Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition

Koya Sakakibara

Applications of Mathematics (2017)

  • Volume: 62, Issue: 4, page 297-317
  • ISSN: 0862-7940

Abstract

top
The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results of numerical experiments, which verify the sharpness of our error estimate.

How to cite

top

Sakakibara, Koya. "Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition." Applications of Mathematics 62.4 (2017): 297-317. <http://eudml.org/doc/294131>.

@article{Sakakibara2017,
abstract = {The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results of numerical experiments, which verify the sharpness of our error estimate.},
author = {Sakakibara, Koya},
journal = {Applications of Mathematics},
keywords = {method of fundamental solutions; biharmonic equation; Almansi-type decomposition},
language = {eng},
number = {4},
pages = {297-317},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition},
url = {http://eudml.org/doc/294131},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Sakakibara, Koya
TI - Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 297
EP - 317
AB - The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results of numerical experiments, which verify the sharpness of our error estimate.
LA - eng
KW - method of fundamental solutions; biharmonic equation; Almansi-type decomposition
UR - http://eudml.org/doc/294131
ER -

References

top
  1. Almansi, E., 10.1007/bf02419286, Annali di Mat. (3) 2 (1897), Italian 1-51 9999JFM99999 30.0331.03. (1897) DOI10.1007/bf02419286
  2. Bock, S., Gürlebeck, K., 10.1002/mma.1033, Math. Methods Appl. Sci. 32 (2009), 223-240. (2009) Zbl1151.74308MR2478914DOI10.1002/mma.1033
  3. Karageorghis, A., 10.1007/s11075-014-9900-6, Numer. Algorithms 68 (2015), 185-211. (2015) Zbl1308.65210MR3296706DOI10.1007/s11075-014-9900-6
  4. Karageorghis, A., Fairweather, G., 10.1016/0021-9991(87)90176-8, J. Comput. Phys. 69 (1987), 434-459. (1987) Zbl0618.65108MR0888063DOI10.1016/0021-9991(87)90176-8
  5. Karageorghis, A., Fairweather, G., 10.1002/nme.1620260714, Int. J. Numer. Methods Eng. 26 (1988), 1665-1682. (1988) Zbl0639.65066MR1016068DOI10.1002/nme.1620260714
  6. Katsurada, M., A mathematical study of the charge simulation method by use of peripheral conformal mappings, Mem. Inst. Sci. Tech. Meiji Univ. 35 (1998), 195-212. (1998) MR0965011
  7. Katsurada, M., Okamoto, H., A mathematical study of the charge simulation method. I, J. Fac. Sci., Univ. Tokyo, Sect. I A 35 (1988), 507-518. (1988) Zbl0662.65100MR0965011
  8. Krakowski, M., Charnes, A., Stokes' Paradox and Biharmonic Flows, Report 37, Carnegie Institute of Technology, Department of Mathematics, Pittsburgh (1953). (1953) 
  9. Langlois, W. E., Deville, M. O., 10.1007/978-3-319-03835-3, Springer, Cham (2014). (2014) Zbl1302.76003MR3186274DOI10.1007/978-3-319-03835-3
  10. Li, Z.-C., Lee, M.-G., Chiang, J. Y., Liu, Y. P., 10.1016/j.cam.2011.03.024, J. Comput. Appl. Math. 235 (2011), 4350-4367. (2011) Zbl1222.65131MR2802010DOI10.1016/j.cam.2011.03.024
  11. Poullikkas, A., Karageorghis, A., Georgiou, G., 10.1007/s004660050320, Comput. Mech. 21 (1998), 416-423. (1998) Zbl0913.65104MR1628005DOI10.1007/s004660050320
  12. Sakakibara, K., 10.1007/s10543-016-0605-1, BIT 56 (2016), 1369-1400. (2016) Zbl06667568MR3576615DOI10.1007/s10543-016-0605-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.