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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Kunisch, K., Schappacher, W., Weeb, G.F.: Nonlinear age-dependent population dynamics with random diffusion. Comput. Math. Appl. 11, 155–173 (1985)
Kolmogorov, A.N.: Zur Theorie der stetigen zufälligen prozesse. Math. Ann. 108, 149–160 (1933)
Kolmogorov, A.N.: Zufällige bewegungen. Ann. of Math. 35, 116–117 (1934)
Genčev, T.G.: Ultraparabolic equations. Dokl. Akad. Nauk SSSR 151, 265–268 (1963)
Deng, Q., Hallam, T.G.: An age structured population model in a spatially heterogeneousenvironment: Existence and uniqueness theory. Nonlinear Anal. 65, 379–394 (2006)
Di Blasio, G., Lamberti, L.: An initial boundary value problem for age-dependent population diffusion. SIAM J. Appl. Math. 35, 593–615 (1978)
Di Blasio, G.: Nonlinear age-dependent diffusion. UJ. Math. Biol. 8, 265–284 (1979)
Tersenov, S.A.: On boundary value problems for a class of ultraparabolic equations and their applications. Matem. Sbornik. 175, 529–544 (1987)
Ashyralyev, A., Yilmaz, S.: Second order of accuracy difference schemes for ultra parabolic equations. In: AIP Conference Proceedings, vol. 1389, pp. 601–604 (2011)
Ashyralyev, A., Yilmaz, S.: An Approximation of ultra-parabolic equations. Abstr. Appl. Anal, Article ID 840621, 14 pages (2012)
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)
Ashyralyev, A., Yilmaz, S.: Modified Crank-Nicholson difference schemes for ultra-parabolic equations. Comput. Math. Appl. 64, 2756–2764 (2012)
Ashyralyev, A., Sobolevskii, P.E.: Well-Posedness of Parabolic Difference Equations. Operator Theory Advances and Applications, vol. 69. Birkhäuser Verlag, Basel (1994)
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)
Samarskii, K.A., Nikolaev, E.S.: Numerical Methods for Grid Equations. Iterative Methods, vol. 2. Birkhäuser, Basel (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ashyralyev, A., Yilmaz, S. (2013). Modified Crank-Nicholson Difference Schemes for Ultra Parabolic Equations with Neumann Condition. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-41515-9_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41514-2
Online ISBN: 978-3-642-41515-9
eBook Packages: Computer ScienceComputer Science (R0)