1 Introduction
Wave phenomenon is encountered in a context of hydrodynamics and thermodynamics nearly in all areas of life. Let’s share the wave phenomenon in our lives with the simplest and most common example of our daily lives: rainbow. Everyone marvels to watch the image of a rainbow shimmering against a dark and stormy sky. How sunlight falling on clear raindrops simultaneously brings out the colors we see. We can see seven colors in the rainbow, including indigo [1]. These colors are related to the wavelengths of light. That is, our eyes interpret these wavelengths as different colors. We see a certain color only if we see a color wavelength or a limited range of wavelengths. We see all wavelengths of visible colors as white, if there are no wavelengths in the visible medium, we see all of them as dark. That is, white light is a smooth mixture of all visible wavelengths. When white light hits a water droplet, it spreads out according to its wavelengths. Dispersion is the spread of white light over the entire spectrum of wavelengths. Technically, dispersion is a process that changes the direction of light with respect to its wavelength. When light enters a transparent material such as a water drop, it dissipates some of its energy as heat energy and loses some of its density, this phenomena is called as “dissipation”. Moreover, dispersion and dissipation are general wave phenomena and can occur for any wave type. Dispersion and dissipation occur in a various biological, chemical and physical conditions in which wave phenomena play a role, from the movement of gas bubbles in a compressible liquid to the formation of black holes. Fundamental fields of science such as applied mathematics, computer science, fluid dynamics, and general relativity are the branches of science in which the wave phenomenon is the most studied one. The wave phenomenon has a vital place in the theory of partial differential equations due to the fact that large-scale complex phenomena are modeled with partial differential equations. The examination of the analytical and approximate solutions for partial differential equations which are nonlinear is one of the important fields of the science. Many scientists have emphasized the challenges of working on partial differential equations that modelling nonlinearity and dispersion/dissipation effects [2‐4]. The KdV and Burgers equations have taken attention together with their nonlinearity and dispersion/dissipation properties on formation of solitons. The papers on KdV–Burgers equation, which is the combination of these two equations, have made a great contribution to science by examining the solutions of the equations by investigating the mentioned phenomena [5‐7]. The KdV–Burgers (KB) equation was investigated by Johnson [8] throughout the study of the wave motion on liquid-filled elastic tubes. The main idea of the study is the derivation of the KdV–Burgers equation, and the solution of the KB equation u(x, t) changes in proportion with the radial perturbation of the tube wall [9]. As it is known, KB equation models a liquid with small bubbles, fluid in an elastic tube, and turbulence [10]. The Burgers Fisher equation is a vital equation in the field of fluid dynamics. Inasmuch as it has a highly nonlinear nature, it represents reaction, diffusion and convection mechanisms and takes an crucial role in the application of gas dynamics, heat conduction and elasticity [11]. For further information, one can read the articles given in the references as [12‐14]
In this study; we have proposed to investigate numerical solutions of a new model named as KdV–Burgers–Fisher (KBF) equation which combines reaction properties of Fisher equation with dispersion properties of KdV and dissipation properties of Burgers equation. Thus, the KBF equation provides an example of DDR (dispersion–dissipation–reaction) systems. The motivation of the paper is based on the fact that it has a wide range of applications of dispersion–dissipation–reaction models on environmental studies such as pollutants in air, water, or soil, chemical engineering studies as chemical reactions occur in reactors, biology studies as population dynamics, disease spread. Moreover, DDR models arise in improving water treatment processes, optimize energy savings, modelling pollutant transport in the atmosphere, identify population migration and predict its growth. Since discovering numerical or analytical solutions of DDR system will assist us to discover the manner in which factors balance one another in a specific event, which factor is dominating, and how systems may be optimized, thus the current article will make an important contribution to the literature. In order to investigate numerical solutions of the KBF equation, we have used collocation method based on finite element approach with the help of quintic trigonometric B-spline basis. The main purpose of the collocation method is based on the idea of creating an approximate solution that provides the governing equation at specific collation points so that the residues in a finite-sized space are zero.
Anzeige
The collocation method is described in several important studies, such as, Jena and Senapati [15] have utilized the cubic spline approach to numerically solve the one-dimensional (1D) heat problem (one spatial and one temporal dimension), employing both explicit and implicit strategies. Additionally, in [16, 17], Jena and co-authors have considered one-dimensional heat, advection–diffusion equation and Klein Gordon, Sine-Gordon equations using improvised cubic B-spline collocation, finite element method and Crank–Nicolson technique, respectively. Iqbal et al. [18] have presented an application of fifth degree basis spline functions for a numerical investigation of the Kuramoto–Sivashinsky equation. Karabenli et al. [19] have focused on obtaining approximate solutions of the fractional Fokker Planck equation using trigonometric quintic B-spline basis functions. Jena and Senapati [20] have handled the complex-valued Schrodinger equation and solved using a nonic B-spline collocation finite element method. The numerical solutions of Emden–Fowler type equations have been investigated in [21]. Moreover, the authors have addressed third and fourth order Singular Boundary Value problems with the help of new Quartic B-splines in [22, 23], respectively. They used typical QBS functions in combination with novel approximations for third and fourth order derivatives to interpolate the solution in spatial domain. Kutluay et al. [24] have employed collocation finite element method where cubic Hermite B-splines are used as trial functions to solve the Modified Equal-Width (MEW) equation, and in [25], one can find numerical solution of the modified regularized long wave equation using a quartic B-spline approach and Butcher’s fifth-order Runge–Kutta (BFRK) scheme. The collocation method combination with octic B-spline are applied to Benjamin–Bona–Mahony–Burgers equation in [26]. Now, let us consider the KBF equation given as [27]When \(\epsilon =0,\) the above equation transforms into the fifth order KdV equation [28, 29] and when \(v=\epsilon =0,\) it turns into modified Kawahara equation [30]. The model given in (1) is studied in [27] at first time, and Koçak have investigated its travelling wave solutions via tanh method.
$$\begin{aligned} & \varphi _{t}\left( x,t\right) +\varepsilon \varphi ^{2}\left( x,t\right) \varphi _{x}\left( x,t\right) -\upsilon \varphi _{xx}\left( x,t\right) +\mu \varphi _{xxx}\left( x,t\right) \\ & \quad -\gamma \varphi _{xxxxx}\left( x,t\right) -\epsilon \varphi \left( x,t\right) \left( 1-\varphi \left( x,t\right) \right) =0. \end{aligned}$$
(1)
The layout of the manuscript can be given as follows: Sect. 2 exhibits the existence and uniqueness of the found solution using the fixed point theory. In Sect. 3 and its subsection, in order to set numerical scheme, discretization procedure of space-time domains, constructing approximate solutions with the help of quintic trigonometric B-spline basis are presented. In Sect. 4, the numerical scheme for the problem is obtained and in subsections, application of boundary conditions and obtaining initial vector are discussed. Section 5 is proposed for stability analysis of the scheme using a useful procedure named as von-Neumann technique, and truncation error of the scheme, respectively. Section 6 contains the numerical results, calculation of the error norms, convergence rates, CPU times and graphical simulations of numerical results for two problem. In the last section of the manuscript, a brief conclusion has been presented.
2 Existence and uniqueness of solution
In the present section, we explore the existence and uniqueness of the solution of KBF equation given in (1). Now, Eq. (1) can be considered as follows:where \(K\left( x,t,\varphi \right) =-\varepsilon \varphi ^{2}\varphi _{x}+\upsilon \varphi _{xx}-\mu \varphi _{xxx}+\gamma \varphi _{xxxxx}+\epsilon \varphi \left( 1-\varphi \right) .\) The foregoing equation is transformed into the following equation by imposing the integral operator both side of the Eq. (2) as:where \(\varphi \left( x,0\right)\) is initial condition. Here, we use the normNow, it is required to show the Lipschitz condition and contraction for \(K\left( x,t,\varphi \right) .\) In order to prove, we consider two functions \(\varphi _{1}\) and \(\varphi _{2},\) then one obtainsLet us to assume \(\varphi \left( x,t\right)\) is bounded and the functions which involving its derivatives satisfy the Lipschitz condition,and so, there are positive constant \(L_{\varepsilon },L_{\upsilon },L_{\mu },L_{\gamma },L_{\epsilon }.\) Thuswhere \(L_{\varepsilon }=\varepsilon C\left( \left\| \varphi \right\| +\left\| \varphi \right\| ^{2}\right) ,\) \(L_{\epsilon }=\epsilon \left( 1+2\,M\right) ,\) we can haveHere, \(L=L_{\varepsilon }+L_{\upsilon }+L_{\mu }+L_{\gamma }+L_{\epsilon }\) is final Lipschitz bound. Moreover, if \(0\le L<1,\) then it leads to contraction. Now, according to (3), we develop the following iterative formulaand the associated initial condition is \(\varphi \left( x,0\right) =\varphi _{0}\left( x,t\right) .\) The difference of successive iterations can be presented as followsOne can easily observe that:Using the iterative formula given in Eq. (5), we getAs we know kernel \(K\left( x,s,\varphi _{n}\right)\) holds the Lipschitz condition, so it givesSuppose that the function \(\varphi \left( x,t\right)\) is bounded and it is already known that the kernel holds the Lipschitz condition and using Eqs. (6) and (9), we getTherefore, the function given in (7) exists and smooth. The next step is to prove that (10) is the solution of Eq. (1). To prove it, we suppose thatTherefore;Applying the same process recursively, at \(t=t_{0},\) one getswhere M is bound for \(\varphi \left( x,t\right) .\) As n approaches one, we can observe that the \(\left\| G_{n}\left( x,t\right) \right\|\) goes to zero.
