zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Numerical solutions to fractional perturbed Volterra equations. (English) Zbl 1259.65210
Summary: A class of perturbed Volterra equations of convolution type with three kernel functions is considered. The kernel functions $g_\alpha = t^{\alpha - 1}/\Gamma(\alpha)$, $t > 0$, $\alpha \in [1, 2]$, correspond to the class of equations interpolating heat and wave equations. The results obtained generalize our previous results from 2010.

MSC:
65R20Integral equations (numerical methods)
45D05Volterra integral equations
45E10Integral equations of the convolution type
26A33Fractional derivatives and integrals (real functions)
WorldCat.org
Full Text: DOI
References:
[1] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, 1993. · doi:10.1007/978-3-0348-8570-6
[2] A. Karczewska and C. Lizama, “Stochastic Volterra equations under perturbations,” preparation. · Zbl 1314.60134
[3] H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser Boston Inc., Boston, Mass, USA, 1996. · Zbl 0860.60045
[4] A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematical Studies, Elsevier, 2006. · Zbl 1092.45003
[5] Leszczyński, An Introduction to Fractional Mechanics, The Publishing Office of Czestochowa University of Technology, Czestochowa, Poland, 2011. · Zbl 1276.34065
[6] R. L. Magin, Fractional Calculus in Bioengineering, Begel House Inc, Redding, Calif, USA, 2006.
[7] F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378, pp. 291-348, Springer, Vienna, Austria, 1997. · Zbl 0917.73004
[8] A. Karczewska, Convolution Type Stochastic Volterra Equations, vol. 10 of Lecture Notes in Nonlinear Analysis, Juliusz Schauder Center for Nonlinear Studies, 2007. · Zbl 1149.60041
[9] A. Karczewska and C. Lizama, “On stochastic fractional Volterra equations in Hilbert space,” Discrete and Continuous Dynamical Systems A, pp. 541-550, 2007. · Zbl 1163.60316 · http://www.aimsciences.org/journals/redirecting.jsp?paperID=2861
[10] A. Karczewska and C. Lizama, “Stochastic Volterra equations driven by cylindrical Wiener process,” Journal of Evolution Equations, vol. 7, no. 2, pp. 373-386, 2007. · Zbl 1120.60062 · doi:10.1007/s00028-007-0302-2
[11] A. Karczewska, “On difficulties appearing in the study of stochastic Volterra equations,” in Quantum probability and related topics, vol. 27, pp. 214-226, World Scientific, 2011. · Zbl 1258.60039
[12] A. Karczewska and C. Lizama, “Solutions to stochastic fractional relaxation equation,” Physica Scripta T, vol. 136, Article ID 014030, 2009. · Zbl 1158.60028 · doi:10.1088/0031-8949/2009/T136/014030
[13] A. Karczewska and C. Lizama, “Strong solutions to stochastic Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 349, no. 2, pp. 301-310, 2009. · Zbl 1158.60028 · doi:10.1016/j.jmaa.2008.09.005
[14] A. Karczewska and C. Lizama, “Solutions to stochastic fractional oscillation equations,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1361-1366, 2010. · Zbl 1202.60099 · doi:10.1016/j.aml.2010.06.032
[15] Y. Fujita, “Integrodifferential equation which interpolates the heat equation and the wave equation,” Osaka Journal of Mathematics, vol. 27, no. 2, pp. 309-321, 1990. · Zbl 0796.45010
[16] W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134-144, 1989. · Zbl 0692.45004 · doi:10.1063/1.528578
[17] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. · Zbl 1248.26011
[18] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[19] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Linghorne Pa, USA, 1993. · Zbl 0818.26003
[20] D. Baleanu and J. I. Trujillo, “A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1111-1115, 2010. · Zbl 1221.34008 · doi:10.1016/j.cnsns.2009.05.023
[21] R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, pp. 223-276, Springer, Vienna, Austria, 1997. · Zbl 0916.34011
[22] J. Sabatier and O. P. Agrawal, Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Berlin, Germany, 2007. · Zbl 1116.00014
[23] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 0998.26002
[24] R. Magin, X. Feng, and D. Baleanu, “Solving the fractional order Bloch equation,” Concepts in Magnetic Resonance Part A, vol. 34, no. 1, pp. 16-23, 2009.
[25] R. L. Magin, “Fractional calculus models of complex dynamics in biological tissues,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1586-1593, 2010. · Zbl 1189.92007 · doi:10.1016/j.camwa.2009.08.039
[26] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods, Springer, Berlin, Germany, 2006.
[27] C. Li, F. Zeng, and F. Liu, “Spectral approximations to the fractional integral and derivative,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 383-406, 2012. · Zbl 1276.26016 · doi:10.2478/s13540-012-0028-x
[28] B. Bandrowski, A. Karczewska, and P. Rozmej, “Numerical solutions to integral equations equivalent to differential equations with fractional time,” International Journal of Applied Mathematics and Computer Science, vol. 20, no. 2, pp. 261-269, 2010. · Zbl 1201.35020 · doi:10.2478/v10006-010-0019-1 · eudml:207985
[29] R. Barrett, M. Berry, T. F. Chan, and et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. · doi:10.1137/1.9781611971538
[30] H. A. van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” Society for Industrial and Applied Mathematics, vol. 13, no. 2, pp. 631-644, 1992. · Zbl 0761.65023 · doi:10.1137/0913035
[31] Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” Society for Industrial and Applied Mathematics, vol. 7, no. 3, pp. 856-869, 1986. · Zbl 0599.65018 · doi:10.1137/0907058
[32] D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, University of Texas at Austin, 2002. · Zbl 0877.65002