Abstract and Applied Analysis

Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

Deniz Agirseven

Full-text: Open access

Abstract

Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 682752, 31 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174043

Digital Object Identifier
doi:10.1155/2012/682752

Mathematical Reviews number (MathSciNet)
MR2926901

Zentralblatt MATH identifier
1246.65145

Citation

Agirseven, Deniz. Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 682752, 31 pages. doi:10.1155/2012/682752. https://projecteuclid.org/euclid.aaa/1365174043


Export citation

References

  • A. N. Al-Mutib, “Stability properties of numerical methods for solving delay differential equations,” Journal of Computational and Applied Mathematics, vol. 10, no. 1, pp. 71–79, 1984.
  • A. Bellen, “One-step collocation for delay differential equations,” Journal of Computational and Applied Mathematics, vol. 10, no. 3, pp. 275–283, 1984.
  • A. Bellen, Z. Jackiewicz, and M. Zennaro, “Stability analysis of one-step methods for neutral delay-differential equations,” Numerische Mathematik, vol. 52, no. 6, pp. 605–619, 1988.
  • K. L. Cooke and I. Győri, “Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments,” Computers & Mathematics with Applications, vol. 28, no. 1–3, pp. 81–92, 1994, Advances in difference equations.
  • L. Torelli, “Stability of numerical methods for delay differential equations,” Journal of Computational and Applied Mathematics, vol. 25, no. 1, pp. 15–26, 1989.
  • A. F. Yeniçerioğlu and S. Yalçinbaş, “On the stability of the second-order delay differential equations with variable coefficients,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 667–673, 2004.
  • A. F. Yeniçerioğlu, “Stability properties of second order delay integro-differential equations,” Com-puters & Mathematics with Applications, vol. 56, no. 12, pp. 3109–3117, 2008.
  • A. F. Yeniçerioğlu, “The behavior of solutions of second order delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1278–1290, 2007.
  • A. Ashyralyev and H. Akca, “On difference shemes for semilinear delay differential equations with constant delay,” in Proceedings of the Conference TSU: Actual Problems of Applied Mathematics, Physics and Engineering, pp. 18–27, Ashgabat, Turkmenistan, 1999.
  • A. Ashyralyev, H. Akça, and U. Guray, “Second order accuracy difference scheme for approximate solutions of delay differential equations,” Functional Differential Equations, vol. 6, no. 3-4, pp. 223–231, 1999.
  • A. Ashyralyev and H. Akça, “Stability estimates of difference schemes for neutral delay differential equations,” Nonlinear Analysis, vol. 44, no. 4, pp. 443–452, 2001.
  • A. Ashyralyev, H. Akça, and F. Yeniçerioğlu, “Stability properties of difference schemes for neutral differential equations,” in Differential Equations and Applications, vol. 3, pp. 27–34, Nova Science, Hauppauge, NY, USA, 2004.
  • J. Liu, P. Dong, and G. Shang, “Sufficient conditions for inverse anticipating synchronization of uni-directional coupled chaotic systems with multiple time delays,” in Proceedings of the Chinese Control and Decision Conference (CCDC '10), pp. 751–756, IEEE, 2010.
  • S. Mohamad, H. Akça, and V. Covachev, “Discrete-time Cohen-Grossberg neural networks with trans-mission delays and impulses,” Tatra Mountains Mathematical Publications, vol. 43, pp. 145–161, 2009.
  • A. Ashyralyev and P. E. Sobolevskii, “On the stability of the linear delay differential and difference equations,” Abstract and Applied Analysis, vol. 6, no. 5, pp. 267–297, 2001.
  • 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.
  • G. Di Blasio, “Delay differential equations with unbounded operators acting on delay terms,” Nonlinear Analysis, vol. 52, no. 1, pp. 1–18, 2003.
