A rapidly convergent modified Green's function for Laplace's equation in a rectangular region

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 455, Issue 1985, 1999, Pages 1739-1766

  Simon, L ^a   Marshal, L ^a  

^a OAK RIDGE NATIONAL LABORATORY   (United States)

Author keywords

Laplace's equation; Modified green's function; Rectangular region

Indexed keywords

EID: 0000938994     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.1999.0378     Document Type: Article

References (39)

1. Bortels, L., Deconinck, J. ; Van Den Bossche, B. 1996 The multidimensional upwinding method as a new simulation tool for the analysis of multi-ion electrolytes controlled by diffusion, convection and migration. I. Steady-state analysis of a parallel plane flow channel. J. Electroanalyt. Chem. 404, 15-26.
2. Breit, S. R. 1991 The potential of a Rankine source between parallel planes and in a rectangular cylinder. J. Engng Math. 25, 151-163.
3. Budak, B. M., Samarskii, A. A. ; Tikhonov, A. N. 1964 A collection of problems on mathematical physics (transi, by A. R. M. Robson), p. 455. Oxford: Pergamon.
4. Dubrovskii, V. V., Kravchenko, V. F. and Semin, N. V. 1995 The Green function of the Laplace operator with mixed boundary conditions on an isosceles triangle. Dokl. Akad. Nauk. 343, 314-316.
5. Dukovic, J. O. 1990 Computation of current distribution in electrodeposition: a review. IBM Jl Res. Devl. 34, 693-705.
6. Fahidy, T. Z. 1985 Principles of electrochemical reactor analysis. Amsterdam: Elsevier.
7. Gadylshin, R. R. 1996 On scattering by a cylinder with a narrow slit and walls of finite thickness. Theor. Mat. Phys. 106, 19-34.
8. Henrici, P. 1974 Applied and computational complex analysis. New York: Wiley-Interscience.
9. Ibl, N. 1983a Fundamentals of transport phenomena in electrolytic cells. In Comprehensive treatise on electrochemistry (ed. E. Yeager et al.), vol. 6, pp. 1-63. New York: Plenum.
10. Ibl, N. 19836 Current distribution. In Comprehensive treatise on electrochemistry (ed. E. Yeager et al.), vol. 6, pp. 239-315. New York: Plenum.
11. Ibl, N. ; Dossenbach, O. 1983 Convective mass transport. In Comprehensive treatise on electrochemistry (ed. E. Yeager et al.), vol. 6, pp. 133-237. New York: Plenum.
12. Kleinman, R. E. and Roach, G. F. 1982 On modified Green's functions in exterior problems for the Helmholtz equation. Proc. R. Soc. Land. A 383, 313-332.
13. Kleinman, R. E. ; Roach, G. F. 1983 Operators of minimal norm via modified Green's functions. Proc. R. Soc. Edinb. A 94, 163-178.
14. Martens, L. ; Hertwig, K. 1995 Modelling of current and potential distributions in louvred and extended metal electrodes. Electrochim. Acta 40, 387-391.
15. Mitrinovic, D. S. and Keckic, J. D. 1985 Cauchy's method of residues and applications. Dordrecht: Kluwer.
16. Morse, P. M. ; Feshbach, I. 1953 Methods of theoretical physics, pp. 1243-1252. New York: McGraw-Hill.
17. Newman, J. 1968 Engineering design of electrochemical systems. Ind. Engng Chem. 60, 13-27.
18. Newman, J. 1973 Electrochemical systems. Englewood Cliffs, NJ: Prentice-Hall.
19. Newman, J. N. 1992 The Green function for potential flow in a rectangular channel. J. Engng Math. 26, 51-59.
20. Parrish, W. R. ; Newman, J. 1969 Current distribution on a plane electrode below the limiting current. J. Electrochem. Soc. 116, 169-172.
21. Parrish, W. R. ; Newman, J. 1970 Current distribution on plane, parallel electrodes in channel flow. J. Electrochem. Soc. 117, 43-48.
22. Pickett, D. J. 1979 Electrochemical reactor design, 2nd edn. Amsterdam: Elsevier.
23. Podlipenko, Yu. K. 1993 Boundary-value problems for the Helmholtz equation in an angular domain. Ukrainian Math. Jl 45, 430-447.
24. Prentice, G. A. ; Tobias, C. W. 1982 A survey of numerical methods and solutions for current distribution problems. J. Electrochem. Soc. 129, 72-78.
25. Roach, G. F. 1970 Green's functions: introductory theory with applications. London: Van Nostrand Reinhold.
26. Smythe, W. R. 1951 Static and dynamic electricity, pp. 82-92. New York: McGraw-Hill.
27. Smythe, W. R. 1967 Static and dynamic electricity, 3rd edn. New York: McGraw-Hill.
28. Sneddon, I. N. 1966 Mixed boundary-value problems in potential theory. New York: Wiley.
29. Spiegel, M. R. 1974 Complex variables (Schaum Outline Series). New York: McGraw-Hill.
30. Stakgold, I. 1967a Boundary-value problems of mathematical physics, vol. 1. New York: Macmillan.
31. Stakgold, I. 19676 Boundary-value problems of mathematical physics, vol. 2. New York: Macmillan.
32. Steffen, B. ; Rousar, I. 1995 Numerical methods for electrochemical modelling. I. Electrochim. Acta 40, 379-386.
33. Stevenson, R. C. 1990 Green's function for the Helmholtz equation of a scalar wave in an isotropic halfspace. SIAM Jl Appl. Math. 50, 199-215.
34. Trefethen, L. N. 1984 Numerical computation of the Schwarz-Christoffel transformation. Z. Angew. Math. Phys. 35, 962-975.
35. Trefethen, L. N. (ed.) 1986 Numerical conformai mapping. New York: Elsevier.
36. Trefethen, L. N. ; Howell, L. H. 1989 A Modified Schwarz-Christoffel transformation for elongated regions. SIAM Jl Sci. Stat. Comp. 11, 928-935.
37. Wagner, C. 1951 Theoretical analysis of the current density distribution in electrolytic cells. J. Electrochem. Soc. 98, 116-128.
38. West, A. C. ; Newman, J. 1992 Determination of current distributions governed by Laplace's equation. In Modern aspects of electrochemistry (ed. B. E. Conway et al.), vol. 23, pp. 101-147.
39. Yang, F. ; Yao, R. 1996 The solution for mixed boundary-value problems of two-dimensional potential theory. Indian Jl Pure Appl. Math. 27, 313-322.