Abstract and Applied Analysis

Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator

Abstract

The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 120192, 16 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174059

Digital Object Identifier
doi:10.1155/2012/120192

Mathematical Reviews number (MathSciNet)
MR2955018

Zentralblatt MATH identifier
1250.35023

Citation

Kalimbetov, Burkhan T.; Temirbekov, Marat A.; Khabibullayev, Zhanibek O. Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 120192, 16 pages. doi:10.1155/2012/120192. https://projecteuclid.org/euclid.aaa/1365174059

References

• S. A. Lomov, Introduction to the General Theory of Singular Perturbations, vol. 112 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1992.
• L. A. Skinner, “Matched expansion solutions of the first-order turning point problem,” SIAM Journal on Mathematical Analysis, vol. 25, no. 5, pp. 1402–1411, 1994.
• A. B. Vasil'eva, “On contrast structures of step type for a system of singularly perturbed equations,” Computational Mathematics and Mathematical Physics, vol. 34, no. 10, pp. 1215–1223, 1994.
• A. B. Vasil'eva, “Contrast structures of step-like type for a second-order singularly perturbed quasilinear differential equation,” Computational Mathematics and Mathematical Physics, vol. 35, no. 4, pp. 411–419, 1995.
• A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, vol. 14, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1995.
• A. G. Eliseev and S. A. Lomov, “The theory of singular perturbations in the case of spectral singularities of a limit operator,” Matematicheskiĭ Sbornik, vol. 131, no. 4, pp. 544–557, 1986.
• A. Ashyralyev, “On uniform difference schemes of a high order of approximation for evolution equations with a small parameter,” Numerical Functional Analysis and Optimization, vol. 10, no. 5-6, pp. 593–606, 1989.
• G. I. Shishkin, J. J. H. Miller, and E. O'Riordan, Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, River Edge, NJ, USA, 1996.
• A. Ashyralyev and H. O. Fattorini, “On uniform difference schemes for second-order singular pertur-bation problems in Banach spaces,” SIAM Journal on Mathematical Analysis, vol. 23, no. 1, pp. 29–54, 1992.
• 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.
• A. Ashyralyev and Y. Sözen, “A note on the parabolic equation with an arbitrary parameter at the derivative,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2565–2572, 2011.
• A. Ashyralyev, “On uniform difference schemes of a higher order of approximation for elliptical equations with a small parameter,” Applicable Analysis, vol. 36, no. 3-4, pp. 211–220, 1990.
• A. Ashyralyev and H. O. Fattorini, “On difference schemes of the high order of accuracy for singular perturbation elliptic equations,” in Investigation of Theory and Approximation Methods for Differential Equations, pp. 80–83, Ashgabat, Turkmenistan, 1991.
• A. Xu and Z. Cen, “Asymptotic behaviors of intermediate points in the remainder of the Euler-Maclaurin formula,” Abstract and Applied Analysis, vol. 2010, Article ID 134392, 8 pages, 2010.
• M. De la Sen, “Asymptotic comparison of the solutions of linear time-delay systems with point and distributed lags with those of their limiting equations,” Abstract and Applied Analysis, vol. 2009, Article ID 216746, 37 pages, 2009.
• F. Wang and A. Yukun, “Positive solutions for singular complementary Lid-stone boundary value problems,” Abstract and Applied Analysis, vol. 2011, Article ID 714728, 13 pages, 2011.
• A. B. Vasile'va and L. V. Kalachev, “Singularly perturbed periodic parabolic equations with alternat-ing boundary layer type solutions,” Abstract and Applied Analysis, vol. 2010, Article ID 52856, 21 pages, 2006.
• H. Šamajová and E. Špániková, “On asymptotic behaviour of solutions to $n$-dimensional systems of neutral differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 791323, 19 pages, 2011.
• I. Gavrea and M. Ivan, “Asymptotic behaviour of the iterates of positive linear operators,” Abstract and Applied Analysis, vol. 2011, Article ID 670509, 11 pages, 2011.