A numerical treatment for the Gilson-Pickering equation using collocation method with error estimati

Abstract

Keywords: Gilson-Pickering Equation; Finite Element Method; Collocation; Septic B-Spline; Semi-Discrete Approximation

Introduction

Almost all physical processes encountered in nature are defined by various types of non-linear partial differential equations (NLPDEs). Many mathematical models are used to represent physical flows in various disciplines such as wave propagation, shallow water waves, reaction - diffusion models, biomechanical waves etc. [1]. Understanding the structure of these NLPDEs and seeking their solutions is of

prime importance for scientists, as their solutions illuminate the way to understand the behavior of systems and help predict the development of the process in nature. Thus, many mathematicians focus their attention on solving NLPDEs. However, usually it is difficult to find their solutions analytically and sometimes it is almost impossible. Therefore many researchers have been working on to find efficient and high accurate numerical algorithms to overcome such

*Corresponding author. *E-mail address: sbgkarakoc@nevsehir.edu.tr This paper was recommended for publication in revised form by Regional Editor Abdullahi Yusuf Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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problems [2]. In this study, the class of partial differential equations (PDEs) of third order, which are not completely linear, is taken into account. This class also includes different examples that have appeared recently in the literature and have been given outstanding motion wave solutions. These solutions consist of the sum of the exponential sums; we observe that these bases are seen in the presence of a linear inequality that appears to be a factor in motion wave reduction. In recent studies, it is seen that much attention has been paid to the development of numerical schemes. In this context, Claire Gilson and Andrew Pickering named the Gilson - Pickering (G-P) equation in 1995 developed a model [3]. Since this model has wide applications, it has recently managed to attract the attention of the scientific world. The model we consider in the study is as follows [4]:

method [13]. Exact travelling wave solutions of the G-P equation are investigated in [14]. Applicability of the first integral method to the regularized Gilson-Pickering equation under a parameter condition has been studied in [15]. H. M. Baskonus has applied an analytical method based on Bernoulli differential equation to extract new complex soliton solutions to the Gilson - Pickering model [16]. The invariance and multiplier approach has been applied to recover a few of the conserved quantities of the equation by G. Ebadi et al. [10,17]. But unfortunately, there are very few literature for G-P equation that focus on numerical studies. T. Ak et al. have discussed the quasiperiodic and chaotic behaviors of the perturbed Gilson-Pickering equation by analyzing phase portrait analysis, time series analysis and Poincare section [18]. The existence of smooth and nonsmooth travelling wave solutions under different parametric conditions are well studied here as well. The finite element method is one of the most effective techniques used in the solution of PDEs. B-spline basis functions have drawn attention in the theory of approximation, solution of boundary- value problems and PDEs by considering numerical properties. These functions are very useful for numerical calculations. Collocation method based on the B-spline basis functions is very effective technique in point of it has a programmable computation approach and easy to apply. Thus require less computational efforts by comparison other existent methods [19]. As collocation points increase in the collocation method, the problem will be met at more points that force the approximate solution to approach the exact solution. In these days, among various numerical methods [20-25], finite element method is also commonly applied to the problems for numerical solutions [26-43]. In this work, we present collocation method with septic B-splines for the numerical solution of the G-P equation. The paper has been designated as follows: In Section 2, we analyze Galerkin semi-discrete approximation for the G-P equation. In Section 3, a short discussion of septic B-splines is given. The proposed higher order B-spline collocation scheme has been implemented to the G-P model in Section 4. Section 5 shows the stability analysis of the numerical scheme and it is followed by Section 6 which contains numerical examples of the behavior of single soliton. Finally, in Section 7, a brief conclusion about the presented method is given.

(1) where ε, κ, α, β are nonzero real constants. This equation includes many other nonlinear models, such as: a. Fuchssteiner-Fokas-Camassa-Holm equation [5,6], with ε = 1, α = −3 and β = 2, which is a completely integrable nonlinear partial differential equation that arises at different levels of approximation in shallow water theory; b. Fornberg-Whitham equation [7,8] with ε = 1, α = −1, κ = 0.5 and β = 3, which was developed to analyze the qualitative characteristics of wave breakage and admits a wave of the highest height; c. Rosenau-Hyman equation [9,10], with ε = 0, α = 1, κ = 0 and β = 3, which arises in the study of the influence of nonlinear dispersion on the structure of patterns in liquid drops. When we use a transformation ξ = x − ct, Eq. (1) which is a PDE, can be written as

