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Year 2019, Volume: 3 Issue: 1, 23 - 31, 24.06.2019

Tayfun Tanbay

https://doi.org/10.38088/jise.570328

Abstract

References

  • [1] Kansa, E.J. (1986). Application of Hardy’s multiquadric interpolation to hydrodynamics, In: R. Crosbie, P. Luker (Eds.), Proceedings of the 1986 Summer Computer Simulation Conference, San Diego, Society for Computer Simulation, 4:111-117.
  • [2] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators. Computers and Mathematics with Applications, 24:121-138.
  • [3] Hardy, R.L. (1971). Multiquadratic equations for topography and other irregular surfaces. Journal of Geophysical Research, 76:1905-1915.
  • [4] Franke, R. (1982). Scattered data interpolation tests of some methods. Mathematics of Computation, 38:181-200.
  • [5] Fasshauer, G.E. (2002). Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43:423-438.
  • [6] Rippa, S. (1999). An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics, 11:193-210.
  • [7] Esmaeilbeigi, M. and Hosseini, M.M. (2014). A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa’s method. Applied Mathematics and Computation, 249:419-428.
  • [8] Koupaei, J.A., Firouznia, M. and Hosseini, M.M. (2018). Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm. Alexandria Engineering Journal, 57:3641-3652.
  • [9] Luh, L.T. (2019). The choice of the shape parameter-A friendly approach. Engineering Analysis with Boundary Elements, 98:103-109.
  • [10] Tanbay, T. and Ozgener, B. (2013). Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method. Mathematical and Computational Applications, 18:399-407.
  • [11] Tanbay, T. and Ozgener, B. (2014). A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations. Engineering Analysis with Boundary Elements, 46:30-40.
  • [12] Tanbay, T. (2018). On the accuracy and stability of the meshless RBF collocation method for neutron diffusion calculations. Journal of Innovative Science and Engineering, 2:8-18.
  • [13] Fedoseyev, A.I., Friedman, M.J. and Kansa, E.J. (2002). Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications, 43:439-455.

Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

Year 2019, Volume: 3 Issue: 1, 23 - 31, 24.06.2019

Tayfun Tanbay

https://doi.org/10.38088/jise.570328

Abstract

The meshless
radial basis function collocation method is an efficient numerical technique
for solving partial differential equations. The multiquadric is the most widely
utilized radial function for this purpose; but it contains a shape parameter,
which has a significant effect on the performance of the method. In this study,
the meshless collocation method employing multiquadric as the radial function with
optimum shape parameters is applied to the numerical solution of the multigroup
neutron diffusion equation. The optimization of the shape parameter is
performed by minimizing the Madych-Nelson function. One external and two
fission source problems are solved to investigate the performance of the
method. The results show that the meshless collocation method with optimized
shape parameters yield a high level of accuracy with an exponential convergence
rate.

Keywords

Meshless RBF Collocation Optimum shape parameter Neutron diffusion

References

  • [1] Kansa, E.J. (1986). Application of Hardy’s multiquadric interpolation to hydrodynamics, In: R. Crosbie, P. Luker (Eds.), Proceedings of the 1986 Summer Computer Simulation Conference, San Diego, Society for Computer Simulation, 4:111-117.
  • [2] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators. Computers and Mathematics with Applications, 24:121-138.
  • [3] Hardy, R.L. (1971). Multiquadratic equations for topography and other irregular surfaces. Journal of Geophysical Research, 76:1905-1915.
  • [4] Franke, R. (1982). Scattered data interpolation tests of some methods. Mathematics of Computation, 38:181-200.
  • [5] Fasshauer, G.E. (2002). Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43:423-438.
  • [6] Rippa, S. (1999). An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics, 11:193-210.
  • [7] Esmaeilbeigi, M. and Hosseini, M.M. (2014). A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa’s method. Applied Mathematics and Computation, 249:419-428.
  • [8] Koupaei, J.A., Firouznia, M. and Hosseini, M.M. (2018). Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm. Alexandria Engineering Journal, 57:3641-3652.
  • [9] Luh, L.T. (2019). The choice of the shape parameter-A friendly approach. Engineering Analysis with Boundary Elements, 98:103-109.
  • [10] Tanbay, T. and Ozgener, B. (2013). Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method. Mathematical and Computational Applications, 18:399-407.
  • [11] Tanbay, T. and Ozgener, B. (2014). A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations. Engineering Analysis with Boundary Elements, 46:30-40.
  • [12] Tanbay, T. (2018). On the accuracy and stability of the meshless RBF collocation method for neutron diffusion calculations. Journal of Innovative Science and Engineering, 2:8-18.
  • [13] Fedoseyev, A.I., Friedman, M.J. and Kansa, E.J. (2002). Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications, 43:439-455.
There are 13 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Articles
Authors

Tayfun Tanbay 0000-0002-0428-3197

Publication Date June 24, 2019
Published in Issue Year 2019Volume: 3 Issue: 1

Cite

APA Tanbay, T. (2019). Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering, 3(1), 23-31. https://doi.org/10.38088/jise.570328
AMA Tanbay T. Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. June 2019;3(1):23-31. doi:10.38088/jise.570328
Chicago Tanbay, Tayfun. “Meshless Solution of the Neutron Diffusion Equation by the RBF Collocation Method Using Optimum Shape Parameters”. Journal of Innovative Science and Engineering 3, no. 1 (June 2019): 23-31. https://doi.org/10.38088/jise.570328.
EndNote Tanbay T (June 1, 2019) Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering 3 1 23–31.
IEEE T. Tanbay, “Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters”, JISE, vol. 3, no. 1, pp. 23–31, 2019, doi: 10.38088/jise.570328.
ISNAD Tanbay, Tayfun. “Meshless Solution of the Neutron Diffusion Equation by the RBF Collocation Method Using Optimum Shape Parameters”. Journal of Innovative Science and Engineering 3/1 (June 2019), 23-31. https://doi.org/10.38088/jise.570328.
JAMA Tanbay T. Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. 2019;3:23–31.
MLA Tanbay, Tayfun. “Meshless Solution of the Neutron Diffusion Equation by the RBF Collocation Method Using Optimum Shape Parameters”. Journal of Innovative Science and Engineering, vol. 3, no. 1, 2019, pp. 23-31, doi:10.38088/jise.570328.
Vancouver Tanbay T. Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. 2019;3(1):23-31.
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