$$\begin{aligned} \varphi _{t}=K\left( x,t,\varphi \right) \end{aligned}$$
(2)
$$\begin{aligned} \varphi \left( x,t\right) -\varphi \left( x,0\right) =\int \limits _{0}^{t}K\left( x,s,\varphi \right) ds \end{aligned}$$
(3)
$$\begin{aligned} \left\| \varphi \left( x,t\right) \right\| =\max _{\left( x,t\right) \in \left[ a,b\right] x\left[ 0,T\right] }\left| \varphi \left( x,t\right) \right| . \end{aligned}$$
$$\begin{aligned} & \left\| K\left( x,t,\varphi _{1}\right) -K\left( x,t,\varphi _{2}\right) \right\| \\ & \quad =\left\| -\varepsilon \varphi _{1}^{2}\varphi _{1x}+\upsilon \varphi _{1xx}-\mu \varphi _{1xxx}+\gamma \varphi _{1xxxxx}+\epsilon \varphi _{1}\left( 1-\varphi _{1}\right) \right. \\ & \qquad \left. -(-\varepsilon \varphi _{2}^{2}\varphi _{2x}+\upsilon \varphi _{2xx}-\mu \varphi _{2xxx}+\gamma \varphi _{2xxxxx}+\epsilon \varphi _{2}\left( 1-\varphi _{2}\right) )\right\| \\ & \quad =\left\| -\varepsilon \left( \varphi _{1}^{2}\varphi _{1x}-\varphi _{2}^{2}\varphi _{2x}\right) +\upsilon \left( \varphi _{1xx}-\varphi _{2xx}\right) -\mu \left( \varphi _{1xxx}-\varphi _{2xxx}\right) \right. \\ & \qquad \left. +\gamma \left( \varphi _{1xxxxx}-\varphi _{2xxxxx}\right) +\epsilon \left( \varphi _{1}\left( 1-\varphi _{1}\right) -\varphi _{2}\left( 1-\varphi _{2}\right) \right) \right\| \\ & \quad \le \varepsilon \left\| \varphi _{1}^{2}\left( \varphi _{1x}-\varphi _{2x}\right) +\left( \varphi _{1}^{2}-\varphi _{2}^{2}\right) \varphi _{2x}\right\| +\upsilon \left\| \varphi _{1xx}-\varphi _{2xx}\right\| \\ & \qquad +\mu \left\| \varphi _{1xxx}-\varphi _{2xxx}\right\| +\gamma \left\| \varphi _{1xxxxx}-\varphi _{2xxxxx}\right\| \\ & \qquad +\epsilon \left\| \left( \varphi _{1}-\varphi _{2}\right) \left( 1-\varphi _{1}\right) +\varphi _{2}\left( \varphi _{2}-\varphi _{1}\right) \right\| . \end{aligned}$$
$$\begin{aligned} & \left\| K\left( x,t,\varphi _{1}\right) -K\left( x,t,\varphi _{2}\right) \right\| \le L_{\varepsilon }\left\| \varphi _{1}-\varphi _{2}\right\| +L_{\upsilon }\left\| \varphi _{1}-\varphi _{2}\right\| +L_{\mu }\left\| \varphi _{1}-\varphi _{2}\right\| \\ & \qquad +L_{\gamma }\left\| \varphi _{1}-\varphi _{2}\right\| +\epsilon \left( 1+2M\right) \left\| \varphi _{1}-\varphi _{2}\right\| \end{aligned}$$
$$\begin{aligned} \left\| K\left( x,t,\varphi _{1}\right) -K\left( x,t,\varphi _{2}\right) \right\| \le L\left\| \varphi _{1}-\varphi _{2}\right\| . \end{aligned}$$
(4)
$$\begin{aligned} \varphi _{n}\left( x,t\right) =\int \limits _{0}^{t}K\left( x,s,\varphi _{n-1}\right) ds \end{aligned}$$
(5)
$$\begin{aligned} & \Psi _{n}\left( x,t\right) =\varphi _{n}\left( x,t\right) -\varphi _{n-1}\left( x,t\right) =\int \limits _{0}^{t}\left( K\left( x,s,\varphi _{n-1}\right) \right. \\ & \quad \left. -K\left( x,s,\varphi _{n-2}\right) \right) ds. \end{aligned}$$
(6)
$$\begin{aligned} \varphi _{n}\left( x,t\right) =\sum \limits _{i=1}^{n}\Psi _{i} . \end{aligned}$$
(7)
$$\begin{aligned} & \left\| \Psi _{n}\left( x,t\right) \right\| =\left\| \int \limits _{0}^{t}\left( K\left( x,s,\varphi _{n-1}\right) -K\left( x,s,\varphi _{n-2}\right) \right) ds\right\| \\ & \quad \le \int \limits _{0}^{t}\left\| \left( K\left( x,s,\varphi _{n-1}\right) -K\left( x,s,\varphi _{n-2}\right) \right) \right\| ds. \end{aligned}$$
(8)
$$\begin{aligned} \left\| \Psi _{n}\left( x,t\right) \right\| \le \int \limits _{0}^{t}L\left\| \varphi _{n-1}-\varphi _{n-2}\right\| ds=\int \limits _{0}^{t}L\left\| \Psi _{n-1}\left( x,s\right) \right\| ds. \end{aligned}$$
(9)
$$\begin{aligned} \left\| \Psi _{n}\left( x,t\right) \right\| \le \left\| \varphi _{n}\left( x,0\right) \right\| L^{n} . \end{aligned}$$
(10)
$$\begin{aligned} \varphi \left( x,t\right) -\varphi \left( x,0\right) =\varphi _{n}\left( x,t\right) -G_{n}\left( x,t\right) . \end{aligned}$$
(11)
$$\begin{aligned} & \left\| G_{n}\left( x,t\right) \right\| =\left\| \int \limits _{0}^{t}\left( K\left( x,s,\varphi \right) -K\left( x,s,\varphi _{n-1}\right) \right) ds\right\| \\ & \quad \le \int \limits _{0}^{t}\left\| \left( K\left( x,s,\varphi \right) -K\left( x,s,\varphi _{n-1}\right) \right) \right\| ds \\ & \quad \le Lt\left\| \varphi -\varphi _{n-1}\right\| . \end{aligned}$$
$$\begin{aligned} \left\| G_{n}\left( x,t\right) \right\| \le L^{n+1}t_{0}^{n+1}M \end{aligned}$$
(12)
For the last step, we are going to demonstrate uniqueness for the solution of mentioned problem. Let us suppose that there existed a different solution \(\varphi ^{*}\left( x,t\right)\) for the Eq. (1), then we havewhen the norm on Eq. (9) is applied, it yields,from above, it is clear that under the condition \(\left( 1-Lt\right) \geqslant 0,\) then \(\left\| \varphi \left( x,t\right) -\varphi ^{*}\left( x,t\right) \right\| =0.\) It demonstrates the uniqueness of the solution.
$$\begin{aligned} & \varphi \left( x,t\right) -\varphi ^{*}\left( x,t\right) =\int \limits _{0}^{t}K\left( x,s,\varphi \right) ds-\int \limits _{0}^{t}K\left( x,s,\varphi ^{*}\right) ds \\ & \quad \le \int \limits _{0}^{t}\left( K\left( x,s,\varphi \right) -K\left( x,s,\varphi ^{*}\right) \right) ds, \end{aligned}$$
(13)
$$\begin{aligned} & \left\| \varphi \left( x,t\right) -\varphi ^{*}\left( x,t\right) \right\| \le \int \limits _{0}^{t}\left\| K\left( x,s,\varphi \right) -K\left( x,s,\varphi ^{*}\right) \right\| ds \\ & \quad \le Lt\left\| \varphi \left( x,t\right) -\varphi ^{*}\left( x,t\right) \right\| \\ & \quad \Rightarrow \left\| \varphi \left( x,t\right) -\varphi ^{*}\left( x,t\right) \right\| \left( 1-Lt\right) \le 0 \end{aligned}$$
Anzeige
3 Description of collocation method on fifth order KdV–Burgers–Fisher equation
The main focus of the present section is to explain the basic steps of the collocation finite element method (CM) while presenting application of the method on KBF equation and to obtain an effective numerical scheme for the model equation given in (1). As it is mentioned in Ref. [31] approximate algorithm can also be regarded as a special kind of Ritz–Galerkin method. Indeed, it does not matter which admissible functions are chosen as trigonometric as in this paper, polynomial, or orthogonal functions, we may enforce numerical solution to be equal to partial differential equation directly at the certain collocation points and the value of residue is expected to be zero. The most attractive property of the CM is its ability to obtain algebraic equations easily.
In order to handle the KdV–Burgers–Fisher equation given in (1), consider the initial and boundary conditions related with equation as followswhere \(\varepsilon ,v,\gamma\) and \(\epsilon\) are real parameters, x and t denote partial derivatives according to spatial and temporal directions, respectively. As it can be seen from the above equation the highest order derivative is fifth order, thus, since B-splines are piecewise polynomials their degree can be chosen as the same or one higher than highest order derivative seen in the equation. To avoid using septic trigonometric B-splines, over calculating cost and memory handicaps, the equation given in (1) will be divided into two equations by an auxiliary variable \(w\left( x,t\right) =\varphi _{xxx}\left( x,t\right) .\) Thus, the Eq. (1) will transform into a coupled equation system having the highest derivative of 3rd order such asNow,to be able to get approximate solutions of coupled equation system given in (16), we can use only fifth order basis functions. For this paper, we will use quintic trigonometric B-splines as basis function for collocation method.
