# Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

Open Mathematics (2010)

- Volume: 8, Issue: 1, page 53-72
- ISSN: 2391-5455

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topYuri Melnikov. "Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients." Open Mathematics 8.1 (2010): 53-72. <http://eudml.org/doc/268950>.

@article{YuriMelnikov2010,

abstract = {Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.},

author = {Yuri Melnikov},

journal = {Open Mathematics},

keywords = {Green’s function; Elliptic equations; Piecewise-constant coefficients; Green's function; elliptic equations; piecewise-constant coefficients},

language = {eng},

number = {1},

pages = {53-72},

title = {Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients},

url = {http://eudml.org/doc/268950},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Yuri Melnikov

TI - Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

JO - Open Mathematics

PY - 2010

VL - 8

IS - 1

SP - 53

EP - 72

AB - Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.

LA - eng

KW - Green’s function; Elliptic equations; Piecewise-constant coefficients; Green's function; elliptic equations; piecewise-constant coefficients

UR - http://eudml.org/doc/268950

ER -

## References

top- [1] Ang W.T., Clements D.L., A boundary-integral equation method for the solution of a class of crack problems, J. Elasticity, 1987, 17, 9–21 http://dx.doi.org/10.1007/BF00042444 Zbl0602.73096
- [2] Clements D.L., Haselgrove M.D., A boundary-integral equation method for a class of crack problems in anisotropic elasticity, Int. J. Comput. Math., 1983, 12, 267-278 Zbl0501.73099
- [3] Courant R., Hilbert D., Methods of Mathematical Physics, vol.2, Interscience, New York, 1953 Zbl0051.28802
- [4] Deutz J.W., Schober H.R., Boundary value problems using Green’s functions, Comput. Phys. Commun., 1983, 30, 87–91 http://dx.doi.org/10.1016/0010-4655(83)90125-X
- [5] Dolgova I.M., Melnikov Yu.A., Construction of Green’s functions and matrices for equations and systems of elliptic type, Translation Russian PMM (J. Appl. Math. Mech.), 1978, 42, 740–746 http://dx.doi.org/10.1016/0021-8928(78)90017-5
- [6] Duffy D., Green’s Functions with Applications, CRC Press, Boca Raton, 2001 Zbl0983.35003
- [7] Embree M., Trefethen L.N., Green’s functions for multiply connected domains via conformal mapping, SIAM Rev., 1999, 41, 745–761 http://dx.doi.org/10.1137/S0036144598349277 Zbl0938.30003
- [8] Gradstein I.S., Ryzhik I.M., Tables of Intergrals, Series and Products, Academic Press, New York, 1980
- [9] Irschik H., Ziegler F., Application of the Green’s function method to thin elastic polygonal plates, Acta Mech., 1981, 39, 155–169 http://dx.doi.org/10.1007/BF01170339 Zbl0459.73047
- [10] Marshall S.L., A rapidly convergent modified Green’s function for Laplace equation in a rectangular region, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1999, 155, 1739–1766 http://dx.doi.org/10.1098/rspa.1999.0378
- [11] Melnikov Yu.A., Some applications of the Green’s function method in mechanics, Internat. J. Solids Structures, 1977, 13, 1045–1058 http://dx.doi.org/10.1016/0020-7683(77)90075-0
- [12] Melnikov Yu.A., Koshnarjova V.A., Green’s matrices and 2-D elasto-potentials for external boundary value problems, Appl. Math. Model., 1994, 18, 161–167 http://dx.doi.org/10.1016/0307-904X(94)90259-3 Zbl0796.73006
- [13] Melnikov Yu.A., Shirley K.L., Matrices of Green’s type for the potential equation on a cylindrical surface joined to a hemisphere, Appl. Math. Comput., 1994, 65, 241–252 http://dx.doi.org/10.1016/0096-3003(94)90180-5 Zbl0815.65119
- [14] Melnikov Yu.A., Green’s Functions in Applied Mechanics, Comput. Mech. Publications, Boston - Southampton, 1995 Zbl0898.73001
- [15] Melnikov Yu.A., Influence functions for 2-D compound regions of complex configuration, Comput. Mech., 1996, 17, 297–305 http://dx.doi.org/10.1007/BF00368552 Zbl0845.35016
- [16] Melnikov Yu.A., Green’s function formalism extended to systems of mechanical differential equations posed on graphs, J. Eng. Math., 1998, 34, 369–386 http://dx.doi.org/10.1023/A:1004396614908 Zbl0928.34024
- [17] Melnikov Yu.A., Influence Functions and Matrices, Marcel Dekker, New York - Basel, 1999
- [18] Melnikov Yu.A., Sheremet V.D., Some new results on the bending of a circular plate subject to point forces, Math. Mech. Solids, 2001, 6, 29–46 http://dx.doi.org/10.1177/108128650100600102 Zbl1024.74034
- [19] Melnikov Yu.A., Matrices of Green’s type of steady-state heat conduction in multiply connected piecewise homogeneous regions, Eng. Anal. Bound. Elements, 2003, 27, 779–787 http://dx.doi.org/10.1016/S0955-7997(03)00044-4 Zbl1045.80001
- [20] Melnikov Yu.A., Influence functions of a point source for perforated compound plates with facial convection, J. Eng. Math., 2004, 49, 253–270 http://dx.doi.org/10.1023/B:ENGI.0000031187.96637.ea Zbl1155.80002
- [21] Morse P.M., Feshbach H., Methods of Theoretical Physics, vol.2, McGraw-Hill, New York - Toronto - London, 1953 Zbl0051.40603
- [22] Pan E., Han F., Green’s functions for transversely isotropic piezoelectric multilayered half-spaces, J. Eng. Math., 2004, 49, 271–288 http://dx.doi.org/10.1023/B:ENGI.0000031183.83519.19 Zbl1068.74541
- [23] Roach G.F., Green’s Functions, Cambridge University Press, New York, 1982
- [24] Sheremet V.D., Handbook of Green’s Functions and Matrices, WITPress, Southampton - Boston, 2002
- [25] Smirnov V.I., A Course of Higher Mathematics, Pergamon Press, Oxford - New York, 1964 Zbl0121.25904
- [26] Stakgold I., Green’s functions and Boundary Value Problems, John Wiley, New York, 1980
- [27] Tewary V.K., Wagoner R.H., Hirth J.P., Elastic Green’s functions for a composite solid with a planar interface, J. Mater. Res., 1989, 4, 113–123 http://dx.doi.org/10.1557/JMR.1989.0113
- [28] Tewary V.K., Elastic Green’s function for a bimaterial composite solid containing a free surface normal to the interface, J. Mater. Res., 1991, 6, 2592–2608 http://dx.doi.org/10.1557/JMR.1991.2592
- [29] Ting T.C.T., Green’s functions for a bimaterial consisting of two orthotropic quarter planes subjected to an antiplane force and a screw dislocation, Math. Mech. Solids, 2005, 10, 197–211 http://dx.doi.org/10.1177/1081286505036318 Zbl1074.74018
- [30] Yang B., Tewary V.K., Efficient Green’s function method of line and surface defects in multilayered elestic and piezoelastic materials, Comput. Model. Eng. Sci., 2006, 15, 165–178
- [31] Yang B., Wong S.-C., Qu S., A micromechanics analysis of nanoscale graphite platelet-reinforced epoxy using defect Green’s function, Comput. Model. Eng. Sci., 2008, 24, 81–94

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