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

Abstract

top
The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as t 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.

How to cite

top

Jangveladze, 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
  1. Amadori, A. L., Karlsen, K. H., Chioma, C. La, 10.1080/10451120410001696289, Stochastics Stochastics Rep. 76 (2004), 147-177. (2004) Zbl1049.60050MR2060349DOI10.1080/10451120410001696289
  2. 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
  3. Coleman, B. D., Gurtin, M. E., 10.1017/S0022112068002430, J. Fluid Mech. 33 (1968), 165-181. (1968) Zbl0207.25302DOI10.1017/S0022112068002430
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. Gripenberg, G., 10.1006/jdeq.1993.1035, J. Differ. Equations 102 (1993), 382-390. (1993) Zbl0780.45012MR1216735DOI10.1006/jdeq.1993.1035
  11. 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
  12. Gurtin, M. E., Pipkin, A. C., 10.1007/BF00281373, Arch. Ration. Mech. Anal. 31 (1968), 113-126. (1968) Zbl0164.12901MR1553521DOI10.1007/BF00281373
  13. (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
  14. (Dzhangveladze), T. A. Jangveladze, Kiguradze, Z. V., 10.1134/S0012266108040083, Differ. Equ. 44 (2008), 538-550. (2008) MR2432866DOI10.1134/S0012266108040083
  15. (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) MR2266515DOI10.1007/s11202-006-0095-5
  16. (Dzhangveladze), T. A. Jangveladze, Kiguradze, Z. V., Estimates of the stabilization rate as t of solutions of the nonlinear integro-differential diffusion system, Appl. Math. Inform. Mech. 8 (2003), 1-19. (2003) MR2072736
  17. (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
  18. (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
  19. Kačur, J., Application of Rothe's method to evolution integrodifferential equations, J. Reine Angew. Math. 388 (1988), 73-105. (1988) Zbl0638.65098MR0944184
  20. Landau, L. D., Lifshitz, E. M., Electrodynamics of Continuous Media, Pergamon Press Oxford-London-New York (1960). (1960) Zbl0122.45002MR0121049
  21. 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
  22. 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
  23. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod/Gauthier-Villars Paris (1969), French. (1969) Zbl0189.40603MR0259693
  24. 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
  25. 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
  26. MacCamy, R. C., 10.1090/qam/452184, Q. Appl. Math. 35 (1977), 1-19. (1977) Zbl0351.45018MR0452184DOI10.1090/qam/452184
  27. 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
  28. 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) 
  29. 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 ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.