Thus we convert the equation to the above non-linear ordinary differential equation (ODE) and the ODE can be written as the following system of autonomous ODEs (2) This

The evolution of such u(ξ) is presented in Figure 1. Because of the great importance of G-P equation in nonlinear equations, many analytical methods have been used such as, Bernoulli sub-equation function method [11], a knot meshless method [12], first integral method [7], G'/G

Galerkin Semi-Discrete Approximation for the G-P Equation In this section, we analyze optimal-order L²-error for a standard spatial Galerkin semi-discrete approximation considering a uniform mesh and smooth splines for the following G-P equation with the initial and boundary values (3)

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Figure 1. Solutions of (2): Here figures on the left (a and c) show initial evolutions for a set of parameters and the figures on the right (b and d) show long ξ behaviour of the solutions for another set of parameters. Here in figures (a and b) we vary k and in figures (c, and d) we vary α to demonstrate the solutions behaviour for long ξ.

(4) (5) where u0(x) is a given L = (b − a) -periodic function smooth enough. We next establish the notation to be used throughout the article. For integer m ≥ 0, we denote Hm (Ω)the Sobolev the subspace of Hm (Ω) space of order m on Ω, consisting of L-periodic functions. We denote Cm (Ω), m ≥ 0, m-times continuously differentiable functions on Ω L-periodic such functions, (•,•) denoted the and

inner product of L2 (Ω) and the corresponding norm by and L∞ (Ω) norms are indicated by

. For integer r ≥ 2, we denote the N-dimensional space of L-periodic smooth spline

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or equal to r − 1 on Ji. The following statement for known [44,45]: Theorem If v is L-periodic and such that then there exists a

(9) where C is a generic positive constant independent of h, but may hold different real values at different occasions. In [46], Thomée et al. demonstrated that there occurs a basis , in fact if v is a smooth

enough L-periodic function, the associated quasi-interpolant

(15) In the same approach as [45], applying an energy procedure we demonstrate an optimal-order L2 estimate for the error of semi-discrete approximation described by the initial-value problem (14)-(15). Theorem Let u(x,t) be sufficiently smooth solutions of , h sufficiently small and r ≥ 3. Then, (3)-(5) in of the semidiscrete nonlinear problem (14)-(15) on [0, T]such that

(16) holds where C is a constant independent of h. Septic B-Spline Approximation In the present work, G-P equation is assumed with the boundary conditions u → 0 while x → ±∞, x and t which generally denote time and space, respectively. To obtain the solution on the interval [a,b] division a = x0 < x1 < ⋯ < xN-1 < xN = b of the space domain is imagined for m scattered uniformly with = 1, (1), N. The set of seventh degree B-spline functions , at the knots xm can be written on problem domain [a,b] as [47]:

Using the approximations (6)-(10), it can be seen that following result also possesses for quasi-interpolant. For and , we have [45] (11)

(13) Noting that and taking integration by parts, we describe standard Galerkin semi-discretization of G-P equation as follows. We seek , for r ≥ 3 satisfying for the equations:

(17) where a = (x − xm−4)7, b = (x − xm−3)7, c = (x − xm−2)7, d = (x − xm−1)7, e = (xm+4 − x)7, f = (xm+3 − x)7, g = (xm+2 − x)7, h = (xm+1 − x)7. B-spline functions have some confidental features like smoothness, local support and ability of handling local phenomena, which make them appropriate to solve linear and nonlinear partial differential equations easily and sensitively. Using septic B-spline basis functions give rise to a global solution for which both the functions and up to their sixth order derivatives are continuous [48]. Collocation method has two excellent advantages: establishing method does not include integrations and resulting matrix system is banded with small band width. Therefore, B-splines when associate with the collocation ensures a

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simple solution procedure of partial differential equations [49]. Approximate solution uN(x,t) for analytical solution u(x,t) is sought in the following equality,

(18) where σm(t) are time dependent unknown coefficients specified from the boundary conditions [50]. The septic B-spline functions are employed to overcome the higher order derivatives in the equation and when the bases are chosen at a high degree, generally better numerical results are obtained [51]. Since each septic B-spline covers eight consecutive elements, each finite element interval is covered with eight septic B-spline functions. In each element, applying the transformation to the spesific region is planned to more easily practicable region [0,1]. Thus, septic B-splines depending on variable ξ over the finite element [0,1] are defined as [52]:

