Numerical Analysis and Applications

, Volume 2, Issue 1, pp 46–57 | Cite as

Implicit difference methods for Hamilton-Jacobi functional differential equations

  • Z. KamontEmail author
  • W. Czernous


Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.

Key words

initial boundary value problem functional differential equation implicit difference method Newton method 


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  1. 1.
    Baranowska, A., Numerical Methods for Nonlinear First-Order Partial Differential Equations with Deviated Variables, Numer. Meth. Partial Diff. Equat., 2006, no. 22, pp. 708–727.Google Scholar
  2. 2.
    Brandi, P., Kamont, Z., and Salvadori, A., Approximate Solutions of Mixed Problems for First Order Partial Differential Functional Equations, Atti Sem. Mat. Fis. Univ. Modena, 1991, no. 39, pp. 277–302.Google Scholar
  3. 3.
    Brandi, P., Kamont, Z., and Salvadori, A., Existence of Generalized Solutions of Hyperbolic Functional Differential Equations, Nonlin. Anal. TMA, 2002, no. 50, pp. 919–940.Google Scholar
  4. 4.
    Brandi, P. and Marcelli, C., On Mixed Problem for First Order Partial Differential Functional Equations, Atti Sem. Mat. Fis. Univ. Modena, 1998, no. 46, pp. 497–510.Google Scholar
  5. 5.
    Czernous, W., Generalized Euler Method for First Order Partial Differential Functional Equations, Mem. Diff. Equat. Math. Phys., 2006, no. 39, pp. 49–68.Google Scholar
  6. 6.
    Godlewski, E. and Raviart, P., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Berlin: Springer-Verlag, 1996zbMATHGoogle Scholar
  7. 7.
    Kamont, Z., Hyperbolic Functional Differential Inequalities and Applications, Dordrecht: Kluwer, 1999.zbMATHGoogle Scholar
  8. 8.
    Kantorovich, L.V. and Akilov, G.P., Functional Analysis, New York: Pergamon, 1982.zbMATHGoogle Scholar
  9. 9.
    Lakshmikantham, V. and Leela, S., Differential Inequalities, vol. 2, New York: Academic Press, 1969.zbMATHGoogle Scholar
  10. 10.
    Pao, C.V., Numerical Methods for Systems of Nonlinear Parabolic Equations with Time Delays, J. Math. Anal. Appl., 1999, no. 240, pp. 249–279.Google Scholar
  11. 11.
    Pao, C.V., Finite Difference Reaction-Diffusion Systems with Coupled Boundary Conditions and Time Delays, J. Math. Anal. Appl., 2002, no. 272, pp. 407–434.Google Scholar
  12. 12.
    Przadka, K., Difference Methods for Non-Linear Partial Differential Functional Equations of the First Order, Math. Nachr., 1988, no. 138, pp. 105–123.Google Scholar
  13. 13.
    Szarski, J., Differential Inequalities, Warszawa: Polish Sci. Publ., 1966.Google Scholar
  14. 14.
    Thomas, J.W., Numerical Partial Differential Equations, New York: Springer, 1999.zbMATHGoogle Scholar
  15. 15.
    Voigt, W., On Finite-Difference Methods for Parabolic Functional Differential Equations on Unbounded Domains, in Numerical Methods and Applications, Sofia: Publ. House Bulg. Acad. Sci., 1989, pp. 559–567.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of GdańskGdańskPoland

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