Abstract and Applied Analysis

An Approximation of Semigroups Method for Stochastic Parabolic Equations

Allaberen Ashyralyev and Mehmet Emin San

Full-text: Open access


A single-step difference scheme for the numerical solution of the nonlocal-boundary value problem for stochastic parabolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In application, the convergence estimates for the solution of the difference scheme are obtained for two nonlocal-boundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

Article information

Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 684248, 24 pages.

First available in Project Euclid: 5 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ashyralyev, Allaberen; San, Mehmet Emin. An Approximation of Semigroups Method for Stochastic Parabolic Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 684248, 24 pages. doi:10.1155/2012/684248. https://projecteuclid.org/euclid.aaa/1365174067

Export citation


  • A. Ashyralyev and I. Hasgur, “Linear stochastic differential equations in a Hilbert space,” Abstract of Statistic Conference-95, Ankara, Turkey, 1–6, 1995.
  • R. F. Curtain and P. L. Falb, “Stochastic differential equations in Hilbert space,” Journal of Differential Equations, vol. 10, pp. 412–430, 1971.
  • G. Da Prato, “Regularity properties of a stochastic convolution integral,” Analisi Matematica, vol. 72, no. 4, pp. 217–219, 1982.
  • G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1992.
  • A. Ashyralyev and G. Michaletsky, “The approximation of solutions of stochastic differential equations in Hilbert space by the difference schemes, Trudy nauchno-prakticheskoy konferencii, `Differencialniye uravneniya i ih prilozheniya',” Ashgabat, vol. 1, pp. 85–95, 1993.
  • E. J. Allen, S. J. Novosel, and Z. Zhang, “Finite element and difference approximation of some linear stochastic partial differential equations,” Stochastics and Stochastics Reports, vol. 64, no. 1-2, pp. 117–142, 1998.
  • E. Hausenblas, “Numerical analysis of semilinear stochastic evolution equations in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 147, no. 2, pp. 485–516, 2002.
  • A. Yurtsever and A. Yazliyev, “High order accuracy difference scheme for stochastic parabolic equation in a Hilbert space,” in Some Problems of Applied Mathematics, pp. 212–220, Fatih University, Istanbul, Turkey, 2000.
  • T. Shardlow, “Numerical methods for stochastic parabolic PDEs,” Numerical Functional Analysis and Optimization, vol. 20, no. 1-2, pp. 121–145, 1999.
  • A. Ashyralyev, “On modified Crank-Nicholson difference schemes for stochastic parabolic equation,” Numerical Functional Analysis and Optimization, vol. 29, no. 3-4, pp. 268–282, 2008.
  • L. Han, L. G. Han, X. B. Gong, Shan Gang-Yi, and J. Cui, “Implicit finite-difference plane wave migration in TTI media,” Chinese Journal of Geophysics, vol. 54, no. 4, pp. 1090–1097, 2011.
  • A. Jentzen, “Higher order pathwise numerical approximations of SPDEs with additive noise,” SIAM Journal on Numerical Analysis, vol. 49, no. 2, pp. 642–667, 2011.
  • A. Jentzen and P. E. Kloeden, “The numerical approximation of stochastic partial differential equations,” Milan Journal of Mathematics, vol. 77, no. 1, pp. 205–244, 2009.
  • A. Ashyralyev and M. Akat, “An approximation of stochastic hyperbolic equations,” AIP Conference Proceedings, vol. 1389, pp. 625–628, 2011.
  • A. Ashyralyev and E. M. San, “Finite difference method for stochastic parabolic equations,” AIP Conference Proceedings, vol. 1389, pp. 589–592, 2011.
  • A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Berlin, Germany, 2004.
  • A. A. Samarskii and E. S. Nikolaev, Nikolaev, Numerical Methods for Grid Equations, Vol. 2: Iterative Methods, Birkhäuser, Basel, Switzerland, 1989.