THE MESHLESS KERNEL-BASED METHOD OF LINES FOR SOLVING THE DISSIPATIVE GENERALIZED SRLW EQUATIONS WITH DAMPING TERM

Year 2020, Volume: 8 Issue: 1, 34 - 48, 28.02.2020

Bahar Karaman Yılmaz Dereli

https://doi.org/10.20290/estubtdb.530370

Abstract

In this study, we dealt with numerical
solutions of the dissipative generalized symmetric regularized long wave
equations with damping term. The problem is a nonlinear partial differential
equations system. Numerical solutions of the problem were evaluated by using
the meshless kernel based method of lines for known initial-boundary conditions
on the given solution domain. This used numerical method is known to be a truly
meshless approximation because any separation method is required. Radial basis
functions are used as kernel functions on the meshless method. The performance
of this meshless method was illustrated on many standard test problems.
Numerical computations were performed by using Gaussian and Wendland’s
functions. Error comparisons for computed numerical results were made in the
sense of L error norm. Graphs of wave simulations for
test problems are plotted in this study. The results show that the used
meshless method is suitable to solve numerically to this type nonlinear
equations system.

Keywords

Meshless Method, Method of Lines, Damping term, Dissipative Symmetric RLW Equation

References

• [1] Seyler, C. E., Fenstermacher, D.L. A symmetric regularized-long-wave equation. Physics of Fluids. 1984; 27(1), 1-15.
• [2] Guo, B. L. The spectral method for symmetric regularized wave equations, Journal of Computational Mathematics, 1987; 5(4), 297-306.
• [3] Duan, G. S., Zao, T.F. Solitary wave solutions for equation of generalized symmetric regular long wave. Journal of Changsha University. 2000; 14(2), 31-32. [4] Montes, A. M. Travelling waves for a generalized symmetric regularized-long-wave model. International Journal of Mathematical Analysis, 2015; 9(33); 1609 – 1625.
• [5] Chen, L. Stability and instability of solitary waves for generalized symmetric regularized long wave equation. Physica D. 1998; 118(1-2), 53-68.
• [6] Shang, Y. Guo, B. Analysis of chebyshev pseduspectral method for multi-dimensional generalized SRLW equations. Applied Mathematics and Mechanics. 2003; 24(10), 1168-1183.
• [7] Yong, C., Biao, L. Travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms. Chinese Physics, 2004, 13(3); 302-306. [8] Zhou, J. Numerical simulation of generalized symmetric regularized long-wave equations with damping term. International Journal of Digital Content Technology and its Applications. 2013; 7(6); 1142- 1149.
• [9] Xu, Y., Hu, B., Xie, X., Hu, J. Mixed finite element analysis for dissipative SRLW equations with damping term. Applied Mathematics and Computation. 2012; 218; 4788–4797.
• [10] Hu, J., Xu, Y., Hu, B. A linear difference scheme for dissipative symmetric regularized long wave equations with damping term. Journal of Boundary Value Problems. 2010; 2010, 1-16.
• [11] Hu, J., Hu, B., Xu, Y. C-N difference schemes for dissipative symmetric regularized long wave equations with damping term. Mathematical Problems in Engineering. 2011; 2011; 1-16.
• [12] Shang, Y. D., Guo, B. L. Exponential attractor for the generalized symmetric regularized long wave equation with damping term. Applied Mathematics and Mechanics, 2005; 26(3); 283-291.
• [13] Schaback, R. The meshless kernel-based method of lines for solving nonlinear evolution equations. Preprint, Göttingen, 2008.
• [14] Wendland, H. Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree. Advances in Computational Mathematics. 1995; 4(1); 389-396.