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Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions
- Doğan Kaya^{1},
- Sema Gülbahar^{1},
- Asıf Yokuş^{2} and
- Mehmet Gülbahar^{3}Email author
https://doi.org/10.1186/s13662-018-1531-0
© The Author(s) 2018
- Received: 14 November 2017
- Accepted: 15 February 2018
- Published: 2 March 2018
Abstract
The exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV–mKdV) equation is obtained by using the \((1/G^{\prime})\) expansion method. To investigate a geometrical surface of the exact solution, we choose \(\gamma=1\). The collocation method is applied to the fractional combined KdV–mKdV equation with the help of radial basis for \(0<\gamma<1\). \(L_{2}\) and \(L_{\infty}\) error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz–Minkowski space.
Keywords
- Collocation method
- Fractional combined Korteweg-de Vries and modified Korteweg-de Vries equation
- Lorentz–Minkowski space
1 Introduction
Later, this frame of derivatives was studied by Liouville, Riemann, Weyl, Lacroix, Leibniz, Grunward, Letnikov, etc. (cf. [1]).
Since the inception of the definition of fractional order derivatives created by Leibniz, fractional partial differential equations have drawn attention of many mathematicians and have also shown an increasing development (cf. [2–14] etc.). Recently, analytical solutions of fractional differential equations have been obtained by the authors in [15, 16]. Furthermore, there exist many various applications of fractional partial differential equations in physics and engineering such as viscoelastic mechanics, power-law phenomenon in fluid and complex network, biology and ecology of allometric measurement legislation, colored noise, the electrode–electrolyte polarization, dielectric polarization, electromagnetic waves, numerical finance, etc. (cf. [17–21]).
In recent years, the collocation method has been a useful alternative tool to obtain numerical solutions since this method yields multiple numerical solutions depending on whether numerical methods such as finite differences, Runge–Kutta and Crank–Nicolson methods yield only numerical solutions. Using a few numbers of collocation points, this method has been widely studied by various authors to obtain high accuracy in numerical analysis (cf. [31–36]). On the other hand, radial basis functions are univariate functions which depend only on the distance between points and they are attractive to high dimensional differential equations. Furthermore, implementation and coding of the collocation method are very practical by using these bases. However, this method usually gives very efficient results as the number of node points is increased. We will see that it is not true when finding numerical solutions of the fractional combined KdV–mKdV equation in the present paper. This situation led us to examine the geometry of numerical solutions.
In the present paper, we obtain the exact solution of the fractional combined KdV–mKdV equation by using the \((1/G^{\prime })\) expansion method. With the help of radial basis functions, we apply the collocation method to this equation and obtain numerical solutions. We recognize that numerical solutions are more accurate for \(h=0.1\) than for \(h=0.01\). Therefore, we investigate the exact solution of the combined KdV–mKdV equation in a Lorentz–Minkowski space. Furthermore, we compute the Gauss curvature and the mean curvature of the exact solution and give a geometrical interpretation of these curvatures at the node points of our numerical solution. Finally, we observe that the exact solution contains some degenerate points in the Lorentz–Minkowski space at \(h=0.01\).
2 Analysis of \((1/G^{\prime})\)-expansion method
The \((1/G^{\prime})\) expansion method is used to obtain traveling wave solutions in nonlinear differential equation. In this section, we shall firstly mention a simple description of the \((1/G^{\prime})\)-expansion method by following [37]. Later, we shall obtain the exact solution of the combined KdV–mKdV equation by using this method.
The method is constructed as follows.
Firstly, if we substitute solution (7) into Eq. (6), then we obtain the second order IODE given in (8). Later, using (8), we have a set of algebraic equations of the same order of \((1/G^{\prime})\) which have to vanish. That is, all coefficients of the same order have to vanish. After we have manipulated these algebraic equations, we can find \(a_{i}\), \(i\geq0\), and V are constants and then, substituting \(a_{i}\) and the general solutions of Eq. (8) into (7), we can obtain solutions of Eq. (5).
Example 2.1
3 Collocation method using radial basis functions
Now, we shall use radial basis functions.
3.1 Stability analysis
In this subsection, we shall investigate the stability of this method with the help of Von-Neumann analysis.
3.2 \(L_{2}\) and \(L_{\infty}\) error norms
3.3 Test problem
We obtain good results for \(h=0.1\). However, we do not get better results for \(h=0.01\). Furthermore, we cannot find any result as the number of nodes increases.
4 Geometry of the exact solution
The geometry of the exact solutions of various equations has been intensely studied by different authors in various ways (cf. [40–44]). In this section, we are going to investigate the exact solution and the numerical solutions in the 3-dimensional space-time known as Lorentz–Minkowski space \(\mathbb{R}_{1}^{3}\). The main reason for choosing to work in this space is that the Lorentz–Minkowski space plays an important role in both special relativity and general relativity with space coordinates and time coordinates.
First, we need to recall some basic facts and notations in \(\mathbb{R}_{1}^{3}\) (cf. [45–49]).
