A numerical solution of the dissipative wave equation by means of the cubic B-spline method

In the past few decades, partial differential equations have drawn considerable attention, owing to their ability to model certain physical phenomena. The aim of this paper is to investigate a cubic B-spline polynomial to obtain a numerical solution of a nonlinear dissipative wave equation. For the numerical procedure, the time derivative is obtained by the usual finite difference scheme. The approximate solution and its principal derivatives over the subinterval is approximated by the combination of the cubic B-spline and unknown element parameters. The accuracy of the proposed method will be shown by computing L∞ error norms for different time levels. By applying Von Neumann stability analysis, the developed method is shown to be conditionally stable for given values of specified parameters. A numerical example is given to illustrate the accuracy of the cubic B-spline polynomial method. The obtained numerical results show that our proposed method maintains good accuracy.


Introduction
The nonlinear partial differential equations arise in a wide variety of physical phenomena in several different aspects of physics, such as water wave theory, fluid dynamics, plasma physics, solid mechanics, and nonlinear optics, etc. There are many methods for solving partial differential equations via numerical solutions. One of these is numerically solving a nonlinear dissipative wave equation by using the Adomian decomposition method [1,2]. The cubic B-spline, used for solving nonlinear partial differential equations, has been employed by many researchers. The most known and well-focused results are those presented by Daǧ et al (2004) who presented a way to solve the Regularised Long Wave (RLW) equation. The numerical results obtained in this paper demonstrate that the method is capable of solving the RLW equation accurately and reliably [3]. Daǧ et al published a paper that described a numerical solution for the one-dimensional Burger's equation in 2005. The comparison of the calculations with the analytic solution shows that a cubic B-spline collocation method is capable of solving Burgers' equation accurately. The proposed method is easy to implement and does not require any inner iteration or corrector to deal with the nonlinear term of Burgers' equation [4]. Khalifa et al (2008) discussed the Modified Regularised Long Wave (MRLW) equation. The collocation method using cubic B-splines was applied to study the solitary waves of the MRLW equation, and it is shown that the scheme is marginally stable. Moreover, despite the fact that the wave does not change, results show that the interaction results in a tail of small amplitude in two, and clearly in three, soliton interactions, and the conservation laws were reasonably satisfied. The appearance of such a tail can be beneficial in further study [5]. In 2008, El Danaf and E I Abdel Alaal constructed a non-polynomial spline-based method to obtain numerical solutions of a dissipative wave equation. The obtained numerical results show that their proposed method maintains good accuracy [6]. Later, Mittal and Jain (2012) argued that some numerical method should be proposed to approximate the solution of the nonlinear parabolic partial differential equation with Neumann's boundary conditions. The numerical results produced by the present method are quite satisfactory and in good agreement with the exact solutions. The computed results justify the advantage of this method. The proposed method can be extended to solve multi-dimensional parabolic equations [7]. In 2015, Zaki developed a new numerical method based on quadratic non-polynomial spline functions, which has three coefficients in each sub interval for solving a dissipative wave equation. The results obtained by the proposed technique show that the approach is easy to implement and computationally attractive. The proposed method is shown to be robust, efficient, and easy to implement for linear and nonlinear problems arising in science and engineering [8]. A year later, El-Danaf et al addressed methods for solving the Generalised Regularised Long Wave (GRLW) equation. The cubic B-splines used to study solitary waves of GRLW equation show that the scheme is unconditionally stable. Also, the obtained approximate numerical solutions maintain good accuracy when compared with the exact solutions [9]. Hepson and Daǧ, in their 2017 research, implemented a numerical technique to obtain approximate solutions of Fisher's equation. The method is capable of producing solutions for Fisher's equation fairly and can be used as an alternative to the method's accompanied B-spline functions [10]. In 2017, Iqbal et al's proposed numerical technique was based on the cubic B-spline collocation method. Their version used a new approximation for the second order derivative. The proposed scheme is based on the cubic B-spline collocation method equipped with a new approximation for second order derivative and produces fifth order accurate results. The proposed method also generates a piecewise spline solution in the presence of the singularity, which can be used to obtain a numerical solution at any point in the domain and is not restricted to the values at the selected knots, unlike existing finite difference methods [11]. A year later, Başhan (2018) studied the numerical solutions of the third-order nonlinear Korteweg-de Vries (KdV) equation by using modified cubic B-splines in five different test problems. The performance and accuracy of the modified cubic B-splines method was shown by calculating and comparing the L 2 and L ∞ error norms with earlier works. The stability analysis has been performed for all of the test problems, and all of the eigenvalues are in convenience with stability criteria. So, MCBC-DQM may be useful in obtaining the numerical solutions of other important nonlinear problems [12]. In research conducted in 2019 by Bashan, a modified cubic B-spline differential quadrature method has successfully been implemented for the numerical solution of nonlinear Kawahara equation. To obtain the first, third, and fifth-order derivative approximation, a modified cubic B-spline differential quadrature method was utilised. Four different test problems have then been investigated separately. These newly obtained results obviously indicate that a modified cubic B-spline differential quadrature method can be used to produce numerical solutions of the Kawahara equation with high accuracy [13]. More recently, Iqbal et al studied the Galerkin method, based on a cubic B-spline function, where the shape and weight functions are applied for the numerical solution of the one-dimensional coupled nonlinear Schrödinger equation. The use of the cubic B-spline Galerkin method produced smooth solutions without numerical smearing in 2020 [14]. In the same year, by Ahmed et al (2020) a Non-polynomial spline function was used to get numerical solutions of a Dissipative Wave equation at middles points for lattice in space direction and at the same time, a finite difference method was used in time direction. The presented method is shown to be conditionally stable. The approximating results showed to have well agreement compared with the true solutions, hence it can be used to set approximate solutions for such type of problems [15]. In the current work, we propose a mathematical treatment for the nonlinear dissipative wave equation, utilising the collocation technique with cubic B-spline shape functions. For the mathematical methodology, the time derivatives will be achieved through the typical finite difference method. The technique will be shown to be conditionally stable by applying the Von Neumann stability investigation procedure. We will test the precision of the proposed strategy by conducting an examination of the mathematical outcomes and the specific arrangement of the condition.

