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Modified Crank-Nicholson Difference Schemes for Ultra Parabolic Equations with Neumann Condition

  • Allaberen Ashyralyev
  • Serhat Yilmaz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

In this paper, our interest is studying the stability of difference schemes for the approximate solution of the initial boundary value problem for ultra-parabolic equations. For approximately solving the given problem, the second-order of accuracy modified Crank-Nicholson difference schemes are presented. Theorem on almost coercive stability of these difference schemes is established. Numerical example is given to illustrate the applicability and efficiency of our method.

Keywords

Ultra parabolic equations difference schemes stability estimates matlab implementation numerical solutions 

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References

  1. 1.
    Dyson, J., Sanches, E., Villella-Bressan, R., Weeb, G.F.: An age and spatially structured model of tumor invasion with haptotaxis. Discrete Continuous Dynam. Systems B. 8, 45–60 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kunisch, K., Schappacher, W., Weeb, G.F.: Nonlinear age-dependent population dynamics with random diffusion. Comput. Math. Appl. 11, 155–173 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kolmogorov, A.N.: Zur Theorie der stetigen zufälligen prozesse. Math. Ann. 108, 149–160 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kolmogorov, A.N.: Zufällige bewegungen. Ann. of Math. 35, 116–117 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Genčev, T.G.: Ultraparabolic equations. Dokl. Akad. Nauk SSSR 151, 265–268 (1963)MathSciNetGoogle Scholar
  6. 6.
    Deng, Q., Hallam, T.G.: An age structured population model in a spatially heterogeneousenvironment: Existence and uniqueness theory. Nonlinear Anal. 65, 379–394 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Di Blasio, G., Lamberti, L.: An initial boundary value problem for age-dependent population diffusion. SIAM J. Appl. Math. 35, 593–615 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Di Blasio, G.: Nonlinear age-dependent diffusion. UJ. Math. Biol. 8, 265–284 (1979)CrossRefzbMATHGoogle Scholar
  9. 9.
    Tersenov, S.A.: On boundary value problems for a class of ultraparabolic equations and their applications. Matem. Sbornik. 175, 529–544 (1987)Google Scholar
  10. 10.
    Ashyralyev, A., Yilmaz, S.: Second order of accuracy difference schemes for ultra parabolic equations. In: AIP Conference Proceedings, vol. 1389, pp. 601–604 (2011)Google Scholar
  11. 11.
    Ashyralyev, A., Yilmaz, S.: An Approximation of ultra-parabolic equations. Abstr. Appl. Anal, Article ID 840621, 14 pages (2012)Google Scholar
  12. 12.
    Ashyralyev, A., Yilmaz, S.: On the numerical solution of ultra-parabolic equations with the Neumann Condition. In: AIP Conference Proceedings, vol. 1470, pp. 240–243 (2012)Google Scholar
  13. 13.
    Ashyralyev, A., Yilmaz, S.: Modified Crank-Nicholson difference schemes for ultra-parabolic equations. Comput. Math. Appl. 64, 2756–2764 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ashyralyev, A., Sobolevskii, P.E.: Well-Posedness of Parabolic Difference Equations. Operator Theory Advances and Applications, vol. 69. Birkhäuser Verlag, Basel (1994)CrossRefGoogle Scholar
  15. 15.
    Alibekov, K.A., Sobolevskii, P.E.: Stability and convergence of difference schemes of a high order for parabolic partial differential equations. Ukrain. Math. Zh. 32, 291–300 (1980)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Samarskii, K.A., Nikolaev, E.S.: Numerical Methods for Grid Equations. Iterative Methods, vol. 2. Birkhäuser, Basel (1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Allaberen Ashyralyev
    • 1
    • 2
  • Serhat Yilmaz
    • 1
  1. 1.Department of MathematicsFatih UniversityIstanbulTurkey
  2. 2.Department of MathematicsITTUAshgabatTurkmenistan

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