$$\begin{aligned} \varphi \left( x,0\right) =f\left( x\right) \quad x\in \left[ x_{l},x_{r}\right] \end{aligned}$$
(14)
$$\begin{aligned} \begin{array}{ll} \varphi \left( x_{l},t\right) =f_{1}\left( t\right) & \varphi \left( x_{r},t\right) =f_{2}\left( t\right) \\ \varphi _{x}\left( x_{l},t\right) =g_{1}\left( t\right) & \varphi _{x}\left( x_{r},t\right) =g_{2}\left( t\right) \end{array}\quad t\in \left[ 0,T\right] \end{aligned}$$
(15)
$$\begin{aligned} & w\left( x,t\right) -\varphi _{xxx}\left( x,t\right) =0, \\ & \varphi _{t}\left( x,t\right) +\varepsilon \varphi ^{2}\left( x,t\right) \varphi _{x}\left( x,t\right) -\upsilon \varphi _{xx}\left( x,t\right) +\mu w\left( x,t\right) \\ & \quad -\gamma w_{xx}\left( x,t\right) -\epsilon \varphi \left( x,t\right) \left( 1-\varphi \left( x,t\right) \right) =0. \end{aligned}$$
(16)
3.1 Derivation of the system of equations
3.1.1 Discretization procedure
Let \(x_{l}\) and \(x_{r}\) are real variables and \(I=[x_{l},x_{r}]\) is an interval which the solution of the problem will be investigated over it. Define a uniform partition of I : where \(\Delta x\) is spatial mesh widths (that is, the length of each element). Additionally, a uniform time mesh-widths can be expressed as followswhere T is final time of process, \(\Delta t\) is time step, n is time level and M is partition number. For simplicity, we will define the values of exact solutions \(\varphi \left( x,t\right) ,\) \(w\left( x,t\right)\) at mesh points \(\left( x_{m},t_{n}\right)\) as
$$\begin{aligned} & x_{l}=x_{0}<x_{1}<x_{2}<\cdots<x_{N-1}<x_{N}=x_{r} \\ & \Delta x=h=x_{m+1}-x_{m} \end{aligned}$$
$$\begin{aligned} & \Delta t=T/M \\ & t_{n}=n\Delta t,\quad n=0,1,2,\ldots ,M \end{aligned}$$
$$\begin{aligned} \varphi \left( x_{m},t_{n}\right) =\varphi _{m}^{n},\quad w\left( x_{m},t_{n}\right) =w_{m}^{n}. \end{aligned}$$
(17)
3.1.2 Approximate solutions
Collocation method involves evaluating an approximate solution as a linear combination of trial functions and unknowns which are depending on time variable. In this subsection, we will chose trial functions as quintic trigonometric B-splines as follows [32‐34]where \(h\left( x_{m-3}\right) =\Im _{1}\left( x\right) ,\) \(h\left( x_{m-2}\right) =\Im _{2}\left( x\right) ,\) \(h\left( x_{m-1}\right) =\Im _{3}\left( x\right) ,\) \(h\left( x_{m}\right) =\Im _{4}\left( x\right) ,\) \(h\left( x_{m+1}\right) =\Im _{5}\left( x\right) ,\) \(h\left( x_{m+2}\right) =\Im _{6}\left( x\right)\) and \(h\left( x_{m+3}\right) =\Im _{7}\left( x\right)\) with \(\chi =h/2,\) \(\Theta =\sin \left( \chi \right) \sin \left( 2\chi \right) \sin \left( 3\chi \right) \sin \left( 4\chi \right) \sin \left( 5\chi \right) .\)
$$\begin{aligned} T_{m}^{5}(x)=\frac{1}{\Theta }\left\{ \begin{array}{ll} \Im _{1}^{3}\left( x\right) \left( \Im _{1}^{2}\left( x\right) -\Im _{1}\left( x\right) \Im _{3}\left( x\right) -\Im _{2}\left( x\right) \Im _{4}\left( x\right) \right) ,& x_{m-3}\le x<x_{m-2} \\ -\Im _{2}^{2}\left( x\right) \left( \Im _{1}^{2}\left( x\right) \Im _{5}\left( x\right) -\Im _{1}\left( x\right) \Im _{2}\left( x\right) \Im _{6}\left( x\right) \right) & \\ & \\ \Im _{1}^{2}\left( x\right) \left( \Im _{1}\left( x\right) \Im _{4}^{2}\left( x\right) +\Im _{2}\left( x\right) \Im _{4}\left( x\right) \Im _{5}\left( x\right) \right) ,& \quad x_{m-2}\le x<x_{m-1} \\ +\Im _{1}^{2}\left( x\right) \Im _{5}^{2}\left( x\right) \left( \Im _{3}\left( x\right) -\Im _{5}\left( x\right) \right) & \\ +\Im _{1}\left( x\right) \Im _{2}\left( x\right) \left( \Im _{2}\left( x\right) \Im _{4}\left( x\right) \Im _{6}\left( x\right) +\Im _{3}\left( x\right) \Im _{5}\left( x\right) \Im _{6}\left( x\right) \right) & \\ +\Im _{1}\left( x\right) \Im _{2}\left( x\right) \left( \Im _{2}^{2}\left( x\right) \Im _{4}\left( x\right) -\Im _{5}^{2}\left( x\right) \Im _{6}\left( x\right) \right) & \\ +\Im _{3}\left( x\right) \Im _{6}\left( x\right) \left( \Im _{1}\left( x\right) \Im _{3}\left( x\right) \Im _{6}\left( x\right) +\Im _{2}\left( x\right) \Im _{3}\left( x\right) \Im _{7}\left( x\right) \right) & \\ +\Im _{3}\left( x\right) \Im _{7}\left( x\right) \left( \Im _{2}^{2}\left( x\right) \Im _{5}\left( x\right) +\Im _{3}^{2}\left( x\right) \Im _{7}\left( x\right) \right) & \\ -\Im _{6}^{2}\left( x\right) \left( \Im _{1}\left( x\right) \Im _{3}\left( x\right) \Im _{5}\left( x\right) +\Im _{1}\left( x\right) \Im _{3}\left( x\right) \Im _{6}\left( x\right) \Im _{7}\left( x\right) \right) & \\ & \\ -\Im _{2}\left( x\right) \Im _{5}\left( x\right) \Im _{7}\left( x\right) \left( \Im _{2}\left( x\right) \Im _{5}\left( x\right) +\Im _{3}\left( x\right) \Im _{6}\left( x\right) \right) ,& \quad x_{m}\le x<x_{m+1}, \\ -\Im _{3}\left( x\right) \Im _{7}\left( x\right) \left( \Im _{2}\left( x\right) \Im _{6}^{2}\left( x\right) +\Im _{3}\left( x\right) \Im _{5}\left( x\right) \Im _{7}\left( x\right) \right) & \\ -\Im _{5}\left( x\right) \Im _{7}\left( x\right) \left( \Im _{3}\left( x\right) \Im _{5}\left( x\right) \Im _{7}\left( x\right) +\Im _{4}\left( x\right) \Im _{7}^{2}\left( x\right) \right) & \\ & \\ \Im _{6}\left( x\right) \Im _{7}\left( x\right) \left( \Im _{2}\left( x\right) \Im _{6}^{2}\left( x\right) +\Im _{3}\left( x\right) \Im _{6}\left( x\right) \Im _{7}\left( x\right) \right) ,& \quad x_{m+1}\le x<x_{m+2}, \\ +\Im _{7}^{3}\left( x\right) \left( \Im _{4}\left( x\right) \Im _{6}\left( x\right) +\Im _{5}\left( x\right) \Im _{6}\left( x\right) \right) & \\ & \\ -\Im _{7}^{5}(x) ,& \quad x_{m+2}\le x<x_{m+3}, \\ & \\ 0 & otherwise \end{array} \right. \end{aligned}$$
Thus, the approximate solutions \(\vartheta \left( x,t\right)\) and \(\varpi \left( x,t\right)\) in place of the exact solutions \(\varphi \left( x,t\right)\) and \(w\left( x,t\right) ,\) respectively can be obtained aswhere \(T_{m}\left( x\right)\) are quintic trigonometric B-splines, \(\delta _{m}\left( t\right)\) and \(\sigma _{m}\left( t\right)\) are time dependent unknown parameters. Due to local support property of the quintic trigonometric B-splines only \(T_{m-2}\left( x\right) ,T_{m-1}\left( x\right) ,T_{m}\left( x\right) ,T_{m+1}\left( x\right)\) and \(T_{m+2}\left( x\right)\) will be nonzero. Hence, the values of the approximate solutions \(\vartheta \left( x,t\right) ,\) \(\varpi \left( x,t\right)\) and their required derivatives at mesh points \(x_{m}\) can be expressed in terms of time dependent unknown parameters \(\delta _{m}\left( t\right)\) and \(\sigma _{m}\left( t\right)\) as followsandwherewhere \(\chi =h/2\) and \(\Theta =\sin \left( \chi \right) \sin \left( 2\chi \right) \sin \left( 3\chi \right) \sin \left( 4\chi \right) \sin \left( 5\chi \right) .\)
$$\begin{aligned} & \vartheta \left( x,t\right) =\sum \limits _{m=-2}^{N+2}T_{m}\left( x\right) \delta _{m}\left( t\right) \\ & \varpi \left( x,t\right) =\sum \limits _{m=-2}^{N+2}T_{m}\left( x\right) \sigma _{m}\left( t\right) \end{aligned}$$
(18)
$$\begin{aligned} & \vartheta ={{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\delta _{m-2}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\delta _{m-1}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{3}\delta _{m}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\delta _{m+1}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\delta _{m+2}\left( t\right) \\ & \vartheta ^{\prime }=\beta _{1}\delta _{m-2}\left( t\right) +\beta _{2}\delta _{m-1}\left( t\right) -\beta _{2}\delta _{m+1}\left( t\right) -\beta _{1}\delta _{m+2}\left( t\right) \\ & \vartheta ^{\prime \prime }=\xi _{1}\delta _{m-2}\left( t\right) +\xi _{2}\delta _{m-1}\left( t\right) +\xi _{3}\delta _{m}\left( t\right) +\xi _{2}\delta _{m+1}\left( t\right) +\xi _{1}\delta _{m+2}\left( t\right) \\ & \vartheta ^{\prime \prime \prime }=\zeta _{1}\delta _{m-2}\left( t\right) +\zeta _{2}\delta _{m-1}\left( t\right) -\zeta _{2}\delta _{m+1}\left( t\right) -\zeta _{1}\delta _{m+2}\left( t\right) \end{aligned}$$
(19)
$$\begin{aligned} & \varpi ={{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\sigma _{m-2}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\sigma _{m-1}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{3}\sigma _{m}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\sigma _{m+1}\left( t\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\sigma _{m+2}\left( t\right) \\ & \varpi ^{\prime }=\beta _{1}\sigma _{m-2}\left( t\right) +\beta _{2}\sigma _{m-1}\left( t\right) -\beta _{2}\sigma _{m+1}\left( t\right) -\beta _{1}\sigma _{m+2}\left( t\right) \\ & \varpi ^{\prime \prime }=\xi _{1}\sigma _{m-2}\left( t\right) +\xi _{2}\sigma _{m-1}\left( t\right) +\xi _{3}\sigma _{m}\left( t\right) +\xi _{2}\sigma _{m+1}\left( t\right) +\xi _{1}\sigma _{m+2}\left( t\right) \\ & \varpi ^{\prime \prime \prime }=\zeta _{1}\sigma _{m-2}\left( t\right) +\zeta _{2}\sigma _{m-1}\left( t\right) -\zeta _{2}\sigma _{m+1}\left( t\right) -\zeta _{1}\sigma _{m+2}\left( t\right) \end{aligned}$$
(20)
$$\begin{aligned} & {{\mathchoice{\lambda }{\lambda }{lambda}{\lambda } } } _{1}=\sin ^{5}(\chi )/\Theta , \\ & {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}=2\sin ^{5}\left( \chi \right) \cos \left( \chi \right) \left( 16\cos ^{2}\left( \chi \right) -3\right) /\Theta , \\ & {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{3}=2\left( 1+48\cos ^{4}\left( \chi \right) -16\cos ^{2}\left( \chi \right) \right) /\Theta , \\ & \beta _{1}=\left( -5/2\right) \sin ^{4}\left( \chi \right) \cos \left( \chi \right) /\Theta , \\ & \beta _{2}=-5\sin ^{4}\left( \chi \right) \cos ^{2}\left( \chi \right) \left( 8\cos ^{2}\left( \chi \right) -3\right) /\Theta , \\ & \xi _{1}=\left( 5/4\right) \sin ^{3}\left( \chi \right) \left( 5\cos ^{2}\left( \chi \right) -1\right) /\Theta \\ & \xi _{2}=\left( 5/2\right) \sin ^{3}\left( \chi \right) \cos \left( \chi \right) \left( -15\cos ^{2}\left( \chi \right) +3+16\cos ^{4}\left( \chi \right) \right) /\Theta , \\ & \xi _{3}=\left( -5/2\right) \sin ^{3}\left( \chi \right) \left( 16\cos ^{6}\left( \chi \right) -5\cos ^{2}\left( \chi \right) +1\right) /\Theta ,\\ & \zeta _{1}=\left( -5/8\right) \sin ^{2}\left( \chi \right) \cos \left( \chi \right) \left( 25\cos ^{2}\left( \chi \right) -13\right) /\Theta , \\ & \zeta _{2}=\left( -5/4\right) \sin ^{2}\left( \chi \right) \cos ^{2}\left( \chi \right) \left( 8\cos ^{4}\left( \chi \right) -35\cos ^{2}\left( \chi \right) +15\right) /\Theta , \end{aligned}$$
4 Numerical scheme
Before obtaining difference equation i.e numerical scheme for (16), we need to use Crank–Nicolson approximation for derivative with respect to spatial variable and forward difference approximation for derivative with respect to temporal variable. Discretization for obtaining the numerical solutions of dynamic systems is a necessary process for the correct expression of the dynamics of nonlinear complex systems. Different linearizing techniques are widely used in differentiating nonlinear engineering and physics problems. In this study, a Rubin–Graves type linearization technique developed by Rubin and Graves [35] in 1975 will be used for nonlinear terms \(\varphi _{x}^{2}\varphi\) and \(\varphi ^{2}\) seen in the Eq. (16). Thus, the complex system, which changes over time, will be transformed into linear equations, facilitating the solution process. The general mathematical expression of the technique can be found in [36] as followsWhen we use the above equality for the nonlinear terms \(\varphi ^{2}\varphi _{x}\) and \(\varphi ^{2}\) in (16) for \(p=2\) and \(p=1,\) \(\varphi _{x}=\varphi ,\) we get following equalities, respectivelyAfter applying finite difference approximations, one obtainsAfter some basic mathematical operations, one can rewrite the above equation asNow, the difference equation at the nodal points \(x_{m}\) associated with coupled equation given in (16) can be expressed asPutting Eqs. (19) and (20) into Eq. (22) and after some arrangements, the difference equation is obtained in terms of quintic trigonometric B-splines and unknowns depending on time variable as followsandwhere \(\Psi _{1}=\left( 2\varepsilon \vartheta ^{n}\vartheta _{x}^{n}-\epsilon +2\epsilon \vartheta ^{n}\right)\) and \(\Psi _{2}=\varepsilon \vartheta ^{n}\vartheta ^{n}.\) Thus, we have derived a numerical scheme for system (16). The system can be written in matrix form as followsFor \(m=0,1,2,\ldots ,N,\) the matrices \(A,B,C,D,\widetilde{A},\widetilde{B},\widetilde{C}\) and \(\widetilde{D}\) have \((N+1)\) rows and \(\left( N+5\right)\) columns and the vectors \(\delta ,\) \(\sigma\) have \(\left( N+5\right)\) rows, one column. Thus, the system given above has got \((2N + 2)\) equations and \((2N + 10)\) unknowns. In order to obtain a solvable system, 8 unknowns must be removed from the system in Eq. (23) or 8 equations must be added to the system in Eq. (23). For this paper, we will chose first path and remove 8 unknown parameters using the boundary conditions in the next subsection.
