing the Caputo fractional derivative for time discretization and QBS functions for spatial discretization. This innovative approach allows for accurate and effective solutions at each node, offering a unique and effective method for addressing the fractional CBBE. This combination improves accuracy and stability by balancing the time step’s implicit and explicit contributions. Compared to current approaches, this methodology has the advantage of improved numerical solution smoothness and precision. This particular time-fractional equation has never before been solved using such a combination, illustrating the method’s superiority in providing an accurate, piecewise cubic, and smooth solution for complex fractional-order equations.

The article is organized as follows: In Section 2, the TFCBBE model is presented along with the necessary initial and boundary conditions (BCs). The preliminaries are described, consisting of two subsections. Caputo fractional derivative is presented in subsection 3.1, and subsection 3.2 provides an overview of the QBS functions at various points. Section 4 discusses the process of discretizing the fractional derivative, whereas Section 5 outlines the steps involved in the suggested numerical technique. Section 6 establishes the starting condition for the approximate solution. Section 7 provides the algorithm of our numerical technique. Sections 8 and 9 examine the stability and convergence analyzis of the proposed methodology. Section 10 presents the numerical test problems. Section 11 outlines the study’s conclusion.

2 Model

The TFCBBE, with real-valued functions U ( x , t ) and V ( x , t ) , takes the following form (Table 1):

with initial conditions (ICs)

and BCs

3 Preliminaries

4 Illustration of the scheme

The technique of forward finite difference is utilized for the discretization of the Caputo fractional derivative. Consider step size is τ = T M and t p = p τ , where p = 0 , 1 , … , M . At point t p , Caputo’s L 1 formula [42] can be defined as follows:

where A s = ( s + 1 ) − λ + 1 − s − λ + 1 and B τ p + 1 is bounded truncation error, i.e. ∣ B τ p + 1 ∣ ≤ Θ τ 2 − λ [43], where Θ is the constant which depends upon U .

The coefficients A s preserve the following properties [44]:

5 Numerical technique

The θ -weighted scheme is used to find the approximate solution of the system (2.1) at t p + 1 th time level, which can be written as follows [45]:

where

Setting the value of θ yields three schemes. First, when θ is set to 0, the scheme mentioned earlier turns into an explicit scheme. The second is that when θ = 1 , θ -weighted scheme transforms into an implicit scheme. Finally, when θ = 1 ⁄ 2 is taken, the scheme transforms to the Crank–Nicolson scheme because it is an unconditionally stable scheme that provides reasonable accuracy. By employing Eqs. (5.1) and (2.1), we obtained:

Eqs. (5.2) and (5.3) contain non-linear terms that we aim to linearize at the ( p + 1 ) th stage. By simplifying complex non-linear terms, this linearization technique [46] facilitates iterative equation solving, which is defined as follows:

Using above relation, Eqs. (5.2) and (5.3) become:

where C = 1 τ λ Γ ( 2 − λ ) . After simplification, Eqs. (5.4) and (5.5) are yield into Eqs. (5.6) and (5.7):

By putting the values of QBS basis function in Eqs. (5.6) and (5.7), then we obtained Eqs. (5.8) and (5.9):

where

Matrix form of Eqs. (5.8) and (5.9) becomes:

After some simplification, we obtain

Here, the square matrix of order ( 2 N + 10 ) × ( 2 N + 10 ) is denoted by E . The column vector I consists of the order ( 2 N + 10 ) and F p + 1 = [ α − 2 p + 1 , α − 1 p + 1 , … , β N + 2 p + 1 , β − 2 p + 1 , β − 1 p + 1 , … , β N + 2 p + 1 ] T contains the control points.

where