  • M. A. Bazarov, “On the structure of fractional spaces,” in Proceedings of the 27th All-Union Scientific Student Conference “The Student and Scientific-Technological Progress” (Novosibirsk, 1989), pp. 3–7, Novo-sibirsk University, Novosibirsk, Russia, 1989.
  • A. O. Ashyralyev and P. E. Sobolevskiĭ, “The theory of interpolation of linear operators and the stability of difference schemes,” Doklady Akademii Nauk SSSR, vol. 275, no. 6, pp. 1289–1291, 1984.
  • U. Erdogan and T. Ozis, “A smart nonstandard finite difference scheme for second order nonlinear boundary value problems,” Journal of Computational Physics, vol. 230, no. 17, pp. 6464–6474, 2011.
  • A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994.
  • A. Ashyralyev, “High-accuracy stable difference schemes for well-posed NBVP,” Operator Theory, vol. 191, no. 2, pp. 229–252, 2009.
  • H. A. Alibekov and P. E. Sobolevskiĭ, “The stability of difference schemes for parabolic equations,” Doklady Akademii Nauk SSSR, vol. 232, no. 4, pp. 737–740, 1977 (Russian).
  • H. A. Alibekov and P. E. Sobolevskiĭ, “Stability and convergence of high-order difference schemes of approximation for parabolic equations,” Akademiya Nauk Ukrainskoĭ SSR, vol. 31, no. 6, pp. 627–634, 1979.
  • H. A. Alibekov and P. E. Sobolevskiĭ, “Stability and convergence of difference schemes of a high order of approximation for parabolic partial differential equations,” Akademiya Nauk Ukrainskoĭ SSR, vol. 32, no. 3, pp. 291–300, 1980.
  • A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. II, Birkhäuser, Basel, Switzerland, 1989.
  • H. Akça, V. B. Shakhmurov, and G. Arslan, “Differential-operator equations with bounded delay,” Nonlinear Times and Digest, vol. 2, no. 2, pp. 179–190, 1995.
  • H. Akca and V. Covachev, “Spatial discretization of an impulsive Cohen-Grossberg neural network with time-varying and distributed delays and reaction-diffusion terms,” Analele Stiintifice Ale Univer-sitatii Ovidius Constanta-Seria Matematica, vol. 17, no. 3, pp. 15–26, 2009.
  • S. J. Liao, “Homotopy analysis method: a new analytical technique for nonlinear problems,” Com-munications in Nonlinear Science and Numerical Simulation, vol. 2, no. 2, pp. 95–100, 1997.
  • S. J. Liao, “On the general boundary element method,” Engineering Analysis with Boundary Elements, vol. 21, no. 1, pp. 39–51, 1998.
  • S. Liao, “General boundary element method: an application of homotopy analysis method,” Communications in Nonlinear Science & Numerical Simulation, vol. 3, no. 3, pp. 159–163, 1998.
  • S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Compu-tation, vol. 147, no. 2, pp. 499–513, 2004.
  • S. J. Liao, “Boundary element method for general nonlinear differential operators,” Engineering Ana-lysis with Boundary Elements, vol. 20, no. 2, pp. 91–99, 1997.
  • S. J. Liao, “On the general Taylor theorem and its applications in solving non-linear problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 3, pp. 135–140, 1997.
  • S. Liao, Beyond Perturbation, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/ CRC, Boca Raton, Fla, 2004.
  • H. Khan, S. J. Liao, R. N. Mohapatra, and K. Vajravelu, “An analytical solution for a nonlinear timedelay model in biology,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 7, pp. 3141–3148, 2009.
  • Y. Y. Wu, S. J. Liao, and X. Y. Zhao, “Some notes on the general boundary element method for highly nonlinear problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 7, pp. 725–735, 2005.
  • A. Ashyralyev and D. Agirseven, “Finite difference method for delay parabolic equations,” in Pro-ceedings of the International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis And Applied Mathematics (ICNAAM '11), vol. 1389 of AIP Conference Proceedings, pp. 573–576, 2011.