Advertisement

Ukrainian Mathematical Journal

, Volume 66, Issue 8, pp 1152–1164 | Cite as

Exponentially Convergent Method for the First-Order Differential Equation in a Banach Space with Integral Nonlocal Condition

  • V. B. Vasylyk
  • V. L. Makarov
Article
  • 70 Downloads

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. The algorithm is based on the representations of operator functions via the Dunford–Cauchy integral along a hyperbola covering the spectrum of A and the quadrature formula containing a small number of resolvents. The efficiency of the proposed algorithm is illustrated by several examples.

Keywords

Banach Space Quadrature Formula Gauss Quadrature Quadrature Point Unbounded Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. A. Samarskii, “Some problems of the theory of differential equations,” Different. Equat., 16, No. 11, 1925–1935 (1980).MathSciNetGoogle Scholar
  2. 2.
    A.W. Leung and G.-S. Chen, “Optimal control of multigroup neutron fission systems,” Appl. Math. Optim., 40, No. 1, 39–60 (1999).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    A.W. Leung and L. A. Ortega, “Existence and monotone scheme for time-periodic nonquasimonotone reaction-diffusion systems: Application to autocatalytic chemistry,” J. Math. Anal. Appl., 221, No. 2, 712–733 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Gordeziani and G. Avalishvili, “Investigation of the nonlocal initial boundary value problems for some hyperbolic equations,” Hiroshima Math. J., 31, No. 3, 345–366 (2001).zbMATHMathSciNetGoogle Scholar
  5. 5.
    W. Huyer, “Approximation of a linear age-dependent population model with spatial diffusion,” Comm. Appl. Anal., 8, No. 1, 87–108 (2004).zbMATHMathSciNetGoogle Scholar
  6. 6.
    E. Sinestrari and G. F. Webb, “Nonlinear hyperbolic systems with nonlocal boundary conditions,” J. Math. Anal. Appl., 121, No. 2, 449–464 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    I. P. Gavrilyuk and V. L. Makarov, “Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces,” SIAM J. Numer. Anal., 43, No. 5, 2144–2171 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    I. P. Gavrilyuk, V. L. Makarov, and V. B. Vasylyk, “Exponentially convergent algorithms for abstract differential equations,” Front. Math., Birkhäuser, Basel (2011), viii+180 p.Google Scholar
  9. 9.
    M. López-Fernández, C. Palencia, and A. Schädle, “A spectral order method for inverting sectorial Laplace transforms,” SIAM J. Numer. Anal., 44, 1332–1350 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    M. López-Fernández, C. Lubich, C. Palencia, and A. Schädle, “Fast Runge–Kutta approximation of inhomogeneous parabolic equations,” Numer. Math., 102, No. 2, 277–291 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    D. Sheen, I. H. Sloan, and V. Thomée, “A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature,” IMA J. Numer. Anal., 23, No. 2, 269–299 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    V. Thomée, “A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature,” Int. J. Numer. Anal. Model., 2, 121–139 (2005).Google Scholar
  13. 13.
    J. A. C. Weideman, “Optimizing Talbot’s contours for the inversion of the Laplace transform,” SIAM J. Numer. Anal., 44, No. 6, 2342–2362 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    I. P. Gavrilyuk, V. L. Makarov, D. O. Sytnyk, and V. B. Vasylyk, “Exponentially convergent method for the m-point nonlocal problem for a first order differential equation in Banach space,” Numer. Funct. Anal. Optim., 31, No. 1-3, 1–21 (2010).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ph. Clément, H. J. A. M. Heijmans, S. Angenent, et al., One-Parameter Semigroups, North-Holland Publ. Co., Amsterdam (1987).zbMATHGoogle Scholar
  16. 16.
    S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).Google Scholar
  17. 17.
    L. N. Trefethen, “Approximation theory and approximation practice,” SIAM, Philadelphia, PA (2013).zbMATHGoogle Scholar
  18. 18.
    F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York etc. (1993).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • V. B. Vasylyk
    • 1
  • V. L. Makarov
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

Personalised recommendations