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On the Numerical Solution of Evolution Equation via Soliton Kernels

Year 2015, Volume: 28 Issue: 4, 631 - 637, 16.12.2015

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.

References

  • 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).
  • 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).
  • 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).
  • Atluri, S. N. and T. L. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22, 117-127, (1998).
  • Buhmann, M. D., Radial basis functions: theory and Cambridge implementations, (2003). University Press,
  • Fasshauer, G. E., Meshfree Approximation Methods with MATLAB, Mathematical Sciences, World Scientific Publishers, Singapore, volume 6, (2007). Interdisciplinary
  • 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).
  • Uddin M, Haq S., On the numerical solution of generalized nonlinear Schrodinger equation using RBFs, Miskolc Mathematical Notes, 14 (3), 1067- 1084, (2013).
  • 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).
  • time-dependent partial
  • Uddin, M., RBF-PS scheme for solving the equal width Computation, 222, 619-631, (2013). Mathematics and
  • Dehghan M., Shokri A., A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn, 50, 111-120, (2007).
  • 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).
  • Ronghua Chen, Zongmin Wu, Solving partial differential equation by using multiquadric quasi- interpolation, Appl. math. Comput. 186, 1502-1510, (2007).

Soliton Kernels

Year 2015, Volume: 28 Issue: 4, 631 - 637, 16.12.2015

Abstract

References

  • 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).
  • 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).
  • 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).
  • Atluri, S. N. and T. L. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22, 117-127, (1998).
  • Buhmann, M. D., Radial basis functions: theory and Cambridge implementations, (2003). University Press,
  • Fasshauer, G. E., Meshfree Approximation Methods with MATLAB, Mathematical Sciences, World Scientific Publishers, Singapore, volume 6, (2007). Interdisciplinary
  • 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).
  • Uddin M, Haq S., On the numerical solution of generalized nonlinear Schrodinger equation using RBFs, Miskolc Mathematical Notes, 14 (3), 1067- 1084, (2013).
  • 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).
  • time-dependent partial
  • Uddin, M., RBF-PS scheme for solving the equal width Computation, 222, 619-631, (2013). Mathematics and
  • Dehghan M., Shokri A., A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn, 50, 111-120, (2007).
  • 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).
  • Ronghua Chen, Zongmin Wu, Solving partial differential equation by using multiquadric quasi- interpolation, Appl. math. Comput. 186, 1502-1510, (2007).
There are 14 citations in total.

Details

Journal Section Mathematics
Authors

Marjan Uddin

Irshad Ali Shah This is me

Hazrat Ali This is me

Publication Date December 16, 2015
Published in Issue Year 2015 Volume: 28 Issue: 4

Cite

APA Uddin, M., Shah, I. A., & Ali, H. (2015). On the Numerical Solution of Evolution Equation via Soliton Kernels. Gazi University Journal of Science, 28(4), 631-637.
AMA Uddin M, Shah IA, Ali H. On the Numerical Solution of Evolution Equation via Soliton Kernels. Gazi University Journal of Science. December 2015;28(4):631-637.
Chicago Uddin, Marjan, Irshad Ali Shah, and Hazrat Ali. “On the Numerical Solution of Evolution Equation via Soliton Kernels”. Gazi University Journal of Science 28, no. 4 (December 2015): 631-37.
EndNote Uddin M, Shah IA, Ali H (December 1, 2015) On the Numerical Solution of Evolution Equation via Soliton Kernels. Gazi University Journal of Science 28 4 631–637.
IEEE M. Uddin, I. A. Shah, and H. Ali, “On the Numerical Solution of Evolution Equation via Soliton Kernels”, Gazi University Journal of Science, vol. 28, no. 4, pp. 631–637, 2015.
ISNAD Uddin, Marjan et al. “On the Numerical Solution of Evolution Equation via Soliton Kernels”. Gazi University Journal of Science 28/4 (December 2015), 631-637.
JAMA Uddin M, Shah IA, Ali H. On the Numerical Solution of Evolution Equation via Soliton Kernels. Gazi University Journal of Science. 2015;28:631–637.
MLA Uddin, Marjan et al. “On the Numerical Solution of Evolution Equation via Soliton Kernels”. Gazi University Journal of Science, vol. 28, no. 4, 2015, pp. 631-7.
Vancouver Uddin M, Shah IA, Ali H. On the Numerical Solution of Evolution Equation via Soliton Kernels. Gazi University Journal of Science. 2015;28(4):631-7.