# Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

Temur A. Jangveladze; Zurab V. Kiguradze

Applications of Mathematics (2010)

- Volume: 55, Issue: 6, page 471-493
- ISSN: 0862-7940

## Access Full Article

top## Abstract

top## How to cite

topJangveladze, Temur A., and Kiguradze, Zurab V.. "Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance." Applications of Mathematics 55.6 (2010): 471-493. <http://eudml.org/doc/116486>.

@article{Jangveladze2010,

abstract = {The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as $t\rightarrow \infty $ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these two cases.},

author = {Jangveladze, Temur A., Kiguradze, Zurab V.},

journal = {Applications of Mathematics},

keywords = {system of nonlinear integro-differential equations; magnetic field; asymptotics for large time; system of nonlinear integro-differential equations; magnetic field; asymptotic},

language = {eng},

number = {6},

pages = {471-493},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance},

url = {http://eudml.org/doc/116486},

volume = {55},

year = {2010},

}

TY - JOUR

AU - Jangveladze, Temur A.

AU - Kiguradze, Zurab V.

TI - Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

JO - Applications of Mathematics

PY - 2010

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 55

IS - 6

SP - 471

EP - 493

AB - The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as $t\rightarrow \infty $ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these two cases.

LA - eng

KW - system of nonlinear integro-differential equations; magnetic field; asymptotics for large time; system of nonlinear integro-differential equations; magnetic field; asymptotic