$$\begin{aligned} \left( \varphi ^{p}\varphi _{x}\right) ^{n+1}=\left( \varphi ^{p}\right) ^{n}\varphi _{x}^{n+1}+p\left( \varphi ^{p-1}\right) ^{n}\varphi _{x}^{n}\varphi ^{n+1}-p\left( \varphi ^{p}\right) ^{n}\varphi _{x}^{n}. \end{aligned}$$
(21)
$$\begin{aligned} & \left( \varphi ^{2}\varphi _{x}\right) ^{n+1}=\left( \varphi ^{2}\right) ^{n}\varphi _{x}^{n+1}+2\varphi ^{n}\varphi _{x}^{n}\varphi ^{n+1}-2\left( \varphi ^{2}\right) ^{n}\varphi _{x}^{n} \\ & \left( \varphi \varphi \right) ^{n+1}=\varphi ^{n}\varphi ^{n+1}+\varphi ^{n}\varphi ^{n+1}-\varphi ^{n}\varphi ^{n}. \end{aligned}$$
$$\begin{aligned} & \frac{1}{2}\left( w^{n+1}+w^{n}-\varphi _{xxx}^{n+1}-\varphi _{xxx}^{n}\right) =0, \\ & \frac{1}{\Delta t}\left( \varphi ^{n+1}-\varphi ^{n}\right) +\frac{1}{2} \varepsilon \left( \left( \varphi ^{2}\right) ^{n}\varphi _{x}^{n+1}+2\varphi ^{n}\varphi _{x}^{n}\varphi ^{n+1}\right. \\ & \quad \left. -\left( \varphi ^{2}\right) ^{n}\varphi _{x}^{n}\right) - \frac{\upsilon }{2}\left( \varphi _{xx}^{n+1}+\varphi _{xx}^{n}\right) \\ & \quad +\frac{\mu }{2}\left( w^{n+1}+w^{n}\right) -\frac{\gamma }{2} \left( w_{xx}^{n+1}+w_{xx}^{n}\right) \\ & -\frac{ \epsilon }{2}\left( \left( \varphi \right) ^{n+1}+\left( \varphi \right) ^{n}\right) \\ & \quad +\frac{\epsilon }{2}\left( \left( \varphi \right) ^{n+1}\left( \varphi \right) ^{n}+\left( \varphi \right) ^{n}\left( \varphi \right) ^{n+1}\right) =0. \end{aligned}$$
$$\begin{aligned} & -\varphi _{xxx}^{n+1}+w^{n+1}=\varphi _{xxx}^{n}-w^{n}, \\ & \left[ 1+\frac{\Delta t}{2}\left( 2\varepsilon \varphi _{x}^{n}\varphi ^{n}-\epsilon +2\epsilon \varphi ^{n}\right) \right] \varphi ^{n+1}+\left[ \frac{\Delta t}{2}\varepsilon \varphi ^{n}\varphi ^{n}\right] \varphi _{x}^{n+1}-\left[ \frac{\Delta t}{2}\upsilon \right] \varphi _{xx}^{n+1} \\ & \qquad +\left[ \frac{\Delta t}{2}\mu \right] w^{n+1}-\left[ \frac{ \Delta t}{2}\gamma \right] w_{xx}^{n+1} \\ & \quad =\left[ 1+\frac{\Delta t}{2}\epsilon \right] \varphi ^{n}+\left[ \frac{\Delta t}{2}\varepsilon \varphi ^{n}\varphi ^{n}\right] \varphi _{x}^{n}+\left[ \frac{\Delta t}{2}\upsilon \right] \varphi _{xx}^{n} \\ & \qquad -\left[ \frac{\Delta t}{2}\mu \right] w^{n}+\left[ \frac{\Delta t }{2}\gamma \right] w_{xx}^{n}. \end{aligned}$$
$$\begin{aligned} & -\left( \vartheta ^{\prime \prime \prime }\right) ^{n+1}+\varpi ^{n+1}=\left( \vartheta ^{\prime \prime \prime }\right) ^{n}-\varpi ^{n}, \\ & \left[ 1+\frac{\Delta t}{2}\left( 2\varepsilon \vartheta _{x}^{n}\vartheta ^{n}-\epsilon +2\epsilon \vartheta ^{n}\right) \right] \vartheta ^{n+1}+\left[ \frac{\Delta t}{2}\varepsilon \vartheta ^{n}\vartheta ^{n}\right] \left( \vartheta ^{\prime }\right) ^{n+1} \\ & \qquad -\left[ \frac{\Delta t}{2}\upsilon \right] \left( \vartheta ^{\prime \prime }\right) ^{n+1}+\left[ \frac{\Delta t}{2}\mu \right] \varpi ^{n+1}-\left[ \frac{\Delta t}{2}\gamma \right] \left( \varpi ^{\prime \prime }\right) ^{n+1} \\ & \quad =\left[ 1+\frac{\Delta t}{2}\epsilon \right] \vartheta ^{n}+ \left[ \frac{\Delta t}{2}\varepsilon \vartheta ^{n}\vartheta ^{n}\right] \left( \vartheta ^{\prime }\right) ^{n}+\left[ \frac{\Delta t}{2}\upsilon \right] \left( \vartheta ^{\prime \prime }\right) ^{n} \\ & \qquad -\left[ \frac{\Delta t}{2}\mu \right] \varpi ^{n}+\left[ \frac{ \Delta t}{2}\gamma \right] \left( \varpi ^{\prime \prime }\right) ^{n}. \end{aligned}$$
(22)
$$\begin{aligned} & -\zeta _{1}\delta _{m-2}^{n+1}\left( t\right) -\zeta _{2}\delta _{m-1}^{n+1}\left( t\right) +\zeta _{2}\delta _{m+1}^{n+1}\left( t\right) +\zeta _{1}\delta _{m+2}^{n+1}\left( t\right) +{\lambda }_{1}\sigma _{m-2}^{n+1}\left( t\right) \\ & \qquad +{\lambda }_{2}\sigma _{m-1}^{n+1}\left( t\right) +{\lambda }_{3}\sigma _{m}^{n+1}\left( t\right) +{\lambda }_{2}\sigma _{m+1}^{n+1}\left( t\right) + {\lambda }_{1}\sigma _{m+2}^{n+1}\left( t\right) \\ & \quad =\zeta _{1}\delta _{m-2}^{n}\left( t\right) +\zeta _{2}\delta _{m-1}^{n}\left( t\right) -\zeta _{2}\delta _{m+1}^{n}\left( t\right) -\zeta _{1}\delta _{m+2}^{n}\left( t\right) \\ & \qquad -\lambda _{1}\sigma _{m-2}^{n}\left( t\right) -\lambda _{2}\sigma _{m-1}^{n}\left( t\right) -\lambda _{3}\sigma _{m}^{n}\left( t\right) -\lambda _{2}\sigma _{m+1}^{n}\left( t\right) -\lambda _{1}\sigma _{m+2}^{n}\left( t\right) \end{aligned}$$
(23)
$$\begin{aligned} & \left[ {\lambda }_{1}+\frac{\Delta t}{2}\left( {\lambda }_{1}\Psi _{1}+\beta _{1}\Psi _{2}-\xi _{1}\upsilon \right) \right] \delta _{m-2}^{n+1}\left( t\right) +\left[ {\lambda }_{2}+\frac{\Delta t}{2}\left( {\lambda }_{2}\Psi _{1}+\beta _{2}\Psi _{2}-\xi _{2}\upsilon \right) \right] \delta _{m-1}^{n+1}\left( t\right) \\ & \qquad +\left[ {\lambda }_{3}+\frac{\Delta t}{2}\left( {\lambda }_{3}\Psi _{1}-\xi _{3}\upsilon \right) \right] \delta _{m}^{n+1}\left( t\right) +\left[ { \lambda }_{2}+\frac{\Delta t}{2}\left( {\lambda }_{2}\Psi _{1}-\beta _{2}\Psi _{2}-\xi _{2}\upsilon \right) \right] \delta _{m+1}^{n+1}\left( t\right) \\ & \qquad +\left[ {\lambda }_{1}+\frac{\Delta t}{2}\left( {\lambda }_{1}\Psi _{1}-\beta _{1}\Psi _{2}-\xi _{1}\upsilon \right) \right] \delta _{m+2}^{n+1}\left( t\right) +\left[ \frac{\Delta t}{2}\left( {\lambda } _{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m-2}^{n+1}\left( t\right) \\ & \qquad +\left[ \frac{\Delta t}{2}\left( {\lambda }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m-1}^{n+1}\left( t\right) +\left[ \frac{\Delta t}{2}\left( { \lambda }_{3}\mu -\xi _{3}\gamma \right) \right] \sigma _{m}^{n+1}\left( t\right) \\ & \qquad +\left[ \frac{\Delta t}{2}\left( {\lambda }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m+1}^{n+1}\left( t\right) +\left[ \frac{\Delta t}{2}\left( { \lambda }_{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m+2}^{n+1}\left( t\right) \\ & \quad =\left[ {\lambda }_{1}+\frac{\Delta t}{2}\left( {\lambda } _{1}\epsilon +\beta _{1}\Psi _{2}+\xi _{1}\upsilon \right) \right] \delta _{m-2}^{n}\left( t\right) +\left[ {\lambda }_{2}+\frac{\Delta t}{2}\left( { \lambda }_{2}\epsilon +\beta _{2}\Psi _{2}+\xi _{2}\upsilon \right) \right] \delta _{m-1}^{n}\left( t\right) \\ & \qquad +\left[ {\lambda }_{3}+\frac{\Delta t}{2}\left( {\lambda } _{3}\epsilon +\xi _{3}\upsilon \right) \right] \delta _{m}^{n}\left( t\right) +\left[ {\lambda }_{2}+\frac{\Delta t}{2}\left( {\lambda } _{2}\epsilon -\beta _{2}\Psi _{2}+\xi _{2}\upsilon \right) \right] \delta _{m+1}^{n}\left( t\right) \\ & \qquad +\left[ {\lambda }_{1}+\frac{\Delta t}{2}\left( {\lambda } _{1}\epsilon -\beta _{1}\Psi _{2}+\xi _{1}\upsilon \right) \right] \delta _{m+2}^{n}\left( t\right) +\left[ -\frac{\Delta t}{2}\left( {\lambda } _{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m-2}^{n}\left( t\right) \\ & \qquad +\left[ -\frac{\Delta t}{2}\left(
{\lambda }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m-1}^{n}\left( t\right) +\left[ -\frac{ \Delta t}{2}\left( {\lambda }_{3}\mu -\xi _{3}\gamma \right) \right] \sigma _{m}^{n}\left( t\right) \\ & \qquad +\left[ -\frac{\Delta t}{2}\left( {\lambda }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m+1}^{n}\left( t\right) +\left[ -\frac{ \Delta t}{2}\left( {\lambda }_{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m+2}^{n}\left( t\right) \end{aligned}$$
(24)
$$\begin{aligned} \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] \left[ \begin{array}{c} \delta ^{n+1} \\ \sigma ^{n+1} \end{array} \right] =\left[ \begin{array}{cc} \widetilde{A} & \widetilde{B} \\ \widetilde{C} & \widetilde{D} \end{array} \right] \left[ \begin{array}{c} \tilde{\delta }^{n} \\ \tilde{\sigma } ^{n} \end{array} \right] . \end{aligned}$$
(25)
4.1 Boundaries
