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Numerical methods for the solution of partial differential equations of fractional order. (English) Zbl 1047.76075

Summary: Anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas. To model this transport mechanism, fractional order space derivative operators can be used. Here, the numerical properties of partial differential equations of fractional order \(\alpha\), \(1 \leqslant \alpha \leqslant 2\), are studied. Two numerical schemes, an explicit and a semi-implicit one, are used in solving these equations. Two different discretization methods of the fractional derivative operator have also been used. The accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
65R20 Numerical methods for integral equations
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[1] Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J., Nature, 363, 31 (1993)
[2] del-Castillo-Negrete, D., Phys. Fluids, 10, 576 (1998) · Zbl 1185.76850
[3] Montroll, E. W.; Shlesinger, M. F., (Lebowitz, J.; Montroll, E., Studies in Statistical Mechanics (1984), North-Holland: North-Holland Amsterdam), 1
[4] Carreras, B. A., IEEE Trans. Plasma Sci., 25, 1281 (1997)
[5] Carreras, B. A.; Lynch, V. E.; Zaslavsky, G. M., Phys. Plasmas, 8, 5096 (2001)
[6] B.A. Carreras, V.E. Lynch, L. Garcia, M. Edelman, G.M. Zaslavsky, Chaos 14 (2003), to be published; B.A. Carreras, V.E. Lynch, L. Garcia, M. Edelman, G.M. Zaslavsky, Chaos 14 (2003), to be published
[7] Cardozo, N. J.L., Plasma Phys. Contr. Fusion, 37, 799 (1995)
[8] Gentle, K.; Bravenec, R. V.; Cima, G.; Gasquet, H.; Hallock, G. A.; Phillips, P. E.; Ross, D. W.; Rowan, W. L.; Wootton, A. J.; Crowley, T. P.; Heard, J.; Ourona, A.; Schoch, P. M.; Watts, C., Phys. Plasmas, 2, 2292 (1995)
[9] Samorodnitsky, G.; Taqqu, M. S., Stable Non-Gaussian Random Processes (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0925.60027
[10] Zaslavsky, G. M., Chaos, 4, 25 (1994) · Zbl 1055.82525
[11] Metzler, R.; Klafter, J., Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[12] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[13] Luise Blank, Nonlinear World, 4, 473-490 (1997)
[14] Alberto Carpinteri; Francesco Mainardi, Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0917.73004
[15] Kai Diethelm, Elec. Trans. Numer. Anal., 5, 1 (1997) · Zbl 0890.65071
[16] Pozrikidis, C., Numerical Computation in Science and Engineering (1998), Oxford University Press: Oxford University Press Oxford · Zbl 0971.65001
[17] del-Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E., Phys. Rev. Lett., 91, 18302 (2003)
[18] Murray, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0682.92001
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