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

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Ukrainian Mathematical Journal Aims and scope

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.

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References

  1. A. A. Samarskii, “Some problems of the theory of differential equations,” Different. Equat., 16, No. 11, 1925–1935 (1980).

    MathSciNet  Google Scholar 

  2. A.W. Leung and G.-S. Chen, “Optimal control of multigroup neutron fission systems,” Appl. Math. Optim., 40, No. 1, 39–60 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  5. W. Huyer, “Approximation of a linear age-dependent population model with spatial diffusion,” Comm. Appl. Anal., 8, No. 1, 87–108 (2004).

    MATH  MathSciNet  Google Scholar 

  6. E. Sinestrari and G. F. Webb, “Nonlinear hyperbolic systems with nonlocal boundary conditions,” J. Math. Anal. Appl., 121, No. 2, 449–464 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  15. Ph. Clément, H. J. A. M. Heijmans, S. Angenent, et al., One-Parameter Semigroups, North-Holland Publ. Co., Amsterdam (1987).

    MATH  Google Scholar 

  16. S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  17. L. N. Trefethen, “Approximation theory and approximation practice,” SIAM, Philadelphia, PA (2013).

    MATH  Google Scholar 

  18. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York etc. (1993).

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    V. B. Vasylyk & V. L. Makarov

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  1. V. B. Vasylyk
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  2. V. L. Makarov
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 8, pp. 1029–1040, August, 2014.

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Vasylyk, V.B., Makarov, V.L. Exponentially Convergent Method for the First-Order Differential Equation in a Banach Space with Integral Nonlocal Condition. Ukr Math J 66, 1152–1164 (2015). https://doi.org/10.1007/s11253-015-1000-9

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  • DOI: https://doi.org/10.1007/s11253-015-1000-9

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