Let’s assume that is linearly interpolated using the Crank-Nicolson finite difference approach and its time derivative is separated by the forward finite difference formula: (22) Hence, the above operations allows us to derive a recursion relationship between and for as [53]: (23) where

(24) (19) Using Eq. (18) and septic B-splines (19), nodal values of um and its derivatives are calculated in terms of element parameters σm in the following form

(20) Implementation of Collocation Method First of all, using (18) and (20) in Eq. (3), the following general form equation is reached for the linearization technique:

If we take a look at the algebraic system (23) we obtained above, the number of linear equations are less than the number of unknown coefficients, that is, the system contains of (N + 1) equation (N + 7) unknown time dependent parameters. The simplest way to find a unique solution is to remove six unknowns from the system. This procedure is applied using the boundary conditions with the values of u and after eliminating unknowns, following matrix-vector system is obtained (25) where . Now, initial parameters σ0 are established using initial condition and derivatives at the boundaries;

Thus, initial vector d0 can be determined in the following system of algebraic equations in matrix form:

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(31) G-P equation has only two invariants given by [56] (32) Dispersion of a single solitary wave The G-P equation has an exact solution of the form [57]

(33) Stability Analysis It is not easy to undertake the stability analysis of nonlinear partial differential equations. Most researchers solve with the problem by linearizing the partial differential equation [54]. We follow the Von-Neumann analysis for the stability of the scheme. In a typical amplitude mode, we can define the growth factor ξ of the error as follows: (26) where , h is element size and k is mode number. Putting the Fourier mode (26) into the iterative system (23), which give the growth factor: (27) where

and β are arbitrary real numbers. We will consider the G-P equation with the boundary-initial conditions which are (34) where u → 0 as x → ±∞

Case 1 For the first numerical calculation, we choose the quantities ε = 1, κ = −0.5, α = −3, β = −1.5, c = 0.5, Δt =

Numerical Applications And Results

In this section, in order to demonstrate the performance of our algorithm, some numerical examples are considered. Two sets of parameters have been used and discussed for numerical simulations of the motion of a single solitary wave with precise solutions. The performance of the proposed method will be checked with the L2 and L∞ error norms given as [55] (30)

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and efficient. In Figure 4, propagation of single solitary wave is displayed. Further, Figure 4 shows that the method we use in our article performs the propagation motion of a single solitary wave to the desired extent, moves at a constant speed, and maintains its shape and amplitude for a forward time as expected. Distribution of specific errors at time t = 1 are graphed in Figure 5.

Figure 3. Error distributions for the parameters of Table 1 at t = 1.

Case 2 For the second numerical calculation, we consider the G-P equation with the parameters ε = 1, h = 0.1, κ = 1, α = −4, β = −1, c = 0.5 and Δt = 0.01, to analyze the error norms and the quantities of the invariants for different space and time steps. In this case, single soliton has amplitude=0.83. Values of the invariants and error norms are presented in Table 2. Thus, we can see the effects of the amount of sorting points on the numerical method more easily. The values of invariants and errors hardly change as time progresses. It is observed from the Table 2 that our method is credible

Table 2. Invariant quantities and relevant error norms for Case 2. t

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a wide class of non-linear evolutional partial differential equations that arise in various scientific and engineering research. Moreover, the present approach of this study can be applied to other nonlinear evolution equations arising in different fields of nonlinear science.

Data Availability Statement

Figure 5. Error distributions for the parameters of Table 2 at t = 1.

Conclusion

In this paper, a finite element collocation scheme based on septic B-spline has been employed to investigate the propagation of non-linear dispersive solitary wave solutions of the G-P model. At first, we study Galerkin semi-discrete approximation for the equation. We have shown that our linearized scheme is unconditionally stable. In order to make numerical experiments, the algorithm has been studied together with the single solitary wave motion considering known analytical solution. The performance and validness of numerical algorithm has been measured by computing both L2, L∞ error norms and I1, I2 invariants. The results of the examples verify that our error norms are satisfactorily small. We conclude that the numerical scheme proposed here to approximate the solutions of G-P model is powerful, efficient and high accurate technique for solving

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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