Abstract and Applied Analysis

A note on the difference schemes for hyperbolic-elliptic equations

A. Ashyralyev, G. Judakova, and P. E. Sobolevskii

Full-text: Open access

Abstract

The nonlocal boundary value problem for hyperbolic-elliptic equation d 2 u ( t ) / d t 2 + A u ( t ) = f ( t ) , ( 0 t 1 ) , d 2 u ( t ) / d t 2 + A u ( t ) = g ( t ) , ( 1 t 0 ) , u ( 0 ) = ϕ , u ( 1 ) = u ( 1 ) in a Hilbert space H is considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.

Article information

Source
Abstr. Appl. Anal., Volume 2006 (2006), Article ID 14816, 13 pages.

Dates
First available in Project Euclid: 30 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1170202977

Digital Object Identifier
doi:10.1155/AAA/2006/14816

Mathematical Reviews number (MathSciNet)
MR2211671

Zentralblatt MATH identifier
1133.65058

Citation

Ashyralyev, A.; Judakova, G.; Sobolevskii, P. E. A note on the difference schemes for hyperbolic-elliptic equations. Abstr. Appl. Anal. 2006 (2006), Article ID 14816, 13 pages. doi:10.1155/AAA/2006/14816. https://projecteuclid.org/euclid.aaa/1170202977


Export citation

References

  • A. Ashyralyev, The stability of difference schemes for partial differential equations of mixed types, Proceedings of the Sec. Int. Symp. on Math. and Comp. Appl., Inal, Baku, 1999, pp. 314--316.
  • --------, On well-posedness of the nonlocal boundary value problems for elliptic equations, Numerical Functional Analysis and Optimization. An International Journal 24 (2003), no. 1-2, 1--15.
  • A. Ashyralyev and A. Hanalyev, Coercive stability of nonlocal boundary value problem for parabolic equations in spaces of smooth functions, Izvestiya Akademii Nauk Turkmenskoj SSR. Seriya Fiziko-Tekhnicheskikh, Khimicheskikh i Geologicheskikh Nauk (1996), no. 3, 3--10 (Turkmen).
  • --------, Coercive estimate in Holder norms for parabolic equations with a dependent operator, Modeling the Processes in Exploration of Gas Deposits and Applied Problems of Theoretical Gas Hydrodynamics, Ilim, Ashgabat, 1998, pp. 154--162.
  • A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, Coercive solvability of the nonlocal boundary value problem for parabolic differential equations, Abstract and Applied Analysis 6 (2001), no. 1, 53--61.
  • A. Ashyralyev and G. Judakova, A note on the nonlocal boundary value problem for hyperbolic-elliptic equations in a Hilbert space, Proceeding of the International Conference ``Modern Mathematical Problems'', Institute of Mathematics of ME and RK, Astana, 2002, pp. 66--70.
  • A. Ashyralyev and I. Karatay, On the second order of accuracy difference schemes of the nonlocal boundary value problem for parabolic equations, Functional Differential Equations 10 (2003), no. 1-2, 45--63.
  • A. Ashyralyev and B. Kendirli, Well-posedness of the nonlocal boundary value problems for elliptic equations, Functional Differential Equations 9 (2002), no. 1-2, 33--55.
  • A. Ashyralyev and I. Muradov, On hyperbolic-parabolic equations in a Hilbert space, Probl. of Math. and of Econ. Mod. of Turkm., TNEI, Ashgabat, 1995, pp. 26--28.
  • A. Ashyralyev and P. E. Sobolevskii, A note on the difference schemes for hyperbolic equations, Abstract and Applied Analysis 6 (2001), no. 2, 63--70.
  • A. Ashyralyev and H. Soltanov, On elliptic-parabolic equations in a Hilbert space, Proceeding of the IMM and CS of Turkmenistan, no. 4, Ilim, Ashgabat, 1995, pp. 101--104.
  • A. Ashyralyev and A. Yurtsever, On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 47 (2001), no. 5, 3585--3592.
  • A. Ashyralyev, A. Yurtsever, and G. Judakova, On hyperbolic-elliptic equations in a Hilbert space, Some Problems of Applied Mathematics, Fatih University, Istanbul, 2000, pp. 70--79.
  • D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylym, Ashgabat, 1995.
  • T. D. Džuraev, Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, ``Fan'', Tashkent, 1979.
  • Yu. S. Èĭdel'man, Well-posedness of the direct and the inverse problem for a differential equation in a Hilbert space, Akademī ya Nauk Ukraï ni. Dopovī dī Doklady. Matematika, Prirodoznavstvo, Tekhnī chnī Nauki (1993), no. 12, 17--21 (Russian).
  • Yu. S. Èĭdel'man and I. V. Tikhonov, On well-posed problems for a special kind of evolution equation, Matematicheskie Zametki 56 (1994), no. 2, 99--113 (Russian).
  • V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator-Differential Equations, ``Naukova Dumka'', Kiev, 1984.
  • S. G. Kreĭn, Linear Differential Equations in a Banach Space, Izdat. ``Nauka'', Moscow, 1967.
  • M. S. Salakhitdinov, Equations of Mixed-Composite Type, Izdat. ``Fan'' Uzbek. SSR, Tashkent, 1974.
  • A. Shyralyev and A. Yurtsever, Difference schemes for hyperbolic-parabolic equations, Functional Differential Equations 7 (2000), no. 3-4, 189--203.
  • A. L. Skubachevskii, Elliptic Functional-Differential Equations and Applications, Operator Theory: Advances and Applications, vol. 91, Birkhäuser Verlag, Basel, 1997.