Numerical solutions to integral equations equivalent to differential equations with fractional time

• Autorzy: Bandrowski B. , Karczewska A. , Rozmej P.
• Języki publikacji: EN

Abstrakty

EN

This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

Słowa kluczowe

EN

fractional equations; Galerkin method; anomalous diffusion

PL

równanie ułamkowe; metoda Galerkina; dyfuzja anomalna

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2010
• Tom: Vol. 20, no 2
• Strony: 261--269
• Opis fizyczny: Bibliogr. 18 poz., tab., wykr.

Twórcy

autor

Bandrowski B.

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Karczewska A.

• Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, ul. Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Rozmej P.

• Institute of Physics University of Zielona Góra, ul. Szafrana 4a, 65-516 Zielona Góra, Poland

Bibliografia

• [1] Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. and der Vorst, H.V. (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, Philadelphia, PA.
• [2] Bazhlekova, E. (2001). Fractional Evolution Equations in Banach Space, Ph.D. dissertation, Eindhoven University of Technology, Eindhoven.
• [3] Ciesielski, M. and Leszczyński, J. (2006). Numerical treatment of an initial-boundary value problem for fractional partial differential equations, Signal Processing 86(10): 2619-2631.
• [4] Fujita, Y. (1990). Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka Journal of Mathematics 27(2): 309-321.
• [5] Gambin, Y., Massiera, G., Ramos, L., Ligoure, C. and Urbach, W. (2005). Bounded step superdiffusion in an oriented hexagonal phase, Physical Review Letters 94(11): 110602.
• [6] Goychuk, I., Heinsalu, E., Patriarca, M., Schmid, G. and Hänggi, P. (2006). Current and universal scaling in anomalous transport, Physical Review E 73(2): 020101'(R)'.
• [7] Guermah, S., Djennoune, S. and Betteyeb, M. (2008). Controllability and observability of linear discrete-time fractional-order systems, International Journal of Applied Mathematics and Computer Science 18(2): 213-222, DOI:10.2478/v10006-008-0019-6.
• [8] Heinsalu, E., Patriarca, M., Goychuk, I. and Hänggi, P. (2009). Fractional Fokker-Planck subdiffusion in alternating fields, Physical Review E 79(4): 041137.
• [9] Heinsalu, E., Patriarca, M., Goychuk, I., Schmid, G. and Hänggi, P. (2006). Fokker-Planck dynamics: Numerical algorithm and simulations, Physical Review E 73(4): 046133.
• [10] Kaczorek, T. (2008). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science 18(2): 223-228, DOI: 10.2478/v10006-008-0020-0.
• [11] Kou, S. and Sunney Xie, X. (2004). Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule, Physical Review Letters 93(18): 180603.
• [12] Labeyrie, G., Vaujour, E., Müller, C., Delande, D., Miniatura, C., Wilkowski, D. and Kaiser, R. (2003). Slow diffusion of light in a cold atomic cloud, Physical Review Letters 91(22): 223904.
• [13] Meltzer, R. and Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports 339(1): 1-77.
• [14] Ratynskaia, S., Rypdal, K., Knapek, C., Kharpak, S., Milovanov, A., Ivlev, A., Rasmussen, J. and Morfill, G. (2006). Superdiffusion and viscoelastic vortex flows in a two dimensional complex plasma, Physical Review Letters 96(10): 105010.
• [15] Rozmej, P. and Karczewska, A. (2005). Numerical solutions to integrodifferential equations which interpolate heat and wave equations, International Journal on Differential Equations and Applications 10(1): 15-27.
• [16] Saad, Y. and Schultz, M. (1986). GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 7(3): 856-869.
• [17] Schneider,W. andWyss,W. (1989). Fractional diffusion and wave equations, Journal of Mathematical Physics 30(4): 134-144.
• [18] Van der Vorst, H. (1992). Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 13(2): 631-644.