zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the numerical solution of fractional hyperbolic partial differential equations. (English) Zbl 1184.65083
Summary: A stable difference scheme for the numerical solution of a mixed problem for a multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of a modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

65M06Finite difference methods (IVP of PDE)
35R11Fractional partial differential equations
65M12Stability and convergence of numerical methods (IVP of PDE)
35L99Hyperbolic equations and systems
Full Text: DOI EuDML
[1] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993. · Zbl 0924.44003 · doi:10.1080/10652469308819017
[3] J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240-268, 1976. · Zbl 0324.44002 · doi:10.1137/1018042
[4] V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal of Mathematics, vol. 18, no. 3, pp. 281-299, 2007. · Zbl 1119.26011 · doi:10.1142/S0129167X07004102
[5] A. E. M. El-Mesiry, A. M. A. El-Sayed, and H. A. A. El-Saka, “Numerical methods for multi-term fractional (arbitrary) orders differential equations,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 683-699, 2005. · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026
[6] A. M. A. El-Sayed and F. M. Gaafar, “Fractional-order differential equations with memory and fractional-order relaxation-oscillation model,” Pure Mathematics and Applications, vol. 12, no. 3, pp. 296-310, 2001. · Zbl 1006.34008
[7] A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “Numerical solution for multi-term fractional (arbitrary) orders differential equations,” Computational & Applied Mathematics, vol. 23, no. 1, pp. 33-54, 2004. · Zbl 1213.34025 · doi:10.1590/S0101-82052004000100002 · http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022004000100002&lng=en&nrm=iso&tlng=en
[8] R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 223-276, Springer, Vienna, Austria, 1997.
[9] D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in System Application, vol. 2, Lille, France, 1996.
[10] 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. · Zbl 1175.26004 · doi:10.1016/j.jmaa.2009.04.012
[11] I. Podlubny and A. M. A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science-Institute of Experimental Physics, 1996.
[12] S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, Russia, 1967.
[13] P. E. Sobolevskii and L. M. Chebotaryeva, “Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 5, pp. 103-116, 1977 (Russian).
[14] A. Ashyralyev, M. Martinez, J. Paster, and S. Piskarev, “Weak maximal regularity for abstract hyperbolic problems in function spaces, further progress in analysis,” in Proceedings of the 6th International ISAAC Congress, pp. 679-689, World Scientific, Ankara, Turkey, August 2007.
[15] A. Ashyralyev and N. Aggez, “A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations,” Numerical Functional Analysis and Optimization, vol. 25, no. 5-6, pp. 439-462, 2004. · Zbl 1065.35021 · doi:10.1081/NFA-200041711
[16] A. Ashyralyev and I. Muradov, “On difference schemes a second order of accuracy for hyperbolic equations,” in Modelling Processes of Explotation of Gas Places and Applied Problems of Theoretical Gasohydrodynamics, pp. 127-138, Ilim, Ashgabat, Turkmenistan, 1998.
[17] 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. · Zbl 1069.65097
[18] A. Ashyralyev and Y. Ozdemir, “On nonlocal boundary value problems for hyperbolic-parabolic equations,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1075-1089, 2007. · Zbl 1140.65039
[19] A. Ashyralyev and O. Yildirim, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations,” Taiwanese Journal of Mathematics, vol. 13, 22 pages, 2009. · Zbl 1201.65128
[20] A. A. Samarskii, I. P. Gavrilyuk, and V. L. Makarov, “Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces,” SIAM Journal on Numerical Analysis, vol. 39, no. 2, pp. 708-723, 2001. · Zbl 1002.65101 · doi:10.1137/S0036142999357221
[21] A. Ashyralyev and P. E. Sobolevskii, “Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 183-213, 2005. · Zbl 1094.65077 · doi:10.1155/DDNS.2005.183 · eudml:126343
[22] A. Ashyralyev and M. E. Koksal, “On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,” Numerical Functional Analysis and Optimization, vol. 26, no. 7-8, pp. 739-772, 2005. · Zbl 1098.65055 · doi:10.1080/01630560500431068
[23] A. Ashyralyev and M. E. Koksal, “On the stability of the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 57491, 26 pages, 2007. · Zbl 1156.65079 · doi:10.1155/2007/57491 · eudml:129660
[24] M. Ashyraliyev, “A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space,” Numerical Functional Analysis and Optimization, vol. 29, no. 7-8, pp. 750-769, 2008. · Zbl 1146.45001 · doi:10.1080/01630560802292069
[25] A. Ashyralyev and P. E. Sobolevskii, “A note on the difference schemes for hyperbolic equations,” Abstract and Applied Analysis, vol. 6, no. 2, pp. 63-70, 2001. · Zbl 1007.65064 · doi:10.1155/S1085337501000501 · eudml:49844
[26] P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975. · Zbl 0333.47010
[27] A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2 of Iterative Methods, Birkhäuser, Basel, Switzerland, 1989. · Zbl 0649.65054