New Cubic B-spline Approximations for Solving Non-linear Third-order Korteweg-de Vries Equation

Objectives: In this work, the approximate solution of non-linear third order Korteweg-de Vries equation has been studied. Methods: The proposed numerical technique engages finite difference formulation for temporal discretization, whereas, the discretization in space direction is achieved by means of a new cubic B-spline approximation. Findings: In order to corroborate this effort, three test problems have been considered and the computational outcomes are compared with the current methods. It is found that the proposed scheme involves straight forward computations and operates superior to the existing methods. Novelty/Improvements: The proposed numerical scheme is novel for Korteweg-de Vries equation and has never been employed for this purpose before. Indian Journal of Science and Technology, Vol 12(6), DOI: 10.17485/ijst/2019/v12i6/141953, February 2019


Introduction
The third order non-linear Korteweg-de Vries (KdV) equation occurs in many physical applications such as non-linear plasma waves which exhibit certain dissipative effects 1 , propagation of waves 2 and propagation of bores in shallow water waves 3 . The KdV equation is given by  In recent years, the KdV equation has gained a considerable research attraction due to its numerous applications in real life phenomena. Especially, the traveling wave solution has been considered extensively. Kutluay et al. 4 employed integral methods with heat balance to study the small time solutions to KdV equation. The numerical solution to third order KdV equation was discussed by Bahadir 5 using exponential finite difference scheme. Ozer and Kultuay 6 proposed a numerical technique for solving KdV type equations. The authors in 7 employed the method of lines for small times solution of KdV equation. Dehghan and Shokri 8 proposed a numerical method based on multi-quadratic radial basis functions for solving KdV equation. Dag and Dereli 9 explored the numerical solution of KdV equation by means of radial basis functions. A mesh free method based on radial basis functions was presented by Khattak and Tirmizi 10 for approximate solution of KdV equation. Xiao et al. 11 investigated the numerical solution to KdV equation using multi-quadric quasi-interpolation operator. Sarboland and Aminataei 12 proposed a numerical scheme based on integrated radial basis functions and multi-quadric quasi-interpolation operator for solving of KdV equation. Rashid et al. 13 solved Hirota-Satsuma coupled KdV equation by Fourier Pseudo-spectral method.
The spline functions are used extensively to solve the initial and boundary value problems. These functions preserve a smoothness at the nodes and have the ability to provide the numerical solution in the entire domain with great accuracy. Irk et al. 14  In this work, the numerical solution of non-linear KdV equation has been considered. The usual finite difference scheme 24 and new Cubic B-Spline (CBS) approximations 25,26 have been used for temporal and spatial discretization respectively.
The roadways of this study is: In section 2, we shall discuss some preliminaries of ordinary CBS interpolation. The numerical method is presented in section 3 and experimental outcomes are given in section 4.

Cubic B-spline Functions
For r > 0 and x x x Using(4), the typical CBS functions are defined as 28 where, c t p ( )' s are, time dependent real constants, yet to be calculated. For simplicity, we express the CBS and T i respectively. The third degree basis spline functions (5) together with (6) yield the following relations Moreover, for second and third order derivatives, we shall use the following new CBS approximations 25,26 Vol 12 (6)

Description of the Numerical Method
In this section, we present the numerical scheme for solving non-linear KdV equation. Applying usual finite difference method and θ weighted scheme, the problem is discretized in time direction as where, ∆t is the step size in time direction, 0 1 Substituting (12) into (11), we get Forθ = 1 2 , the relation (13) can be rearranged as Substituting the approximation for u and its derivatives at the knot x i , equation (14) where, w      (31) The above system can be expressed in matrix form as The unknown column vector C 0 is determined by wellknown Thomas algorithm. The numerical computations are executed in Mathematica 9.

Numerical Results
In this section, the approximate solution to (1)-(2) is presented. The accuracy and validity of the proposed numerical method is tested by three error norms L ∞ , L 2 and Root Mean Square (RMS), which are calculated as , , , The error norms L ∞ , L 2 and RMS are listed in Tables 1-3, when n = 200 and ∆t = 0.01. It is revealed that the proposed numerical scheme produces more reliable and accurate results as compared to MQRBF 8 , MQ 10 , IMQ 10 , MQQI 11 and IMQQI 12 . Figure 1 shows a very close agreement of the numerical solution with closed form solution for t = 1,3,5. Three dimensional plots of exact and approximate solutions are shown in Figures 2 and 3. The absolute computational error using n t = = 200 0 01 , . ∆ is displayed in Figure 4. Table 1. Absolute numerical error for Example 1, when 0 40 The exact solution is u x t x t , sech The computational error norms L ∞ , L 2 and RMS are listed in Table 4 when n = 200 and ∆t = 0.01. Figure 5 shows the approximate and exact solution at t = 0.2,0.4,0.6,0.8,1. The three dimensional plots of analytical and approximate solutions are displayed in Figures 6 and 7. The absolute computational error is portrayed in Figure 8 using n = 200 and ∆t = 0.01.

Conclusion
In this paper, numerical solution of non-linear third order KdV equation has been explored. We conclude the outcomes of this research as: 1. The presented algorithm is based on usual finite difference scheme and CBS collocation method. 2. The proposed technique is novel for third order nonlinear KdV equation. 3. Usual finite difference scheme has been employed for temporal discretization. 4. The new CBS approximations have been used to interpolate the solution in space direction. 5. Due to straightforward and simple application, it outperforms the MQRBF 8 , MQ 10 , IMQ 10 , MQQI 11 and IMQQI 12 approaches.

Acknowledgments
This study was fully supported by