Abstract and Applied Analysis

An Approximation of Ultra-Parabolic Equations

Allaberen Ashyralyev and Serhat Yılmaz

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The first and second order of accuracy difference schemes for the approximate solution of the initial boundary value problem for ultra-parabolic equations are presented. Stability of these difference schemes is established. Theoretical results are supported by the result of numerical examples.

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Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 840621, 14 pages.

First available in Project Euclid: 5 April 2013

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Ashyralyev, Allaberen; Yılmaz, Serhat. An Approximation of Ultra-Parabolic Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 840621, 14 pages. doi:10.1155/2012/840621. https://projecteuclid.org/euclid.aaa/1365174047

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  • J. Dyson, E. Sanchez, R. Villella-Bressan, and G. F. Webb, “An age and spatially structured model of tumor invasion with haptotaxis,” Discrete and Continuous Dynamical Systems B, vol. 8, no. 1, pp. 45–60, 2007.
  • K. Kunisch, W. Schappacher, and G. F. Webb, “Nonlinear age-dependent population dynamics with random diffusion,” Computers & Mathematics with Applications, vol. 11, no. 1–3, pp. 155–173, 1985, Hyperbolic partial differential equations, II.
  • A. N. Kolmogorov, “Zur Theorie der stetigen zufälligen Prozesse,” Mathematische Annalen, vol. 108, pp. 149–160, 1933.
  • A. N. Kolmogorov, “Zufällige Bewegungen,” Annals of Mathematics, vol. 35, pp. 116–117, 1934.
  • T. G. Genčev, “On ultraparabolic equations,” Doklady Akademii Nauk SSSR, vol. 151, pp. 265–268, 1963.
  • Q. Deng and T. G. Hallam, “An age structured population model in a spatially heterogeneous environment: existence and uniqueness theory,” Nonlinear Analysis, vol. 65, no. 2, pp. 379–394, 2006.
  • G. di Blasio and L. Lamberti, “An initial-boundary value problem for age-dependent population diffusion,” SIAM Journal on Applied Mathematics, vol. 35, no. 3, pp. 593–615, 1978.
  • G. di Blasio, “Nonlinear age-dependent population diffusion,” Journal of Mathematical Biology, vol. 8, no. 3, pp. 265–284, 1979.
  • S. A. Tersenov, “Boundary value problems for a class of ultraparabolic equations and their applications,” Matematicheskiĭ Sbornik, vol. 133(175), no. 4, pp. 529–544, 1987.
  • G. Akrivis, M. Crouzeix, and V. Thomée, “Numerical methods for ultraparabolic equations,” Calcolo, vol. 31, no. 3-4, pp. 179–190, 1994.
  • A. Ashyralyev and S. Y\ilmaz, “On the approximate solution of ultra parabolic equations,” in Proceedings of the 2nd International Symposium on Computing in Science & Engineering, M. Güneş, A. K. Ç\inar, and I. Gürler, Eds., pp. 533–535, Gediz University, Izmir, Turkey, 2011.
  • A. Ashyralyev and S. Y\ilmaz, “Second order of accuracy difference schemes for ultra parabolic equations,” AIP Conference Proceedings, vol. 1389, pp. 601–604, 2011.
  • A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations Operator Theory Advance and Applications, vol. 69, Birkh äuser, Basel, Switzerland, 1994.
  • P. E. Sobolevskii, “On the Crank-Nicolson difference scheme for parabolic equations,” Nonlinear Oscillations and Control Theory, pp. 98–106, 1978.
  • A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. 2: Iterative Methods, Birkhäuser, Basel, Switzerland, 1989.