Let us recall the boundary conditionsand associated conditions with approximate solutions, for \(m=0,x_{l}=x_{0}\) such thatfor \(m=N,x_{R}=x_{N}\) such thatBy using simple arithmetic operations in (26), we can define \(\delta _{-2}\left( t\right)\) and \(\delta _{-1}\left( t\right)\) in terms of \(\delta _{0}\left( t\right)\) and \(\delta _{1}\left( t\right) .\) With the same idea, using arithmetic operations in (27), we can define \(\delta _{N+1}\left( t\right)\) and \(\delta _{N+2}\left( t\right)\) in terms of \(\delta _{N-1}\left( t\right)\) and \(\delta _{N}\left( t\right) .\) The same procedure is applied for \(\sigma _{-2}\left( t\right) ,\sigma _{-1}\left( t\right) ,\sigma _{N+1}\left( t\right)\) and \(\sigma _{N+2}\left( t\right) .\) Consequently, now we can eliminate those parameters which are going to be eliminated from system (23). Applying the boundary conditions results in changing the first two and last two rows of the matrices. We can express matrices clearly as follows;For the first equation each sub-matrix is expressed independently as follows;For the second equation, we getandAlso, the matrix \(\tilde{C}\) is in the same form as the matrix C and one can see the elements of the matrix \(\tilde{C}\) in the “Appendix”. The known and unknown vectors can be expresses asAdditional matrices \(\Phi\) and \(\widetilde{\Phi }\) formed by the application of boundary conditions can now be expressed as followsHence, we obtained a system involving \(\left( 2N+2\right)\) equation, \(\left( 2N+2\right)\) unknowns. Now, we have a solvable system. A more clear expression of the elements of the matrices can be found in the “Appendix”.
$$\begin{aligned} \begin{array}{ll} \varphi \left( x_{l},t\right) =f_{1}\left( t\right) & \quad \varphi \left( x_{r},t\right) =f_{2}\left( t\right) , \\ \varphi _{x}\left( x_{l},t\right) =g_{1}\left( t\right) & \quad \varphi _{x}\left( x_{r},t\right) =g_{2}\left( t\right) \end{array}\quad t\in \left[ 0,T\right] \end{aligned}$$
$$\begin{aligned} & {\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{1}\delta _{-2}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{2}\delta _{-1}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{3}\delta _{0}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{2}\delta _{1}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{1}\delta _{2}\left( t\right) =f_{1}\left( t\right) , \\ & \beta _{1}\delta _{-2}\left( t\right) +\beta _{2}\delta _{-1}\left( t\right) -\beta _{2}\delta _{1}\left( t\right) -\beta _{1}\delta _{2}\left( t\right) =g_{1}\left( t\right) \end{aligned}$$
(26)
$$\begin{aligned} & {\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{1}\delta _{N-2}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{2}\delta _{N-1}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{3}\delta _{N}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{2}\delta _{N+1}\left( t\right) +{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } _{1}\delta _{N+2}\left( t\right) =f_{2}\left( t\right) , \\ & \beta _{1}\delta _{N-2}\left( t\right) +\beta _{2}\delta _{N-1}\left( t\right) -\beta _{2}\delta _{N+1}\left( t\right) -\beta _{1}\delta _{N+2}\left( t\right) =g_{2}\left( t\right) . \end{aligned}$$
(27)
$$\begin{aligned} \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] \left[ \begin{array}{c} \delta ^{n+1} \\ \sigma ^{n+1} \end{array} \right] =\left[ \begin{array}{cc} \widetilde{A} & \widetilde{B} \\ \widetilde{C} & \widetilde{D} \end{array} \right] \left[ \begin{array}{c} \delta ^{n} \\ \sigma ^{n} \end{array} \right] +\left[ \begin{array}{c} \Phi \\ \widetilde{\Phi } \end{array} \right] . \end{aligned}$$
(28)
$$\begin{aligned} & A=-\widetilde{A}=\left[ \begin{array}{ccccccccccc} a_{11} & a_{12} & a_{13} & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & 0 \\ a_{21} & a_{22} & a_{23} & \zeta _{1} & 0 & \ldots & 0 & 0 & 0 & 0 & 0 \\ -\zeta _{1} & -\zeta _{2} & 0 & \zeta _{2} & \zeta _{1} & \ldots & 0 & 0 & 0 & 0 & 0 \\ 0 & -\zeta _{1} & -\zeta _{2} & 0 & \zeta _{2} & \ldots & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \ldots & -\zeta _{2} & 0 & \zeta _{2} & \zeta _{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & \ddots & -\zeta _{1} & -\zeta _{2} & 0 & \zeta _{2} & \zeta _{1} \\ 0 & 0 & 0 & 0 & 0 & \ldots & 0 & -\zeta _{1} & -a_{23} & -a_{22} & -a_{21} \\ 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & -a_{13} & -a_{12} & -a_{11} \end{array} \right] , \end{aligned}$$
(29)
$$\begin{aligned} & B=-\widetilde{B}=\left[ \begin{array}{ccccccccccc} b_{11} & b_{12} & b_{13} & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & 0 \\ b_{21} & b_{22} & b_{23} & \lambda _{1} & 0 & \ldots & 0 & 0 & 0 & 0 & 0 \\ \lambda _{1} & \lambda _{2} & \lambda _{3} & \lambda _{2} & \lambda _{1} & \ldots & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \ddots & \lambda _{1} & \lambda _{2} & \lambda _{3} & \lambda _{2} & \lambda _{1} \\ 0 & 0 & 0 & 0 & 0 & \ldots & 0 & \lambda _{1} & b_{23} & b_{22} & b_{21} \\ 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & b_{13} & b_{12} & b_{11} \end{array} \right] . \end{aligned}$$
(30)
$$\begin{aligned} C=\left[ \begin{array}{ccccccccccccc} c_{11} & c_{12} & c_{13} & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ c_{21} & c_{22} & c_{23} & c_{23} & c_{24} & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & \ddots & c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \ddots & 0 & c_{31} & c_{32} & c_{33} & c_{34} & c_{35} \\ 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & c_{n-1,n-3} & c_{n-1,n-2} & c_{n-1,n-1} & c_{n-1,n} \\ 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & c_{n,n-2} & c_{n,n-1} & c_{n,n} \end{array} \right] , \end{aligned}$$
(31)
$$\begin{aligned} D=-\widetilde{D}=\left[ \begin{array}{cccccccccccccc} d_{11} & d_{12} & d_{13} & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ d_{21} & d_{22} & d_{23} & d_{24} & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ d_{31} & d_{32} & d_{33} & d_{32} & d_{31} & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & d_{31} & d_{32} & d_{33} & d_{32} & d_{31} & 0 & \ldots & 0 & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ddots & d_{31} & d_{32} & d_{33} & d_{32} & d_{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & d_{31} & d_{32} & d_{33} & d_{32} & d_{31} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & d_{24} & d_{23} & d_{22} & d_{21} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & d_{13} & d_{12} & d_{11} \end{array} \right] . \end{aligned}$$
(32)
$$\begin{aligned} \begin{array}{cc} \left[ \delta \right] =\left[ \begin{array}{c} \delta _{0} \\ \delta _{1} \\ \vdots \\ \delta _{n-1} \\ \delta _{n} \end{array} \right] ,& \quad \left[ \sigma \right] =\left[ \begin{array}{c} \sigma _{0} \\ \sigma _{1} \\ \vdots \\ \sigma _{n-1} \\ \sigma _{n} \end{array} \right] . \end{array} \end{aligned}$$
(33)
$$\begin{aligned} \Phi =\left[ \begin{array}{c} \epsilon _{11} \\ \epsilon _{12} \\ 0 \\ \vdots \\ 0 \\ \epsilon _{n-1,n} \\ \epsilon _{n,n} \end{array} \right] ,\quad \widetilde{\Phi }=\left[ \begin{array}{c} \tilde{\epsilon }_{11} \\ \tilde{\epsilon }_{12} \\ 0 \\ \vdots \\ 0 \\ \tilde{\epsilon }_{n-1,n} \\ \tilde{\epsilon }_{n,n} \end{array} \right] . \end{aligned}$$
4.2 Initial vector
The equation system attained at the previous subsection is based on an iterative process that commences via an initial approximation at \(t=0.\) This initial approximation results in a vector called as “initial vector”. This basic process, i.e. deriving initial vector, consists of using initial condition given in (14). Let us recall that the initial condition is as followsAs it is seen above, x values will scan whole interval, the values of \(\varphi \left( x,0\right)\) will be considered at all mesh points. So, when we combine the initial condition with approximate solution, the following relationship is obtainedThe initial vector may be stated in matrix form as below:The initial vector for \(\sigma\) can be written in the same way. It is clear from (34) that the system has \(\left( N+1\right)\) equation and \(\left( N+5\right)\) unknown parameters. As stated in Sect. 4.1, removing 4 unknown parameters \(\delta _{-2}\left( t\right) ,\) \(\delta _{-1}\left( t\right) ,\delta _{N+1}\left( t\right)\) and \(\delta _{N+2}\left( t\right)\) can be carried out with any combination of boundary conditions. In the final part of the section “Derivation of the system of equations“, one obtains a system of equations consisting of \(\left( 2N+2\right)\) equation and \(\left( 2N+2\right)\) unknown parameters with initial vector. Solving this system will carry us to construct approximate solution at desired time level.