UR - http://eudml.org/doc/116486

ER -

## References

top- Amadori, A. L., Karlsen, K. H., Chioma, C. La, 10.1080/10451120410001696289, Stochastics Stochastics Rep. 76 (2004), 147-177. (2004) Zbl1049.60050MR2060349DOI10.1080/10451120410001696289
- Chadam, J. M., Yin, H. M., 10.1216/JIE-1989-2-1-31, J. Integral Equations Appl. 2 (1990), 31-47. (1990) Zbl0701.45004MR1033202DOI10.1216/JIE-1989-2-1-31
- Coleman, B. D., Gurtin, M. E., 10.1017/S0022112068002430, J. Fluid Mech. 33 (1968), 165-181. (1968) Zbl0207.25302DOI10.1017/S0022112068002430
- Dafermos, C. M., 10.1016/0022-0396(70)90101-4, J. Differ. Equations 7 (1970), 554-569. (1970) MR0259670DOI10.1016/0022-0396(70)90101-4
- Dafermos, C., 10.1007/BFb0103245, Proc. Int. Conf. Equadiff 82, Würzburg 1982. Lect. Notes Math. Vol. 1017 (1983), 140-147. (1983) Zbl0547.35014MR0726578DOI10.1007/BFb0103245
- Dafermos, C. M., Nohel, J. A., A nonlinear hyperbolic Volterra equation in viscoelasticity. Contributions to analysis and geometry, Suppl. Am. J. Math. (1981), 87-116. (1981) MR0648457
- Engler, H., 10.1090/S0002-9947-96-01472-9, Trans. Am. Math. Soc. 348 (1996), 267-290. (1996) Zbl0848.45002MR1321573DOI10.1090/S0002-9947-96-01472-9
- Engler, H., 10.1007/BFb0103248, Proc. Int. Conf. Equadiff 82, Würzburg 1982. Lect. Notes Math. Vol. 1017 (1983), 161-167. (1983) Zbl0539.35074MR0726581DOI10.1007/BFb0103248
- Gordeziani, D. G., (Dzhangveladze), T. A. Jangveladze, Korshiya, T. K., Existence and uniqueness of the solution of certain nonlinear parabolic problems, Differ. Equations 19 (1983), 887-895. (1983) MR0708616
- Gripenberg, G., 10.1006/jdeq.1993.1035, J. Differ. Equations 102 (1993), 382-390. (1993) Zbl0780.45012MR1216735DOI10.1006/jdeq.1993.1035
- Gripenberg, G., Londen, S.-O., Staffans, O., Volterra Integral and Functional Equations. Encyclopedia of Mathematics and Its Applications, Vol. 34, Cambridge University Press Cambridge (1990). (1990) MR1050319
- Gurtin, M. E., Pipkin, A. C., 10.1007/BF00281373, Arch. Ration. Mech. Anal. 31 (1968), 113-126. (1968) Zbl0164.12901MR1553521DOI10.1007/BF00281373
- (Dzhangveladze), T. A. Jangvelazde, On the solvability of the first boundary value problem for a nonlinear integro-differential equation of parabolic type, Soobsch. Akad. Nauk Gruz. SSR 114 (1984), 261-264 Russian. (1984) MR0782476
- (Dzhangveladze), T. A. Jangveladze, Kiguradze, Z. V., 10.1134/S0012266108040083, Differ. Equ. 44 (2008), 538-550. (2008) MR2432866DOI10.1134/S0012266108040083
- (Dzhangveladze), T. A. Jangveladze, Kiguradze, Z. V., 10.1007/s11202-006-0095-5, Sib. Mat. Zh. 47 (2006), 1058-1070 Russian English translation: Sib. Math. J. 47 (2006), 867-878. (2006) DOI10.1007/s11202-006-0095-5
- (Dzhangveladze), T. A. Jangveladze, Kiguradze, Z. V., Estimates of the stabilization rate as $t\to \infty $ of solutions of the nonlinear integro-differential diffusion system, Appl. Math. Inform. Mech. 8 (2003), 1-19. (2003) MR2072736
- (Dzhangvelazde), T. A. Jangveladze, Kiguradze, Z. V., 10.1134/S0012266107060110, Differ. Equ. 43 (2007), 854-861 Translation from Differ. Uravn. 43 (2007), 833-840 Russian. (2007) MR2383832DOI10.1134/S0012266107060110
- (Dzhangveladze), T. A. Jangveladze, Lyubimov, B. Ya., Korshiya, T. K., Numerical solution of a class of non-isothermal diffusion problems of an electromagnetic field, Tr. Inst. Prikl. Mat. Im. I. N. Vekua 18 (1986), 5-47 Russian. (1986) MR0897501
- Kačur, J., Application of Rothe's method to evolution integrodifferential equations, J. Reine Angew. Math. 388 (1988), 73-105. (1988) Zbl0638.65098MR0944184
- Landau, L. D., Lifshitz, E. M., Electrodynamics of Continuous Media, Pergamon Press Oxford-London-New York (1960). (1960) Zbl0122.45002MR0121049
- Laptev, G., Mathematical singularities of a problem on the penetration of a magnetic field into a substance, Zh. Vychisl. Mat. Mat. Fiz. 28 (1988), 1332-1345 Russian English translation: U.S.S.R. Comput. Math. Math. Phys. 28 (1990), 35-45. (1990) MR0967528
- Laptev, G., Quasilinear parabolic equations which contains in coefficients Volterra's operator, Math. Sbornik 136 (1988), 530-545 Russian English translation: Sbornik Math. 64 (1989), 527-542. (1989) MR0965891
- Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod/Gauthier-Villars Paris (1969), French. (1969) Zbl0189.40603MR0259693
- Long, N. T., Dinh, A. P. N., 10.1002/mma.1670160404, Math. Methods Appl. Sci. 16 (1993), 281-295. (1993) Zbl0797.35099MR1213185DOI10.1002/mma.1670160404
- Long, N. T., Dinh, A. P. N., 10.1016/0898-1221(95)00068-A, Comput. Math. Appl. 30 (1995), 63-78. (1995) Zbl0834.35070MR1336663DOI10.1016/0898-1221(95)00068-A
- MacCamy, R. C., An integro-differential equation with application in heat flow, Q. Appl. Math. 35 (1977), 1-19. (1977) Zbl0351.45018MR0452184
- Renardy, M., Hrusa, W. J., Nohel, J. A., Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 35, Longman Scientific & Technical/John Wiley & Sons Harlow/New York (1987). (1987) MR0919738
- Vishik, M., Über die Lösbarkeit von Randwertaufgaben für quasilineare parabolische Gleichungen höherer Ordnung (On solvability of the boundary value problems for higher order quasilinear parabolic equations), Mat. Sb. N. Ser. 59 (1962), 289-325 Russian. (1962)
- Yin, H. M., 10.1216/JIE-1988-1-2-249, J. Integral Equations Appl. 1 (1988), 249-263. (1988) Zbl0671.45004MR0978743DOI10.1216/JIE-1988-1-2-249

## NotesEmbed ?

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