全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

Mixed Problem with an Integral Two-Space-Variables Condition for a Class of Hyperbolic Equations

DOI: 10.1155/2013/957163

Full-Text   Cite this paper   Add to My Lib

Abstract:

This paper is devoted to the proof of the existence and uniqueness of the classical solution of mixed problems which combine Neumann condition and integral two-space-variables condition for a class of hyperbolic equations. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered. 1. Introduction The integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow, and population dynamics. Cannon was the first who drew attention to these problems with an integral one-space-variable condition [1], and their importance has been pointed out by Samarskii [2]. The existence and uniqueness of the classical solution of mixed problem combining a Dirichlet and integral condition for the equation of heat demonstrated by cannon [1] using the potential method. Always using the potential method, Kamynin established in [3] the existence and uniqueness of the classical solution of a similar problem with a more general representation. Subsequently, more works related to these problems with an integral one-space-variable have been published, among them, we cite the work of Benouar and Yurchuk [4], Cannon and Van Der Hoek [5, 6], Cannon-Esteva-Van Der Hoek [7], Ionkin [8], Jumarhon and McKee [9], Kartynnik [10], Lin [11], Shi [12] and Yurchuk [13]. In these works, mixed problems related to one-dimensional parabolic equations of second order combining a local condition and an integral condition was discussed. Also, by referring to the articles of Bouziani [14–16] and Bouziani and Benouar [17–19], the authors have studied mixed problems with integral conditions for some partial differential equations, specially hyperbolic equation with integral condition which has been investigated in Bouziani [20]. The present paper is devoted to the study of problems with a boundary integral two-space-variables condition for second-order hyperbolic equation. 2. Setting of the Problem In the rectangle , with , we consider the hyperbolic equation: where the coefficient is a real-valued function belonging to such that in the rest of the paper, , , , denote strictly positive constants. we adjoin to (1) the initial conditions the Neumann condition and the integral condition where and are known functions. We will assume that the function and satisfy a compatibility conditions with (5), that is, The presence of integral terms in boundary conditions can, in general, greatly complicate the application of standard functional or numerical

References

[1]  J. R. Cannon, “The solution of the heat equation subject to the specification of energy,” Quarterly of Applied Mathematics, vol. 21, pp. 155–160, 1963.
[2]  A. A. Samarskii, “Some problems of the theory of differential equations,” Differential'nye Uravneniya, vol. 16, no. 11, pp. 1925–1935, 1980.
[3]  N. I. Kamynin, “A boundary value problem in the theory of the heat conduction with non-classical boundary condition,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 4, no. 6, pp. 1006–1024, 1964.
[4]  N. E. Benouar and N. I. Yurchuk, “Mixed problem with an integral condition for parabolic equations with a bessel operator,” Differential'nye Uravneniya, vol. 27, no. 12, pp. 2094–2098, 1991.
[5]  J. R. Cannon and J. van der Hoek, “The existence of and a continuous dependence result for the solution of the heat equation subject to the specification of energy,” Bollettino dell'Unione Matematica Italiana, no. 1, pp. 253–282, 1981.
[6]  J. R. Cannon and J. van der Hoek, “An implicit finite difference scheme for the diffusion equation subject to the specification of mass in a portion of the domain,” in Numerical Solutions of Partial Differential Equations, J. Noye, Ed., pp. 527–539, North-Holland, Amsterdam, The Netherlands, 1982.
[7]  J. R. Cannon, S. Pérez Esteva, and J. van der Hoek, “A Galerkin procedure for the diffusion equation subject to the specification of mass,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 499–515, 1987.
[8]  N. I. Ionkin, “The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition,” Differential'nye Uravneniya, vol. 13, no. 2, pp. 294–304, 1977.
[9]  B. Jumarhon and S. McKee, “On the heat equation with nonlinear and nonlocal boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 190, no. 3, pp. 806–820, 1995.
[10]  A. V. Kartynnik, “Three points boundary value problern with an integral space variables conditions for second order parabolic equations,” Differential'nye Uravneniya, vol. 26, no. 9, pp. 1568–1575, 1990.
[11]  Y. Lin, Parabolic partial differential equation subject to nonlocal boundary conditions [Ph.D. thesis], Washington State University, Pullman, Wash, USA, 1988.
[12]  P. Shi, “Weak solution to an evolution problem with a nonlocal constraint,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 46–58, 1993.
[13]  N. I. Yurchuk, “A mixed problem with an integral condition for some parabolic equations,” Differential'nye Uravneniya, vol. 22, no. 12, pp. 2117–2126, 1986.
[14]  A. Bouziani, “Mixed problem for certain nonclassical equations with a small parameter,” Académie Royale de Belgique, vol. 5, no. 7–12, pp. 389–400, 1994.
[15]  A. Bouziani, “Mixed problem with boundary integral conditions for a certain parabolic equation,” Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 3, pp. 323–330, 1996.
[16]  A. Bouziani, “Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations,” Hiroshima Mathematical Journal, vol. 27, no. 3, pp. 373–390, 1997.
[17]  A. Bouziani and N. E. Benouar, “Problème mixte avec conditions intégrales pour une classe d'équations paraboliques,” Comptes Rendus de l'Académie des Sciences Série 1, vol. 321, no. 9, pp. 1177–1182, 1995.
[18]  A. Bouziani and N.-E. Benouar, “Mixed problem with integral conditions for a third order parabolic equation,” Kobe Journal of Mathematics, vol. 15, no. 1, pp. 47–58, 1998.
[19]  A. Bouziani and N.-E. Benouar, “Problème aux limites pour une classe d'équations de type non classique pour une structure pluri-dimensionnelle,” Polish Academy of Sciences, vol. 43, no. 4, pp. 317–328, 1995.
[20]  A. Bouziani, “Solution forte d'un problème mixte avec une condition non locale pour une classe d'équations hyperboliques,” Académie Royale de Belgique, vol. 8, no. 1–6, pp. 53–70, 1997.
[21]  L. G?rding, Cauchy's Problem for Hyperbolic Equations, University of Chicago Lecture Notes, 1957.

Full-Text

Contact Us

[email protected]

QQ:3279437679

WhatsApp +8615387084133