Abstract
Large time behavior of the solution to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. Furthermore, the rate of convergence is given. Initial-boundary value problem with mixed boundary conditions is considered.
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Aptsiauri M., Jangveladze T., Kiguradze Z., Large time behavior of solutions and numerical approximation of nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance, J. Appl. Math. Inform. Mech., 2008, 13(2), 3–17
Aptsiauri M., Jangveladze T., Kiguradze Z., On asymptotic behavior of solution of one nonlinear one-dimensional integro-differential analogue of Maxwell’s system, Rep. Enlarged Sess. Semin. I. Vekua Appl. Math., 2009, 23, 3–6
Bai Y., Backward solutions to nonlinear integro-differential systems, Cent. Eur. J. Math., 2010, 8(4), 807–815
Bai Y., Zhang P., On a class of Volterra nonlinear equations of parabolic type, Appl. Math. Comput., 2010, 216(1), 236–240
Dzhangveladze T.A., Investigation of the First Boundary Value Problem for Some Nonlinear Integro-Differential Equations of Parabolic Type, Tbilisi State University, Tbilisi, 1983 (in Russian)
Dzhangveladze T.A., The first boundary value problem for a nonlinear equation of parabolic type, Dokl. Akad. Nauk SSSR, 1983, 269(4), 839–842 (in Russian)
Dzhangveladze T.A., A nonlinear integro-differential equation of parabolic type, Differ. Uravn., 1985, 21(1), 41–46 (in Russian)
Dzhangveladze T.A., Kiguradze Z.V., On the stabilization of solutions of an initial-boundary value problem for a nonlinear integro-differential equation, Differ. Uravn., 2007, 43(6), 833–840 (in Russian)
Dzhangveladze T.A., Lyubimov B.Ya., Korshiya T.K., On the numerical solution of a class of nonisothermic problems of the diffusion of an electromagnetic field, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy, 1986, 18, 5–47 (in Russian)
Gordeziani D.G., Dzhangveladze T.A., Korshiya T.K., Existence and uniqueness of the solution of a class of nonlinear parabolic problems, Differ. Uravn., 1983, 19(7), 1197–1207 (in Russian)
Gripenberg G., Londen S.-O., Staffans O., Volterra Integral and Functional Equations, Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990
Jangveladze T.A., On one class of nonlinear integro-differential parabolic equations, Semin. I. Vekua Inst. Appl. Math. Rep., 1997, 23, 51–87
Jangveladze T., Kiguradze Z., The asymptotic behavior of the solutions of one nonlinear integro-differential parabolic equation, Rep. Enlarged Sess. Semin. I. Vekua Appl. Math., 1995, 10(1), 36–38
Jangveladze T.A., Kiguradze Z.V., Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance, Sibirsk. Mat. Zh., 2006, 47(5), 1058–1070 (in Russian)
Jangveladze T., Kiguradze Z., Asymptotics of solution and finite difference scheme to a nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance, WSEAS Trans. Math., 2009, 8(8), 467–477
Jangveladze T., Kiguradze Z., Neta B., Finite difference approximation of a nonlinear integro-differential system, Appl. Math. Comput., 2009, 215(2), 615–628
Jangveladze T., Kiguradze Z., Neta B., Large time asymptotic and numerical solution of a nonlinear diffusion model with memory, Comput. Math. Appl., 2010, 59(1), 254–273
Kiguradze Z., Finite difference scheme for a nonlinear integro-differential system, Proc. I. Vekua Inst. Appl. Math., 2000–2001, 50–51, 65–72
Kiguradze Z., The asymptotic behavior of the solutions of one nonlinear integro-differential model, Semin. I. Vekua Inst. Appl. Math. Rep., 2004, 30, 21–32
Ladyzhenskaya O.A., New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Trudy Mat. Inst. Steklov., 1967, 102, 85–104 (in Russian)
Lakshmikantham V., Rana Mohana Rao M., Theory of Integro-Differential Equations, Stability Control Theory Methods Appl., 1, Gordon and Breach, Lausanne, 1995
Landau L.D., Lifshitz E.M., Electrodynamics of Continuous Media, Course of Theoretical Physics, 8, Pergamon Press, Oxford-London-New York-Paris, 1960
Laptev G.I., Quasilinear parabolic equations that have a Volterra operator in the coefficients, Mat. Sb., 1988, 136(178)(4), 530–545 (in Russian)
Laptev G.I., Mathematical singularities of a problem on the penetration of a magnetic field into a substance, Zh. Vychisl. Mat. i Mat. Fiz., 1988, 28(9), 1332–1345 (in Russian)
Lin Y., Yin H.-M., Nonlinear parabolic equations with nonlinear functionals. J. Math. Anal. Appl., 1992, 168(1), 28–41
Lions J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969
Long N.T., Dinh A.P.N., Nonlinear parabolic problem associated with the penetration of a magnetic field into a substance, Math. Methods Appl. Sci., 1993, 16(4), 281–295
Vishik M., Solubility of boundary-value problems for quasi-linear parabolic equations of higher orders, Mat. Sb., 1962, 59(101) suppl., 289–325 (in Russian)
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Jangveladze, T., Kiguradze, Z. Large time behavior of the solution to an initial-boundary value problem with mixed boundary conditions for a nonlinear integro-differential equation. centr.eur.j.math. 9, 866–873 (2011). https://doi.org/10.2478/s11533-011-0036-9
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DOI: https://doi.org/10.2478/s11533-011-0036-9