## Abstract and Applied Analysis

### A note on the difference schemes for hyperbolic-elliptic equations

#### Abstract

The nonlocal boundary value problem for hyperbolic-elliptic equation ${d^{2}u(t)/dt^{2}} +Au(t) = f(t)$, $(0\leq t \leq 1)$, $-{d^{2}u(t)/dt^{2}}+Au(t)=g(t)$, $(-1\leq t \leq 0)$, $u(0)=\varphi$, $u(1)=u(-1)$ 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

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

#### 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.