## Abstract and Applied Analysis

### Stability of Difference Schemes for Fractional Parabolic PDE with the Dirichlet-Neumann Conditions

Zafer Cakir

#### Abstract

The stable difference schemes for the fractional parabolic equation with Dirichlet and Neumannboundary conditions are presented. Stability estimates and almost coercive stability estimates with ln $(1/\mathrm{(\tau }+|h|))$ for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes of one-dimensional fractional parabolic partial differential equations.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 463746, 17 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174058

Digital Object Identifier
doi:10.1155/2012/463746

Mathematical Reviews number (MathSciNet)
MR2955022

Zentralblatt MATH identifier
1246.65200

#### Citation

Cakir, Zafer. Stability of Difference Schemes for Fractional Parabolic PDE with the Dirichlet-Neumann Conditions. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 463746, 17 pages. doi:10.1155/2012/463746. https://projecteuclid.org/euclid.aaa/1365174058

#### References

• I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
• S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
• A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 2006.
• J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976.
• V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal of Mathematics, vol. 18, no. 3, pp. 281–299, 2007.
• R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, pp. 223–276, Springer, Vienna, Austria, 1997.
• D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineeringin System Application 2, Lille, France, 1996.
• A. B. Basset, “On the descent of a sphere in a viscous liquid,” Quarterly Journal of Mathematics, vol. 42, pp. 369–381, 1910.
• F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378, pp. 291–348, Springer, Vienna, Austria, 1997.
• A. Ashyralyev, F. Dal, and Z. Pinar, “On the numerical solution of fractional hyperbolic partial differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 730465, 11 pages, 2009.
• A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009.
• A. Ashyralyev, F. Dal, and Z. P\inar, “A note on the fractional hyperbolic differential and difference equations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654–4664, 2011.
• F. Dal, “Application of variational iteration method to fractional hyperbolic partial differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 824385, 10 pages, 2009.
• E. Cuesta, C. Lubich, and C. Palencia, “Convolution quadrature time discretization of fractional diffusion-wave equations,” Mathematics of Computation, vol. 75, no. 254, pp. 673–696, 2006.
• P. E. Sobolevskii, “Some properties of the solutions of differential equations in fractional spaces,” Trudy Naucno-Issledovatel'skogi Instituta Matematiki VGU, vol. 14, pp. 68–74, 1975.
• G. Da Prato and P. Grisvard, “Sommes d'opérateurs linéaires et équations différentielles opérationnelles,” Journal de Mathématiques Pures et Appliquées, vol. 54, no. 3, pp. 305–387, 1975.
• A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition,” in Proceedings of the 2nd International Symposium on Computing in Science and Engineering (ISCSE '11), M. Gunes, Ed., pp. 529–530, 2011.
• A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011.
• A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009.
• A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. II, Birkhäuser, Basel, Switzerland, 1989.