Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 975601, 13 pages
doi:10.1155/2009/975601
Research Article

A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations

A. L. Marhoune1 and F. Lakhal2

1Laboratory Equations Différentielles, Departement of Mathematics, University Mentouri Constantine, Constantine 25017, Algeria
2Department of Mathematics, Science University of 08 Mai 45, P.O. Box 401, Guelma 24000, Algeria

Received 27 January 2009; Revised 14 May 2009; Accepted 13 October 2009

Academic Editor: Sergiu Aizicovici

Copyright © 2009 A. L. Marhoune and F. Lakhal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. A. Samarskiĭ, “Some problems of the theory of differential equations,” Differentsial'nye Uravneniya, vol. 16, no. 11, pp. 1221–1228, 1980. View at MathSciNet
  2. A. L. Marhoune, “A three-point boundary value problem with an integral two-space-variables condition for parabolic equations,” Computers & Mathematics with Applications, vol. 53, no. 6, pp. 940–947, 2007. View at Zentralblatt MATH · View at MathSciNet
  3. R. E. Ewing and T. Lin, “A class of parameter estimation techniques for fluid flow in porous media,” Advances in Water Resources, vol. 14, no. 2, pp. 89–97, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. V. Kartynnik, “A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations,” Differential Equations, vol. 26, no. 9, pp. 1160–1166, 1990. View at MathSciNet
  6. G. W. Batten Jr., “Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations,” Mathematics of Computation, vol. 17, pp. 405–413, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Zentralblatt MATH · View at MathSciNet
  8. 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. View at MathSciNet
  9. J. R. Cannon, The One-Dimensional Heat Equation, vol. 23 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Mento Park, Calif, USA, 1984. View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. N. I. Ionkin, “The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition,” Differentsial'nye Uravneniya, vol. 13, no. 2, pp. 294–304, 1977. View at MathSciNet
  12. L. I. Kamynin, “A boundary-value problem in the theory of heat conduction with non-classical boundary conditions,” U.S.S.R. Computational Mathematics and Mathematical Physicsi, vol. 4, pp. 33–59, 1964. View at MathSciNet
  13. P. Shi and M. Shillor, “Design of contact patterns in one dimentional thermoelasticity,” in Theoretical Aspects of Industrial Design, SIAM, Philadelphia, Pa, USA, 1992.
  14. A. L. Marhoune and M. Bouzit, “High order differential equations with integral boundary condition,” Far East Journal of Mathematical Sciences, vol. 18, no. 3, pp. 341–350, 2005. View at Zentralblatt MATH · View at MathSciNet
  15. A. L. Marhoune and A. Hameida, “Mixed problem with an integral space variable condition for a third order parabolic equation of mixed type,” Far East Journal of Mathematical Sciences, vol. 29, no. 2, pp. 409–418, 2008. View at Zentralblatt MATH · View at MathSciNet
  16. N. I. Yurchuk, “Mixed problem with an integral condition for some parabolic equations,” Differential Equations, vol. 22, pp. 1457–1463, 1986.
  17. A. L. Marhoune and C. Latrous, “A strong solution of a high-order mixed type partial differential equation with integral conditions,” Applicable Analysis, vol. 87, no. 6, pp. 625–634, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. Fairweather and R. D. Saylor, “The reformulation and numerical solution of certain nonclassical initial-boundary value problems,” SIAM Journal on Scientific and Statistical Computing, vol. 12, no. 1, pp. 127–144, 1991. View at Zentralblatt MATH · View at MathSciNet
  19. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Press, Cambridge, UK, 1934.