The Riemann–Hilbert problem in a complicated domain for a model of magnetic reconnection in a plasma

S. I. Bezrodnykh   ; V. I. Vlasov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 42 (2002) no. 3, pp. 277-312

Bibliographie

[1] Gakhov F. D., Kraevye zadachi, Nauka, M., 1968 | MR | Zbl

[2] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR

[3] Shtaerman I. Ya., Kontaktnaya zadacha teorii uprugosti, Gostekhteorizdat, M., 1949

[4] Bitsadze A. V., Uravneniya smeshannogo tipa, Izd-vo AN SSSR, M., 1959

[5] Sedov L. I., Ploskie zadachi gidrodinamiki i aerodinamiki, Nauka, M., 1966 | MR

[6] Muskhelishvili N. I., Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti, Nauka, M., 1966

[7] Bitsadze A. V., Kraevye zadachi dlya ellipticheskikh uravnenii vtorogo poryadka, Nauka, M., 1966

[8] Vekua I. N., Obobschennye analiticheskie funktsii, Nauka, M., 1988 | MR | Zbl

[9] Wegmann R., “Keplerian discs arround magnetised neutron stars – a free boundary problem”, Meth. Verfahren Math. Phys., 37 (1991), 233–253 | MR | Zbl

[10] Wegert E., Nonlinear boundary value problems for holomorphic functions and singular equations, Acad. Verl., Berlin, 1992 | MR

[11] Efendiev M. A., Wendland W. L., “Nonlinear Riemann–Hilbert problem for muliply connected domains”, J. Nonlinear Analys. TMA, 27 (1996), 37–58 | DOI | MR | Zbl

[12] Wegert E., Qestreich D., “On a non-simmetric problem in electrochemical mashining”, Math. Meth. Appl. Sci., 20 (1997), 841–854 | 3.0.CO;2-L class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[13] Somov B. V., Titov V. S., Verneta A. I., “Magnitnoe peresoedinenie v solnechnykh vspyshkakh”, Itogi nauki i tekhn. Astronomiya, 34, VINITI AN SSSR, M., 1987, 136–237

[14] Russell C. T., “The configuration of the magnetosphere”, Crit. Probl. Magnetosph. Phys., Proc. Symp. COSPAR, IAGA, and URSI (Madrid, 1972), Washington D.C., 1972, 1–15

[15] Kadomtsev B. B., “O neustoichivosti sryvov v tokamakakh”, Fiz. plazmy, 1:5 (1975), 710–715

[16] Markovskii S. A., Somov B. V., “Model magnitnogo peresoedineniya v tokovom sloe s udarnymi volnami”, Fiz. solnechnoi plazmy, 1989, 456–472, Nauka, M.

[17] Syrovatskii S. I., “O vozniknovenii tokovykh sloev v plazme s vmorozhennym silnym magnitnym polem”, Zh. eksperim. i teor. fiz., 60:3 (1971), 1726–1741

[18] Roberts B., Priest E. R., “On the maximum rate of magnetic-field reconnection for Petschek's mechanism”, J. Plasma Phys., 14:3 (1975), 417–431 | DOI

[19] Vlasov V. I., Markovskii C. A., Somov B. V., Ob analiticheskoi modeli magnitnogo peresoedineniya v plazme, Dep. v VINITI, 6.01.1989, No 769-V89, 19 pp.

[20] Biskamp D., “Magnetic reconnection via current sheets”, Phys. Fluids, 29:5 (1986), 1520–1531 | DOI | Zbl

[21] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1973 | MR

[22] Koppenfels V., Shtalman F., Praktika konformnykh otobrazhenii, Izd-vo inostr. lit., M., 1963

[23] Markushevich A. I., Teoriya analiticheskikh funktsii, v. I, II, Nauka, M., 1968 | Zbl

[24] Kantorovich L. V., Krylov V. I., Priblizhennye metody vysshego analiza, Fizmatgiz, L., 1962 | MR

[25] Henrici P., Applied and computational complex analysis, v. 3, John Wiley and Sons, New York, 1991 | MR

[26] Vlasov V. I., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, VTs AN SSSR, M., 1987 | MR

[27] Vlasov V. I., Skorokhodov S.L., “Analiticheskoe reshenie zadachi Dirikhle dlya uravneniya Puassona v nekotorom klasse poligonalnykh oblastei”, Soobsch. po prikl. matem., VTs AN SSSR, M., 1988 | MR | Zbl

[28] Vlasov V. I., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, Dis. $\dots$ dokt. fiz.-matem. nauk, VTs AN SSSR, M., 1990 | MR

[29] Trefethen L. N., “Numerical computation of the Schwarz–Christoffel transformation”, SIAM J. Sci. Statist. Comput., 1 (1980), 82–102 | DOI | MR | Zbl

[30] Zemach C., “A conformal map formula for difficult cases”, J. Comput. Appl. Math., 14 (1986), 207–215 | DOI | MR | Zbl

[31] Krikeles B. C., Rubin R. L., “On the crowding of parameters associated with Schwarz–Christoffel transformation”, Appl. Math. Comput., 28:4 (1988), 297–308 | DOI | MR | Zbl

[32] Trefethen L. N., Driscoll T. A., “Schwarz–Christoffel mapping in the computer era”, Proc. Internat. Congress Math. (Berlin, 1998) | MR

[33] Menikoff R., Zemach C., “Methods for numerical conformal mapping”, J. Comput. Phys., 36:3 (1980), 366–410 | DOI | MR | Zbl

[34] Howell L. H., Trefethen L. N., “A modified Schwarz–Christoffel transformation for elongated regions”, SIAM J. Statist. Comput., 11 (1990), 928–949 | DOI | MR | Zbl

[35] Trefethen L. N., “Numerical construction of comformal maps”, App. to: Fundamentals of Complex Analys. for Math., Science, and Engng., Prentice Hall, 1993

[36] Driscoll T. A., “A MATLAB toolbox for Schwarz–Christoffel mapping”, ACM Transactions Math. Soft., 22 (1996), 168–186 | DOI | Zbl

[37] Skorokhodov S. L., Vlasov V. I., “An efficient analytic-numerical method to conformal mapping of circular arc polygons”, Comput. Meth. and Function Theory 2001, Internat. Conf. in Aveiro, Abstracts (August 24–27, 1998), 116–117

[38] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973

[39] Shalashilin V. I., Kuznetsov E. B., Metod prodolzheniya po parametru i nailuchshaya parametrizatsiya, URSS, M., 1999 | MR

[40] Watson L. T., “Globally convergent homotopy methods a tutorial”, Appl. Math. Comput., 31 (1989), 369–396 | DOI | MR | Zbl

[41] Caratheodory C., Gesammelte Mathematische Schriften, Dritter Band, München, 1955; Vierter Band, 1956