Advances in Difference Equations

, 2006:043092 | Cite as

On discrete analogues of nonlinear implicit differential equations

  • Pham Ky Anh
  • Le Cong Loi
Open Access
Research Article


This paper deals with some classes of nonlinear implicit difference equations obtained via discretization of nonlinear differential-algebraic or partial differential-algebraic equations. The unique solvability of discretized problems is proved and the compatibility between index notions for nonlinear differential-algebraic equations and nonlinear implicit difference equations is studied.


Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Functional Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Anh PK, Du NH, Loi LC: Connections between implicit difference equations and differential-algebraic equations. Acta Mathematica Vietnamica 2004,29(1):23–39.MathSciNetMATHGoogle Scholar
  2. 2.
    Anh PK, Yen HTN: On the solvability of initial-value problems for nonlinear implicit difference equations. Advances in Difference Equations 2004,2004(3):195–200. 10.1155/S1687183904402015CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bakhvalov NS, Zhidkov NP, Kobel'kov GM: Numerical Methods. Nauka, Moscow; 1987:599.MATHGoogle Scholar
  4. 4.
    Berger MS: Nonlinearity and Functional Analysis. Lectures on Nonlinear Problems in Mathematical Analysis, Pure and Applied Mathematics. Academic Press, New York; Harcourt Brace Jovanovich, London; 1977:xix+417.Google Scholar
  5. 5.
    Griepentrog E, März R: Differential-Algebraic Equations and Their Numerical Treatment, Teubner Texts in Mathematics. Volume 88. BSB B. G. Teubner, Leipzig; 1986:220.MATHGoogle Scholar
  6. 6.
    Loi LC, Du NH, Anh PK: On linear implicit non-autonomous systems of difference equations. Journal of Difference Equations and Applications 2002,8(12):1085–1105. 10.1080/1023619021000053962MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lucht W, Debrabant K: On quasi-linear PDAEs with convection: applications, indices, numerical solution. Applied Numerical Mathematics 2002,42(1–3):297–314.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lucht W, Strehmel K, Eichler-Liebenow C: Indexes and special discretization methods for linear partial differential algebraic equations. BIT 1999,39(3):484–512. 10.1023/A:1022370703243MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    März R: On linear differential-algebraic equations and linearizations. Applied Numerical Mathematics 1995,18(1–3):267–292.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ortega JM, Rheinboldt WC: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York; 1970:xx+572.MATHGoogle Scholar

Copyright information

© P. K. Anh and L. C. Loi. 2006

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam

Personalised recommendations