## Abstract and Applied Analysis

### Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces

Okan Gercek

#### Abstract

A first order of accuracy difference scheme for theapproximate solution of abstract nonlocal boundary value problem $-{d}^{2}u(t)/d{t}^{2}+\text{sign}(t)Au(t)=g(t)$, $(0\le t\le 1)$, $du(t)/dt+\text{sign}(t)Au(t)=f(t)$, $(-1\le t\le 0)$, $u(0+)=u(0-),{u}^{\prime }(0+)={u}^{\prime }(0-),$ $\text{and\hspace\{0.17em\}}u(1)=u(-1)+\mu$ for differential equations in a Hilbert space $H$ with a self-adjoint positive definite operator A is considered. The well-posedness of this difference scheme in Hölder spaces without a weight is established. Moreover, as applications, coercivity estimates in Hölder normsfor the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 237657, 12 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174070

Digital Object Identifier
doi:10.1155/2012/237657

Mathematical Reviews number (MathSciNet)
MR2970010

Zentralblatt MATH identifier
1253.35083

#### Citation

Gercek, Okan. Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 237657, 12 pages. doi:10.1155/2012/237657. https://projecteuclid.org/euclid.aaa/1365174070

#### References

• M. S. Salakhitdinov, Equations of Mixed-Composite Type, Fan, Tashkent, Uzbekistan, 1974.
• T. D. Dzhuraev, Boundary Value Problems for Equations of Mixed and Mixed Composite Types, Fan, Tashkent, Uzbekistan, 1979.
• D. G. Gordeziani, On Methods of Resolution of a Class of Nonlocal Boundary Value Problems, Tbilisi University Press, Tbilisi, Georgia, 1981.
• V. N. Vragov, Boundary Value Problems for Nonclassical Equations of Mathematical Physics, Textbook for Universities, NGU, Novosibirsk, Russia, 1983.
• D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylim, Ashgabat, Turkmenistan, 1995.
• A. M. Nakhushev, Equations of Mathematical Biology, Vysshaya Shkola, Moskow, Russia, 1995.
• S. N. Glazatov, “Nonlocal boundary value problems for linear and nonlinear equations of variable type,” Sobolev Institute of Mathematics SB RAS, no. 46, p. 26, 1998.
• A. Ashyralyev and H. A. Yurtsever, “On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations,” Nonlinear Analysis-Theory, Methods and Applications, vol. 47, no. 5, pp. 3585–3592, 2001.
• D. Guidetti, B. Karasözen, and S. Piskarev, “Approximation of abstract differential equations,” Journal of Mathematical Sciences, vol. 122, no. 2, pp. 3013–3054, 2004.
• J. I. Díaz, M. B. Lerena, J. F. Padial, and J. M. Rakotoson, “An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator,” Journal of Differential Equations, vol. 198, no. 2, pp. 321–355, 2004.
• A. Ashyralyev, “A note on the nonlocal boundary value problem for elliptic-parabolic equations,” Nonlinear Studies, vol. 13, no. 4, pp. 327–333, 2006.
• A. S. Berdyshev and E. T. Karimov, “Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type,” Central European Journal of Mathematics, vol. 4, no. 2, pp. 183–193, 2006.
• A. Ashyralyev and Y. Ozdemir, “On stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1100–1118, 2009.
• V. B. Shakhmurov, “Regular degenerate separable differential operators and applications,” Potential Analysis, vol. 35, no. 3, pp. 201–222, 2011.
• J. Martín-Vaquero, A. Queiruga-Dios, and A. H. Encinas, “Numerical algorithms for diffusion-reaction problems with non-classical conditions,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5487–5495, 2012.
• A. Ashyralyev and H. Soltanov, “On elliptic-parabolic equations in a Hilbert space,” in Proceedings of the IMM of CS of Turkmenistan, pp. 101–104, Ashgabat, Turkmenistan, 1995.
• A. Ashyralyev and O. Gercek, “On second order of accuracy difference scheme of the approximate solution of nonlocal elliptic-parabolic problems,” Abstract and Applied Analysis, vol. 2010, Article ID 705172, 17 pages, 2010.
• A. Ashyralyev and O. Gercek, “Finite difference method for multipoint nonlocal elliptic-parabolic problems,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 2043–2052, 2010.
• A. Ashyralyev and O. Gercek, “Nonlocal boundary value problems for elliptic-parabolic differential and difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 904824, 16 pages, 2008.
• A. Ashyralyev, “On the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 49, pp. 1–16, 2011.
• P. E. Sobolevskii, “The theory of semigroups and the stability of difference schemes,” in Operator Theory in Function Spaces, pp. 304–337, Nauka, Novosibirsk, Russia, 1977.
• A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994.
• H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.
• A. Ashyralyev, Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations [Doctor of Sciences Thesis], Kiev, Ukraine, 1992.
• P. E. Sobolevskii, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971.
• A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004.
• P. E. Sobolevskii, On Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.