$$\begin{aligned} \varphi \left( x,0\right) =f\left( x\right) \quad x\in \left[ x_{l},x_{r}\right] . \end{aligned}$$
$$\begin{aligned} & {\lambda }_{1}\delta _{-2}\left( t\right) +{\lambda }_{2}\delta _{-1}\left( t\right) +{\lambda }_{3}\delta _{0}\left( t\right) +{\lambda }_{2}\delta _{1}\left( t\right) +{\lambda }_{1}\delta _{2}\left( t\right) =f\left( x_{0}\right) , \\ & {\lambda }_{1}\delta _{-1}\left( t\right) +{\lambda }_{2}\delta _{0}\left( t\right) +{\lambda }_{3}\delta _{1}\left( t\right) +{\lambda }_{2}\delta _{2}\left( t\right) +{\lambda }_{1}\delta _{3}\left( t\right) =f\left( x_{1}\right) , \\ & \vdots \quad \quad \vdots \quad \quad \vdots \quad \quad \vdots \quad \quad \quad \vdots \\ & {\lambda }_{1}\delta _{N-3}\left( t\right) +{\lambda }_{2}\delta _{N-2}\left( t\right) +{\lambda }_{3}\delta _{N-1}\left( t\right) +{\lambda } _{2}\delta _{N}\left( t\right) +{\lambda }_{1}\delta _{N+1}\left( t\right) =f\left( x_{N-1}\right) , \\ & {\lambda }_{1}\delta _{N-2}\left( t\right) +{\lambda }_{2}\delta _{N-1}\left( t\right) +{\lambda }_{3}\delta _{N}\left( t\right) +{\lambda } _{2}\delta _{N+1}\left( t\right) +{\lambda }_{1}\delta _{N+2}\left( t\right) =f\left( x_{N}\right) . \end{aligned}$$
(34)
$$\begin{aligned} \left[ \begin{array}{cccccccccccc} \lambda _{1} & \lambda _{2} & \lambda _{3} & \lambda _{2} & \lambda _{1} & 0 & \ldots & 0 & 0 & 0 & 0 & 0 \\ 0 & \lambda _{1} & \lambda _{2} & \lambda _{3} & \lambda _{2} & \lambda _{1} & \ldots & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & 0 & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & \ldots & \lambda _{2} & \lambda _{3} & \lambda _{2} & \lambda _{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \ldots & \lambda _{1} & \lambda _{2} & \lambda _{3} & \lambda _{2} & \lambda _{1} \end{array} \right] \left[ \begin{array}{c} \delta _{-2} \\ \delta _{-1} \\ \delta _{0} \\ \vdots \\ \delta _{n} \\ \delta _{n+1} \\ \delta _{n+2} \end{array} \right] =\left[ \begin{array}{c} f\left( x_{0}\right) \\ f\left( x_{0}\right) \\ \vdots \\ f\left( x_{n-1}\right) \\ f\left( x_{n}\right) \end{array} \right] . \end{aligned}$$
(35)
5 Stability analysis
In numerical approaches, it is important to know, how the method influence the stability of the numerical scheme. It was already confirmed with von-Neumann stability technique to deliver reliable assessment of properties of the scheme. In this method, the solution is taken as a discrete fourier mode for the problem of the following formwhere \(m=0,1,2,\ldots ,N\) and superscript term on the \(\psi\) is a multiplicative exponent. The claim of the method is that a necessary condition for the stability is obtained by the following conditionIn the previous section, the KBF equation is split into two equation. Due to the fact that the von-Neumann technique works only for linear differential equations, we will consider the discretized form of linearized KBF equation asandwhere \(\Psi _{1}=\epsilon \left( \tilde{\varphi }-1\right) ,\) \(\Psi _{2}=\varepsilon \left( \tilde{\varphi }\right) ^{2},\) \(i=\sqrt{-1}\) and \(\tilde{\varphi }=\max\limits_{0\le m\le N}\left| \varphi _{m}^{n}\right| .\) Before applying the von-Neumann technique, we need to indicate that there are two systems which influence each other. Hence, the stability analysis should be considered between two system, simultaneously. Now, let us assume the solutions of the Eqs. (36)–(37) of the following formwhere q is the amplification factor, P, W are amplitudes of waves and \(\phi\) is mode factor. Putting Eq. (38) in Eqs. (36)–(37), after some calculations, we obtainwhereafter some arrangements, Eq. (39) can be written asWe know that the amplification factor q needs to be investigated, the determinant of the system given in (40) yields as a quadratic equation according to q as followsWhen one calculates the discriminant, one can see easily that \(\sqrt{\Delta } =2\lambda _{2}\left( \lambda _{2}-i\lambda _{4}\right) .\) Therefore, after some simplification processes, the following results can be obtainedHence, the proposed technique is unconditionally stable according to von-Neumann stability technique.
$$\begin{aligned} \varphi _{m}^{n}=\psi ^{n}e^{imh\phi } \end{aligned}$$
$$\begin{aligned} \left| \psi \right| \le 1. \end{aligned}$$
$$\begin{aligned} & -\zeta _{1}\delta _{m-2}^{n+1}\left( t\right) -\zeta _{2}\delta _{m-1}^{n+1}\left( t\right) +\zeta _{2}\delta _{m+1}^{n+1}\left( t\right) +\zeta _{1}\delta _{m+2}^{n+1}\left( t\right) \\ & \qquad +{{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } }_{1}\sigma _{m-2}^{n+1}\left( t\right) +{{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } } _{2}\sigma _{m-1}^{n+1}\left( t\right) +{\alpha }_{3}\sigma _{m}^{n+1}\left( t\right) +{{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } }_{2}\sigma _{m+1}^{n+1}\left( t\right) +{{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } } _{1}\sigma _{m+2}^{n+1}\left( t\right) \\ & \quad =\zeta _{1}\delta _{m-2}^{n}\left( t\right) +\zeta _{2}\delta _{m-1}^{n}\left( t\right) -\zeta _{2}\delta _{m+1}^{n}\left( t\right) -\zeta _{1}\delta _{m+2}^{n}\left( t\right) \\ & \qquad -{{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } }_{1}\sigma _{m-2}^{n}\left( t\right) -{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\sigma _{m-1}^{n}\left( t\right) -{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{3}\sigma _{m}^{n}\left( t\right) -{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\sigma _{m+1}^{n}\left( t\right) -{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\sigma _{m+2}^{n}\left( t\right) \end{aligned}$$
(36)
$$\begin{aligned} & \left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}+ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\Psi _{1}+\beta _{1}\Psi _{2}-\xi _{1}\upsilon \right) \right] \delta _{m-2}^{n+1}\left( t\right) +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}+ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\Psi _{1}+\beta _{2}\Psi _{2}-\xi _{2}\upsilon \right) \right] \delta _{m-1}^{n+1}\left( t\right) \\ & \qquad +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{3}+ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{3}\Psi _{1}-\xi _{3}\upsilon \right) \right] \delta _{m}^{n+1}\left( t\right) +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}+ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\Psi _{1}-\beta _{2}\Psi _{2}-\xi _{2}\upsilon \right) \right] \delta _{m+1}^{n+1}\left( t\right) \\ & \qquad +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}+ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\Psi _{1}-\beta _{1}\Psi _{2}-\xi _{1}\upsilon \right) \right] \delta _{m+2}^{n+1}\left( t\right) \\ & \left[ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m-2}^{n+1}\left( t\right) +\left[ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m-1}^{n+1}\left( t\right) +\left[ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{3}\mu -\xi _{3}\gamma \right) \right] \sigma _{m}^{n+1}\left( t\right) \\ & \left[ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m+1}^{n+1}\left( t\right) +\left[ \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m+2}^{n+1}\left( t\right) \\ & \quad =\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}- \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\Psi _{1}-\beta _{1}\Psi _{2}-\xi _{1}\upsilon \right) \right] \delta _{m-2}^{n}\left( t\right) +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}- \frac{\Delta t}{2}\left( {{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } }_{2}\Psi _{1}-\beta _{2}\Psi _{2}-\xi _{2}\upsilon \right) \right] \delta _{m-1}^{n}\left( t\right) \\ & \qquad +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{3}- \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{3}\Psi _{1}-\xi _{3}\upsilon \right) \right] \delta _{m}^{n}\left( t\right) +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}- \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{2}\Psi _{1}+\beta _{2}\Psi _{2}-\xi _{2}\upsilon \right) \right] \delta _{m+1}^{n}\left( t\right) \\ & \qquad +\left[ {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}- \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } } _{1}\Psi _{1}+\beta _{1}\Psi _{2}-\xi _{1}\upsilon \right) \right] \delta _{m+2}^{n}\left( t\right) \\ & \qquad +\left[ -\frac{\Delta q}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m-2}^{n}\left( t\right) +\left[ - \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m-1}^{n}\left( t\right) +\left[ - \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{3}\mu -\xi _{3}\gamma \right) \right] \sigma _{m}^{n}\left( t\right) \\ & \qquad +\left[ -\frac{\Delta q}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{2}\mu -\xi _{2}\gamma \right) \right] \sigma _{m+1}^{n}\left( t\right) +\left[ - \frac{\Delta t}{2}\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda } } }_{1}\mu -\xi _{1}\gamma \right) \right] \sigma _{m+2}^{n}\left( t\right) \end{aligned}$$
(37)
$$\begin{aligned} \delta _{m}^{n}=Pq^{n}e^{imh\phi }\sigma _{m}^{n}=Wq^{n}e^{imh\phi } \end{aligned}$$
(38)