The governing equation and the derivation of the proposed method
This paper is worried about applying the cubic B-spline method to build up a mathematical strategy for approximating the specific arrangement of the nonlinear dissipative wave equation [1] of the structure: Under the boundary and initial conditions: Let the cubic B-spline basis functions AE x i ( ) given as: 3   2  3  2  1  3  2  1  1  2  1  3  1  3  2  1  1  2  1  3  1   2  3  1 1 are the basis for the function defined over the interval a b , , [ ] this implies that the estimations of the cubic B-spline AE x i ( ) and its derivatives vanish outside the interval where w t i ( ) are the time dependent parameters which can be resolved utilizing the boundary conditions: Applying the finite difference method, we have:

The initial state
In this section, we apply the first initial condition: Table 1. The values of AE x i ( ) and their derivative within the interval - The initial conditions can be communicated as: From graphs ( figure 1-8), it is clearly seen that the present method produces numerical results in good agreement with the exact solutions. The results indicate that, the proposed algorithm is substantially more stable and efficient than that of [15]. x  To find the second initial condition using Taylor expansion to U x t , Subbing equations (2.1) into (3.6) we get: After simplifying, equation (3.7) becomes: To complete this system, differentiate (3.9) with respect to x, and compute its value at the ends of the range, which gives us the following system: while H has the form: , ..........., , .

Stability analysis
The Von Neumann stability analysis for system (2.9) takes effect after linearizing the nonlinear term as: 6 , 6 .  Equation (4.9) is a quadratic in e and hence will have two roots, that is if h is small enough, thus the method is conditionally stable.

Numerical illustration
We apply cubic B-spline method to obtain numerical solution of the dissipative equation for one standard issue. The precision of our proposed mathematical technique estimated by registering the ¥ L error norm. The exact solution of the dissipative equation (2.1) which obtained in [1] given by: cos sin , 0 , 0.

( )
We use the following conditions: We put the acquired mathematical outcomes in the accompanying tables 2-7. From tables 3-7, we observe that the smaller the Δ = t k (than the value of h), the better the accuracy. The numerical approximations is still acceptable within the large time.

Conclusion
In this paper, a numerical solution for the nonlinear dissipative wave equation, utilising a collocation strategy with the cubic B-splines, is proposed. To illustrate our method and to demonstrate its convergence and applicability of our presented methods computationally, we will apply the Von Neumann stability method. The stability analysis investigation will show that the method is conditionally stable. The performance and accuracy of the present method have been shown by calculating and comparing the L ∞ error norms with earlier works. The obtained invariants are considered acceptable when compared with some earlier works. The numerical results produced by the present method are quite satisfactory and show good agreement with the exact solutions.
The computed results justify the advantage of this method. As seen in tables 3 and 4, the present results are better than [15]. The estimated mathematical arrangements that achieve great precision with the specific arrangements, particularly when Δt is more modest than the estimation of h.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).