- (i)
a timelike vector if \(\langle X,X \rangle<0\),
- (ii)
a spacelike vector if \(\langle X,X \rangle>0\),
- (iii)
a lightlike \(( \text{or degenerate} ) \) vector if \(\langle X,X \rangle=0\) and \(X\neq0\).
- (i)
\(\Vert X \Vert =\sqrt{ \langle X,X \rangle}\) if X is a spacelike vector,
- (ii)
\(\Vert X \Vert =-\sqrt{ \langle X,X \rangle}\) if X is a timelike vector.
- (i)
a timelike future pointing vector if \(\langle X,e \rangle>0\),
- (ii)
a timelike past pointing vector if \(\langle X,e \rangle <0\).
- (i)
a timelike surface if N is spacelike,
- (ii)
a spacelike surface if N is timelike,
- (iii)
a lightlike (or degenerate) surface if N is lightlike.
We note that a point is called regular if \(N\neq0\) and singular if \(N=0\).
As a consequence of the above facts, we immediately get the following.
Corollary 4.1
Classification of \(r(x,t)\) surface at node points
Node points | 〈N,N〉 | Class |
---|---|---|
x = 0 | −1166.57 | spacelike |
x = 0.1 | −233.172 | spacelike |
x = 0.2 | −73.2727 | spacelike |
x = 0.3 | −29.2972 | spacelike |
x = 0.4 | −13.4868 | spacelike |
x = 0.5 | −6.72612 | spacelike |
x = 0.6 | −3.46118 | spacelike |
x = 0.7 | −1.73585 | spacelike |
x = 0.8 | −0.758654 | spacelike |
x = 0.9 | −0.173935 | spacelike |
x ≈ 0.94 | 0 | lightlike |
x = 1 | 0.191883 | timelike |
Remark 4.2
From Table 2, we see that the surface \(r(x,t)\) contains at least one degenerate point near \(x=0.94\). As the number of node points increases, we approach degenerate points. Therefore, numerical solutions become instable when the number of node points increases.
5 Gaussian curvature of node points
Another important fact for a surface is to compute the Gaussian curvature which is an intrinsic character of it. The Gaussian curvature is the determinant of the shape operator. For a surface \(r ( x,t ) \), we shall apply the following useful way to compute the Gaussian curvature:
- (i)
\(K ( p ) >0\) means that the surface \(r ( x,t ) \) is shaped like an elliptic paraboloid near p. In this case, p is called an elliptic point.
- (ii)
\(K ( p ) <0\) means that the surface \(r ( x,t ) \) is shaped like a hyperbolic paraboloid near p. In this case, p is called a hyperbolic point.
- (iii)
\(K ( p ) =0\) means that the surface \(r ( x,t ) \) is shaped like a parabolic cylinder or a plane near p. In this case, p is called a parabolic point.
As a consequence of the above facts, we get the following corollary:
Corollary 5.1
Curvatures of \(r(x,t)\) surface at node points
Node points | Mean curvature | Gauss curvature |
---|---|---|
x = 0 | −0.00423076 | −0.0000174161 |
x = 0.1 | −0.0142516 | −0.000039501 |
x = 0.2 | −0.034402 | −0.0000723291 |
x = 0.3 | −0.0699955 | −0.000119721 |
x = 0.4 | −0.130085 | −0.000190356 |
x = 0.5 | −0.233077 | −0.000304767 |
x = 0.6 | −0.423579 | −0.000517959 |
x = 0.7 | −0.838338 | −0.00101754 |
x = 0.8 | −2.11606 | −0.00283336 |
x = 0.9 | −14.4824 | −0.0304277 |
x = 0.94 | −1408.05 | −12.6644 |
x = 1 | −19.2403 | 0.0148218 |
Remark 5.2
From Table 3, we see that if x approaches 0.94, then the values of Gauss curvature and mean curvature change remarkably. Therefore, there exists the maximum external influence near the point 0.94.
6 Conclusions
Using the \((1/G')\) expansion method, the exact solution \(r(x,t)\) of the fractional combined KdV–mKdV equation is obtained. The numerical solutions of the fractional combined KdV–mKdV equation are shown by using the collocation method. These solutions are compared with the exact solution. The computational efficiency and effectiveness of the proposed method were tested on a problem. The error norms \(L_{2}\) and \(L_{\infty}\) have been calculated. The obtained results show that the error norms are small during all computer runs for all bases except for MQ basis. It was proved that the present method is a particularly successful numerical scheme to solve the fractional combined KdV–mKdV equation. However, numerical solutions are more accurate for \(h=0.1\) than for \(h=0.01\). Therefore, casual character of the exact solution was expressed at the nodal points. From Tables 1 and 2, it was realized that the most accurate numerical solution occurred in the timelike case of \(r(x,t)\), and there exists at least one degenerate point near \(x=0.94\). Furthermore, from Table 3, it was realized that the most accurate numerical solution occurred at the elliptic points of \(r(x,t)\), and the ideal node point of \(r(x,t)\) is \(x=0\).
Declarations
Acknowledgements
The authors are thankful to the referees for their valuable comments and constructive suggestions towards the improvement of the paper.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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