$$\begin{aligned} & Pq\left\{ i\eta _{1}\right\} +Wq\left\{ \eta _{2}\right\} =P\left\{ -\eta _{1}\right\} -W\left\{ \eta _{2}\right\} , \\ & Pq\left\{ \eta _{2}+\eta _{3}-i\eta _{4}\right\} +Wq\left\{ \eta _{5}\right\} =P\left\{ \eta _{2}-\eta _{3}+i\eta _{4}\right\} -W\left\{ \eta _{2}\right\} \end{aligned}$$
(39)
$$\begin{aligned} & \eta _{1}=2\zeta _{1}\cos \left( 2\phi h\right) +2\zeta _{2}\cos \left( \phi h\right) +\zeta _{3}, \\ & \eta _{2}=2{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{1}\cos \left( 2\phi h\right) +2{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{2}\cos \left( \phi h\right) +{{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{3}, \\ & \eta _{3}=\frac{\Delta t}{2}\left( 2\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{1}\Psi _{1}-\xi _{1}\upsilon \right) \right) \cos \left( 2\phi h\right) +2\left( {{ \mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{2}\Psi _{1}-\xi _{2}\upsilon \right) \cos \left( \phi h\right) +\left( {{ \mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{3}\Psi _{1}-\xi _{3}\upsilon \right) , \\ & \eta _{4}=\frac{\Delta
t}{2}\left( 2{\beta }_{1}\Psi _{2}\sin \left( 2\phi h\right) +2{\beta }_{2}\Psi _{2}\sin \left( \phi h\right) \right) , \\ & \eta _{5}=\frac{\Delta t}{2}\left( 2\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{1}\mu -\xi _{1}\gamma \right) \right) \cos \left( 2\phi h\right) +2\left( {{ \mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{2}\mu -\xi _{2}\gamma \right) \cos \left( \phi h\right) +\left( {{\mathchoice{\lambda }{\lambda }{\lambda }{\lambda }}}_{3}\mu -\xi _{3}\gamma \right) \end{aligned}$$
$$\begin{aligned} & P\left\{ i\lambda _{1}\left( q+1\right) \right\} +W\left\{ \lambda _{2}\left( q+1\right) \right\} =0, \\ & P\left\{ \left( \lambda _{2}-i\lambda _{4}\right) \left( q-1\right) +\lambda _{3}\left( q+1\right) \right\} +W\left\{ \lambda _{5}\left( q+1\right) \right\} =0. \end{aligned}$$
(40)
$$\begin{aligned} & q^{2}\left( i\left( \lambda _{1}\lambda _{5}+\lambda _{2}\lambda _{4}\right) -\lambda _{2}\lambda _{3}-\lambda _{2}^{2}\right) +2q\left( i\lambda _{1}\lambda _{5}-\lambda _{2}\lambda _{3}\right) \\ & \quad +\left( i\left( \lambda _{1}\lambda _{5}-\lambda _{2}\lambda _{4}\right) -\lambda _{2}\lambda _{3}+\lambda _{2}^{2}\right) =0. \end{aligned}$$
$$\begin{aligned} & q_{1}=-1\Rightarrow \left| q_{1}\right| =1, \\ & q_{2}=\frac{\lambda _{2}^{2}-\lambda _{2}\lambda _{3}-i\left( \lambda _{2}\lambda _{4}-\lambda _{1}\lambda _{5}\right) }{\lambda _{2}^{2}+\lambda _{2}\lambda _{3}-i\left( \lambda _{1}\lambda _{5}+\lambda _{2}\lambda _{4}\right) }\Rightarrow \left| q_{2}\right| \le 1. \end{aligned}$$
6 Computational results
In the present paper, we present numerical analysis of the fifth order KdV–Burgers–Fisher equation with two examples to illustrates the efficiency of the newly presented numerical scheme given in (23) with given boundary conditions. In the present investigation, the spatial lengths of finite elements are chosen as \(h=0.1,0.05,0.025\) and 0.0125 on interval \(I=\left[ -30,30\right]\) and time steps are reduced by halving from 0.2 to 0.0125. The error norms \(L_{2}\) and \(L_{\infty }\) are calculated using the following formulaeWhile investigating numerical solutions, the rate of convergence indicates how quickly numerical solution converges to the exact solution in terms of some characteristics such as time and step sizes. Thus, for the two examples, the rate of convergence is derived using the following formula based on the calculated error norms \(L_{2}\) and \(L_{\infty },\) as well
$$\begin{aligned} \begin{array}{ll} L_{2}=\sum \limits _{m=0}^{N}\sqrt{\left(\varphi_{m} \left( x,t\right) - \vartheta _{m}\left( x,t\right) \right) ^{2}}, & \left( x,t\right) \in [x_{l},x_{r}]\times [0,T] \\ & \\ L_{\infty }=\max\limits_{0\le m\le N} \left| \varphi _{m}\left( x,t\right) -\vartheta_{m} \left( x,t\right) \right| . & \end{array} \end{aligned}$$
$$\begin{aligned} RoC=\frac{Log\left( \left( L_{\infty }\right) _{1}/\left( L_{\infty }\right) _{2}\right) }{Log\left( \left( \Delta t_{1}\right) /\left( \Delta t_{2}\right) \right) }. \end{aligned}$$
Example 1
Consider KBF equation given in (1) with the following initial and boundary conditions, respectivelyandThe exact solution of the Example 1 is \(\varphi \left( x,t\right) =3\text{sech}^{2}\left( kx+ct+\xi _{0}\right) /2\) where \(\mu =\sqrt{10\varepsilon \gamma }/2k=\sqrt{5\mu /\gamma }/10,c=\left( -\varepsilon /25\right) \sqrt{5\mu /\gamma },\epsilon =-\upsilon \mu /5\gamma\) for \(\xi _{0}=0,\) \(\varepsilon =1,\) \(\gamma =1,\upsilon =0.02.\) In Tables 1 and 2, we present the error norms \(L_{2}\) and \(L_{\infty }\) for varying values of spatial step size h and time step size \(\Delta t.\) The first two tables include a change with respect to time step size for two fixed values of \(h= 0.1, 0.05,0.025\) and 0.0125. We can pursuit the effect of step size when Table 1 and 2 are considered together. Table 3 involves values of exact solution \(\varphi \left( x,t\right) ,\) numerical solution \(\vartheta \left( x,t\right)\) and absolute error for various mesh points at \(T=1.\) It can be seen from the Tables 1 and 2, as expected, the error norms are reducing while decreasing values of time and spatial step sizes. Table 4 illustrates the CPU time usage of numerical scheme for the first example. It is clear that the increase in time and location step numbers increases the accuracy of the solution, while CPU time requirement also increases. It is important to state that all the results given in the tables were calculated with symbolic programming language Matlab 2018a. The behavior of the numerical solutions is depicted in Figs. 1, 2, and 3. Specially, Fig. 1 shows a comparison of numerical and exact solutions for the values given in the figure caption. It can be seen from Fig. 1 and its zoomed part that exact and numerical solutions are almost overlapped to each other. Figure 2 presents the movement of the wave which is obtained from numerical solutions at different time levels. Lastly, Fig. 3 is a 3 dimensional representation of the wave obtained from the numerical solution at \(T=10.\)
$$\begin{aligned} \varphi \left( x,0\right) =\frac{3}{2}\text{sech}^{2}\left( kx+\xi _{0}\right) \quad x\in \left[ x_{l},x_{r}\right] \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \varphi \left( x_{l},t\right) =\frac{3}{2}\text{sech}^{2}\left( kx_{l}+ct+\xi _{0}\right) , \\ \varphi \left( x_{r},t\right) =\frac{3}{2}\text{sech}^{2}\left( kx_{r}+ct+\xi _{0}\right) , \\ \varphi _{x}\left( x_{l},t\right) =-3k\text{sinh} (kx_{l}+ct+\xi _{0}))/\cosh ^{3}(kx_{l}+ct+\xi _{0}), \\ \varphi _{x}\left( x_{r},t\right) =-3k\text{sinh} (kx_{r}+ct+\xi _{0}))/\cosh ^{3}(kx_{r}+ct+\xi _{0}). \end{array}\quad t\in \left[ 0,1\right] . \end{aligned}$$
Table 1
The error norms of Example 1 for various values of time step \(\Delta t\) and \(h=0.1,0.05\) at \(T=1\)
\(\Delta t\) | \(h=0.1\) | \(h=0.05\) | ||||
|---|---|---|---|---|---|---|
\(L_{2}\) | \(L_{\infty }\) | \(RoC\left( L_{\infty }\right)\) | \(L_{2}\) | \(L_{\infty }\) | \(RoC\left( L_{\infty }\right)\) | |
0.2 | \(2.20195 \times 10^{-4}\) | \(9.25503 \times 10^{-5}\) | – | \(2.19405 \times 10^{-4}\) | \(9.23422 \times 10^{-5}\) | – |
0.1 | \(5.58706 \times 10^{-5}\) | \(2.38736 \times 10^{-5}\) | 1.9548 | \(5.50664 \times 10^{-5}\) | \(2.36736 \times 10^{-5}\) | 1.9637 |
0.05 | \(1.62734 \times 10^{-5}\) | \(6.89286 \times 10^{-6}\) | 1.7922 | \(1.55425 \times 10^{-5}\) | \(6.70779 \times 10^{-6}\) | 1.8194 |
0.025 | \(4.68282 \times 10^{-6}\) | \(2.08638 \times 10^{-6}\) | 1.7241 | \(3.86748 \times 10^{-6}\) | \(1.65061 \times 10^{-6}\) | 2.0228 |
0.0125 | \(1.98441 \times 10^{-6}\) | \(9.18057 \times 10^{-7}\) | 1.1843 | \(9.86556 \times 10^{-7}\) | \(4.14601 \times 10^{-7}\) | 1.9932 |
Table 2
The error norms of Example 1 for various values of time step \(\Delta t\) and \(h=0.025,0.0125\) at \(T=1\)
\(\Delta t\) | \(h=0.025\) | \(h=0.0125\) | ||||
|---|---|---|---|---|---|---|
\(L_2\) | \(L_{\infty }\) | \(RoC(L_{\infty })\) | \(L_2\) | \(L_{\infty }\) | \(RoC(L_{\infty })\) | |
0.2 | \(2.19356 \times 10^{-4}\) | \(9.23630 \times 10^{-5}\) | – | \(2.19353 \times 10^{-4}\) | \(9.23653 \times 10^{-5}\) | – |
0.1 | \(5.50125 \times 10^{-5}\) | \(2.36659 \times 10^{-5}\) | 1.9645 | \(5.50069 \times 10^{-5}\) | \(2.36633 \times 10^{-5}\) | 1.9647 |
0.05 | \(1.54931 \times 10^{-5}\) | \(6.69824 \times 10^{-6}\) | 1.8210 | \(1.54870 \times 10^{-5}\) | \(6.69730 \times 10^{-6}\) | 1.8210 |
0.025 | \(3.82234 \times 10^{-6}\) | \(1.64057 \times 10^{-6}\) | 2.0296 | \(3.81902 \times 10^{-6}\) | \(1.63996 \times 10^{-6}\) | 2.0299 |
0.0125 | \(9.39692 \times 10^{-7}\) | \(4.03520 \times 10^{-7}\) | 2.0235 | \(9.36735 \times 10^{-7}\) | \(4.02817 \times 10^{-7}\) | 2.0255 |
Table 3
Approximate and analytical solutions with absolute error of Example 1 for the values of \(h=0.0125\) and \(\Delta t=0.00625\) at \(T=1\)
x | \(\varphi \left( x,t\right)\) | \(\vartheta \left( x,t\right)\) | \(\left| \varphi \left( x,t\right) -\vartheta \left( x,t\right) \right|\) |
