| Title:
|
Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition (English) |
| Author:
|
Sakakibara, Koya |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2017 |
| Pages:
|
297-317 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
method of fundamental solutions |
| Keyword:
|
biharmonic equation |
| Keyword:
|
Almansi-type decomposition |
| MSC:
|
31A30 |
| MSC:
|
49M27 |
| MSC:
|
65N80 |
| idZBL:
|
Zbl 06770046 |
| idMR:
|
MR3686419 |
| DOI:
|
10.21136/AM.2017.0018-17 |
| . |
| Date available:
|
2017-08-31T12:43:28Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146830 |
| . |
| Reference:
|
[1] Almansi, E.: Sull'integrazione dell'equazione differentiale $\triangle^{2n}=0$.Annali di Mat. (3) 2 (1897), Italian 1-51 \99999JFM99999 30.0331.03. 10.1007/bf02419286 |
| Reference:
|
[2] Bock, S., Gürlebeck, K.: On a spatial generalization of the Kolosov-Muskhelishvili formulae.Math. Methods Appl. Sci. 32 (2009), 223-240. Zbl 1151.74308, MR 2478914, 10.1002/mma.1033 |
| Reference:
|
[3] Karageorghis, A.: The method of fundamental solutions for elliptic problems in circular domains with mixed boundary conditions.Numer. Algorithms 68 (2015), 185-211. Zbl 1308.65210, MR 3296706, 10.1007/s11075-014-9900-6 |
| Reference:
|
[4] Karageorghis, A., Fairweather, G.: The method of fundamental solutions for the numerical solution of the biharmonic equation.J. Comput. Phys. 69 (1987), 434-459. Zbl 0618.65108, MR 0888063, 10.1016/0021-9991(87)90176-8 |
| Reference:
|
[5] Karageorghis, A., Fairweather, G.: The Almansi method of fundamental solutions for solving biharmonic problems.Int. J. Numer. Methods Eng. 26 (1988), 1665-1682. Zbl 0639.65066, MR 1016068, 10.1002/nme.1620260714 |
| Reference:
|
[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. MR 0965011 |
| Reference:
|
[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. Zbl 0662.65100, MR 0965011 |
| Reference:
|
[8] Krakowski, M., Charnes, A.: Stokes' Paradox and Biharmonic Flows.Report 37, Carnegie Institute of Technology, Department of Mathematics, Pittsburgh (1953). |
| Reference:
|
[9] Langlois, W. E., Deville, M. O.: Slow Viscous Flow.Springer, Cham (2014). Zbl 1302.76003, MR 3186274, 10.1007/978-3-319-03835-3 |
| Reference:
|
[10] Li, Z.-C., Lee, M.-G., Chiang, J. Y., Liu, Y. P.: The Trefftz method using fundamental solutions for biharmonic equations.J. Comput. Appl. Math. 235 (2011), 4350-4367. Zbl 1222.65131, MR 2802010, 10.1016/j.cam.2011.03.024 |
| Reference:
|
[11] Poullikkas, A., Karageorghis, A., Georgiou, G.: Methods of fundamental solutions for harmonic and biharmonic boundary value problems.Comput. Mech. 21 (1998), 416-423. Zbl 0913.65104, MR 1628005, 10.1007/s004660050320 |
| Reference:
|
[12] Sakakibara, K.: Analysis of the dipole simulation method for two-dimensional Dirichlet problems in Jordan regions with analytic boundaries.BIT 56 (2016), 1369-1400. Zbl 06667568, MR 3576615, 10.1007/s10543-016-0605-1 |
| . |
