On the Numerical Solution of Evolution Equation via Soliton Kernels

Abstract

In application of kernel-based methods, some particular types of

PDEs need some special types of kernels for their approximations.

For example some nonlinear evolution equations describing wave

processes in dispersive and dissipative media. These models may have

soliton like solutions for example KdV equation. In such a situation

some special types of kernels may perform better than standards

kernels for example soliton kernels.

Keywords

• Soliton kernels, Meshless technique, RBF-PS scheme, evolution equations

Soliton Kernels

Abstract

References

1. Fasshauer, G. E., Positive definite kernels: Past, present and future. In M. Buhmann, S. D. Marchi, and Plonka-Hoch, editors, Kernel Functions and Meshless Methods, volume 4 of Dolomites Research Notes on Approximation, Special Issue. 21-63, (2011).
2. R. Schaback, R., S. De Marchi, Nonstandard Kernels and their Applications, Dolomites Research Notes on Approximation 2, (2009) (http: drna.di.univr.it).[3] Drazin P. G., Johnson R. S., Solitons: an introduction, Cambridge University Press (1989).
3. Belytschko, T., Y. Krongauz, D. J. Orgam, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments, Comput. Meth. Appl. Mech. and Eng., special issue, 139, 3-47, (1996).
4. Atluri, S. N. and T. L. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22, 117-127, (1998).
5. Buhmann, M. D., Radial basis functions: theory and Cambridge implementations, (2003). University Press,
6. Fasshauer, G. E., Meshfree Approximation Methods with MATLAB, Mathematical Sciences, World Scientific Publishers, Singapore, volume 6, (2007). Interdisciplinary
7. Shu, C., Ding, H., Yeo, S., Local radial basis function- based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput. Method. Appl. Mech. Eng. 192, 941-954, (2003).
8. Uddin M, Haq S., On the numerical solution of generalized nonlinear Schrodinger equation using RBFs, Miskolc Mathematical Notes, 14 (3), 1067- 1084, (2013).

9. Uddin, M., On the selection of a good value of shape parameter differential equations using RBF approximation method, Applied Mathematical Modelling, 38, 135- 144, (2014).
10. time-dependent partial
11. Uddin, M., RBF-PS scheme for solving the equal width Computation, 222, 619-631, (2013). Mathematics and
12. Dehghan M., Shokri A., A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn, 50, 111-120, (2007).
13. Shen Q., A meshless method of lines for the numerical solution of KdV equation using radial basis functions, Engineering Analysis with Boundary Elements, 33, 1171-1180, (2009).
14. Ronghua Chen, Zongmin Wu, Solving partial differential equation by using multiquadric quasi- interpolation, Appl. math. Comput. 186, 1502-1510, (2007).