|---|---|---|---|
\(-25\) | 0.0000037610 | 0.0000037577 | 0.0000000034 |
\(-20\) | 0.0000625208 | 0.0000625200 | 0.0000000008 |
\(-15\) | 0.0010398780 | 0.0010398866 | 0.0000000086 |
\(-10\) | 0.0172084922 | 0.0172085088 | 0.0000000166 |
\(-5\) | 0.2622014044 | 0.2622013087 | 0.0000000957 |
\(-3\) | 0.6730885788 | 0.6730885588 | 0.0000000200 |
0 | 1.4811852653 | 1.4811851947 | 0.0000000706 |
3 | 0.9164908921 | 0.9164909665 | 0.0000000745 |
5 | 0.3905728370 | 0.3905728750 | 0.0000000379 |
10 | 0.0268969258 | 0.0268969123 | 0.0000000136 |
15 | 0.0016303536 | 0.0016303386 | 0.0000000150 |
20 | 0.0000980483 | 0.0000980372 | 0.0000000110 |
25 | 0.0000058992 | 0.0000058924 | 0.0000000068 |
Table 4
CPU times of numerical scheme for Example 1 at \(T=1\)
CPU (s) | ||||
|---|---|---|---|---|
\(\Delta t\) | \(h=0.1\) | \(h=0.05\) | \(h=0.025\) | \(h=0.0125\) |
0.2 | 0.364084 | 0.955737 | 3.653997 | 23.720387 |
0.1 | 0.530086 | 1.454284 | 7.018478 | 38.963774 |
0.05 | 0.813310 | 2.676578 | 14.191663 | 80.939778 |
0.025 | 1.397405 | 5.175012 | 30.163960 | 169.418466 |
0.0125 | 2.509887 | 10.014613 | 61.588257 | 335.821115 |
Fig. 1
1-Dimensional simulations of numerical(–) and exact(o) solution of Example 1 for values of \(h=0.4,\) \(\Delta t=0.1,\) \(T=10,\) \(\epsilon =1,\) \(\gamma =1,\) \(\nu =0.02\) over \([-40,40]\)
Fig. 2
1-Dimensional simulations of numerical solutions of Example 1 for values of \(h=0.025,\) \(\Delta t=0.00625,\) \(\epsilon =1,\) \(\gamma =1,\) \(\nu =0.02\) over \([-40,40]\) at \(T=1(1)10\)
Fig. 3
3-Dimensional simulations of numerical solutions of Example 1 for values of \(h=0.025,\) \(\Delta t=0.00625,\) \(T=10,\) \(\epsilon =1,\) \(\gamma =1,\) \(\nu =0.02\) over \([-40,40]\)
×
![]()
×
![]()
×
![]()
Example 2
For the second example, we are going to take into consideration the KBF equation with the following initial and boundary conditionsandThe exact solution of the problem given as Example 2 is \(\varphi \left( x,t\right) =\left( 3\text{tanh}^{2}\left( kx+ct+\xi _{0}\right) -1\right) /2\) where \(\mu =-\sqrt{10\varepsilon \gamma }/2,\) \(k=\sqrt{-5\mu /\gamma }/10,\) \(c=\left( -\varepsilon /25\right) \sqrt{-5\mu /\gamma },\) \(\epsilon =-\upsilon \mu /5\gamma\) for \(\xi _{0}=0,\) \(\varepsilon =0.15,\) \(\gamma =\upsilon =0.1.\)
$$\begin{aligned} \varphi \left( x,0\right) =\left( 3\text{tanh}^{2}\left( kx+\xi _{0}\right) -1\right) /2\quad x\in \left[ x_{l},x_{r}\right] \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \varphi \left( x_{l},t\right) =\left( 3\text{tanh}^{2}\left( kx_{l}+ct+\xi _{0}\right) -1\right) /2, \\ \varphi \left( x_{r},t\right) =\left( 3\text{tanh}^{2}\left( kx_{r}+ct+\xi _{0}\right) -1\right) /2, \\ \varphi _{x}\left( x_{l},t\right) =-3k\text{tanh} (kx_{l}+ct+\xi _{0})(\tanh ^{2}(kx_{l}+ct+\xi _{0})-1), \\ \varphi _{x}\left( x_{r},t\right) =-3k\text{tanh}(kx_{r}+ct+\xi _{0})(\tanh ^{2}(kx_{r}+ct+\xi _{0})-1). \end{array}\quad t\in \left[ 0,1\right] . \end{aligned}$$
We repeated reducing the spatial and time step sizes by halving and reported the error norms in Tables 5 and 6. Additionally, Table 7 presents numerical solutions, exact solutions and absolute error for different mesh points at \(T=1.\) It is easy to see from the tables that for \(h=0.1, \Delta t=0.2,\) the error norms are \(L_{2}=1.25910\times 10^{-6}\) and \(L_{\infty }=8.53358\times 10^{-7}.\) When we chose \(\Delta t=h=0.0125,\) the error norms \(L_{2}=4.46325\times 10^{-8}\) and \(L_{\infty }=1.09876\times 10^{-8},\) the percentage decrease is approximately \(\%96.46\) and \(\%98.7\) for the error norms \(L_{2}\) and \(L_{\infty },\) respectively. Tables show us numerical results are in a good agreement with exact ones. Additionally, Table 8 displays the CPU time requirements of the numerical scheme. During simulation framework, numerical and exact solutions are presented in Fig. 4 for one dimensional illustrations. Figure 5 is a 3 dimensional representation of numerical solutions, simulations runs for final time \(T=10.\)
Table 5
The error norms of Example 2 for various values of time step \(\Delta t\) and \(h=0.1,0.05\) at \(T=1\)
\(\Delta t\) | \(h=0.1\) | \(h=0.05\) | ||||
|---|---|---|---|---|---|---|
\(L_{2}\) | \(L_{\infty }\) | \(RoC\left( L_{\infty }\right)\) | \(L_{2}\) | \(L_{\infty }\) | \(RoC\left( L_{\infty }\right)\) | |
0.2 | \(1.25910\times 10^{-6}\) | \(8.53358\times 10^{-7}\) | – | \(1.10238\times 10^{-6}\) | \(7.88217\times 10^{-7}\) | – |
0.1 | \(5.83436\times 10^{-7}\) | \(2.64415\times 10^{-7}\) | 1.6903 | \(2.80229\times 10^{-7}\) | \(2.00196\times 10^{-7}\) | 1.9772 |
0.05 | \(4.90710\times 10^{-7}\) | \(1.33777\times 10^{-7}\) | 0.9830 | \(7.84975\times 10^{-8}\) | \(5.32193\times 10^{-8}\) | 1.9114 |
0.025 | \(4.79343\times 10^{-7}\) | \(1.22889\times 10^{-7}\) | 0.1225 | \(3.65968\times 10^{-8}\) | \(1.66229\times 10^{-8}\) | 1.6788 |
0.0125 | \(4.77046\times 10^{-7}\) | \(1.21644\times 10^{-7}\) | 0.0147 | \(3.03631\times 10^{-8}\) | \(8.28761\times 10^{-9}\) | 1.0041 |
Table 6
The error norms of Example 2 for various values of time step \(\Delta t\) and \(h=0.025,0.0125\) at \(T=1\)
\(\Delta t\) | \(h=0.025\) | \(h=0.0125\) | ||||
|---|---|---|---|---|---|---|
\(L_{2}\) | \(L_{\infty }\) | \(RoC\left( L_{\infty }\right)\) | \(L_{2}\) | \(L_{\infty }\) | \(RoC\left( L_{\infty }\right)\) | |
0.2 | \(1.09806\times 10^{-6}\) | \(7.85205\times 10^{-7}\) | – | \(1.09830\times 10^{-6}\) | \(7.85524\times 10^{-7}\) | – |
0.1 | \(2.75016\times 10^{-7}\) | \(1.96545\times 10^{-7}\) | 1.9982 | \(2.76214\times 10^{-7}\) | \(1.97236\times 10^{-7}\) | 1.9937 |
0.05 | \(6.96794\times 10^{-8}\) | \(4.95782\times 10^{-8}\) | 1.9871 | \(7.28715\times 10^{-8}\) | \(5.10241\times 10^{-8}\) | 1.9507 |
0.025 | \(1.89078\times 10^{-8}\) | \(1.30327\times 10^{-8}\) | 1.9276 | \(3.51717\times 10^{-8}\) | \(1.69300\times 10^{-8}\) | 1.5916 |
0.0125 | \(1.25213\times 10^{-8}\) | \(4.98650\times 10^{-9}\) | 1.3860 | \(4.46325\times 10^{-8}\) | \(1.09876\times 10^{-8}\) | 0.6237 |
Table 7
Approximate and analytical solutions with absolute error of Example 2 for \(h=0.0125\) and \(\Delta t=0.00625\) at \(T=1\)
x | \(\varphi \left( x,t\right)\) | \(\vartheta \left( x,t\right)\) | \(\left| \varphi \left( x,t\right) -\vartheta \left( x,t\right) \right|\) |
|---|---|---|---|
\(-25\) | 0.9999989925 | 0.9999989883 | 0.0000000042 |
\(-20\) | 0.9999772875 | 0.9999772796 | 0.0000000079 |
\(-15\) | 0.9994898299 | 0.9994898191 | 0.0000000108 |
\(-10\) | 0.9885838665 | 0.9885838536 | 0.0000000129 |
\(-5\) | 0.7633672834 | 0.7633672685 | 0.0000000148 |
\(-3\) | 0.3230961572 | 0.3230961422 | 0.0000000150 |
0 | \(-\,0.4994772533\) | \(-\,0.4994772687\) | 0.0000000154 |
3 | 0.2850529119 | 0.2850528966 | 0.0000000152 |
5 | 0.7466375992 | 0.7466375852 | 0.0000000141 |
10 | 0.9877023032 | 0.9877022901 | 0.0000000130 |
15 | 0.9994502783 | 0.9994502673 | 0.0000000110 |
20 | 0.9999755260 | 0.9999755179 | 0.0000000081 |
25 | 0.9999989143 | 0.9999989099 | 0.0000000044 |
Table 8
CPU times of numerical scheme for Example 2 at \(T=1\)
CPU (s) | ||||
|---|---|---|---|---|
\(\Delta t\) | \(h=0.1\) | \(h=0.05\) | \(h=0.025\) | \(h=0.0125\) |
0.2 | 0.343504 | 0.893239 | 3.541823 | 21.428189 |
0.1 | 0.484449 | 1.575547 | 6.737044 | 54.002954 |
0.05 | 0.750325 | 2.715588 | 13.077740 | 89.354218 |
0.025 | 1.279236 | 5.207845 | 26.518607 | 174.219470 |
0.0125 | 2.627722 | 10.087041 | 64.157237 | 335.586161 |
0.00625 | 4.370999 | 20.190889 | 105.520583 | 679.977016 |
0.003125 | 8.961022 | 46.802687 | 205.815083 | 1321.358029 |
Fig. 4
1-Dimensional simulations of numerical(–) and exact(o) solution of Example 2 for values of \(h=0.4,\) \(\Delta t=0.1,\) \(T=10,\) \(\upsilon =1,\) \(\gamma =1,\) \(\nu =0.02\) over \([-40,40]\)
Fig. 5
3-Dimensional simulations of numerical solutions of Example 2 for values of \(h=0.025,\) \(\Delta t=0.05,\) \(T=10,\) \(\upsilon =0.15,\) \(\gamma =0.1,\) \(\nu =0.1\) over \([-40,40]\)
×
![]()
×
![]()
7 Conclusion
This paper presents a numerical approach to the fifth order KdV–Burgers–Fisher equation which has been used as a model for dispersion–dissipation–reaction relationship. To implement the numerical approach, a numerical scheme is utilized with combination of quintic trigonometric B-spline basis functions and collocation method. On the other hand, linearization process is considered a Rubin–Graves type linearization technique. The theoretical analysis of the proposed method includes the existence and uniqueness of the numerical solution, further proving its validity. To confirm the efficiency and effectiveness of the collocation method with quintic trigonometric B-splines, two numerical examples are considered, where the error norms, the rate of convergence, and CPU times are evaluated. It is observed that the proposed method produces quite accurate solutions and it can be a powerful technique for solving nonlinear partial differential equations utilizing different linearization techniques.
Declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.