zbMATH — the first resource for mathematics

Spectral method for solution of the fractional transport equation. (English) Zbl 1237.82041
Summary: The Chebyshev polynomials expansion method is applied to find both an analytical solution of the fractional transport equation in the one-dimensional plane geometry and its numerical approximations. The idea of the method is in reducing of the fractional transport equation to a system of the linear fractional differential equations for the unknown coefficients of the Chebyshev polynomials expansion. The obtained system of equations is then solved by using the operational method for the Caputo fractional derivative.

82C70 Transport processes in time-dependent statistical mechanics
Full Text: DOI
[1] Bogolyubov, N.N., ()
[2] Hopf, E., ()
[3] Davison, B., Neutron transport, Oxford, (1957) · Zbl 0077.22505
[4] Chandrasekhar, S., ()
[5] Duderstadt, J.J.; Martin, W.R., ()
[6] Garcia, R.D.M., A review of the facile (F_N) method in particle transport theory, Transport theory and statistical physics,, 14, 39, (1985) · Zbl 0624.65142
[7] Ganapol, B.D.; Kornreich, D.E.; Dahl, J.A.; Nigg, D.W.; Jahshan, S.N.; Wemple, C.A., The searchlight problem for neutrons in a semi-infinite medium, Nucl. sci. engr., 118, 38, (1994)
[8] Ganapol, B.D., Distributed neutron sources in a semi-infinite medium, Nucl. sci. engr., 110, 275, (1992)
[9] Barros, R.C.; Larsen, E.W., Transport theory and statistical physics, 20, (1991)
[10] Barros, R.C.; Larsen, E.W., A numerical method for one-group slab geometry discrete ordinate problem without spatial truncation error, Nucl. sci. engr., 104, 199, (1990)
[11] Vilhena, M.T.; Barichello, L.B., A new analytical approach to solve the neutron transport equation, Kerntechnik, 56, 334, (1991)
[12] Barichello, L.B.; Vilhena, M.T., A general approach to one group one dimensional transport equation, Kerntechnik, 58, (1993)
[13] Vilhena, M.T.; Streck, E.E., An approximated analytical solution of the one-group neutron transport equation, Kerntechnik, 58, 182, (1993)
[14] Cardona, A.V.; Vilhena, M.T., A solution of linear transport equation using Walsh function and Laplace transform, Ann. nucl. energ., 21, 495, (1994)
[15] Kadem, A., Solving the one-dimensional neutron transport equation using Chebyshev polynomials and the sumudu transform, Anal. univ. oradea, fasc. math., 12, 153-171, (2005) · Zbl 1164.82331
[16] Cardona, A.V.; Vilhena, M.T., Analytical solution for the AN approximation, Progr. nucl. energy, 31, 219, (1997)
[17] Barros, R.C.; Cardona, A.V.; Vilhena, M.T., Analytical numerical methods applied to linear discontinuous angular approximations of the transport equation in slab geometry, Kerntechnik, 61, 11, (1996)
[18] Seed, T.J.; Albrecht, R.W., Application of Walsh functions to neutron transport problems—I. theory, Nucl. sci. eng., 60, 337, (1976)
[19] Kharroubi, M.M., Mathematical topics in neutron transport theory, () · Zbl 0997.82047
[20] Gottlieb, D.; Orszag, S.A., ()
[21] Lewis, E.E.; Miller, W.F., ()
[22] Vilhena, M.T.; Barichello, L.B.; Zabadal, J.; Segatto, C.F.; Cardona, A.V., General solution of one-dimensional approximations to the transport equation, Progr. nucl. energy, 33, 99, (1998)
[23] A. Kadem: Solving a transport equation using a fractional derivative and a Chebyshev polynomials, submitted. · Zbl 1190.45009
[24] A. Kadem and D. Baleanu: Fractional radiative equation within Chebyshev spectral approach, to appear in Comput. Math. Appl. · Zbl 1189.35359
[25] A. Kadem and D. Baleanu: Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation, to appear in Commun. Nonlin. Sci. Numer. Simul. · Zbl 1221.45008
[26] Jaffel, L.B.; Vidal-Madjar, A., New developments in the discrete ordinate method for the resolution of the radiative transfer equation, Astron. astrophys., 220, 306-312, (1989)
[27] Ben Jaffel, L.; Vidal-Madjar, A., A new method for the resolution of the radiative transfer equation in three-dimensional geometry. I—theory, Astrophys. J., 350, 801-818, (1990)
[28] Oldham, K.; Spanier, J., ()
[29] Podublny, I., ()
[30] Diethelm, K.; Ford, N.; Freed, A.; Luchko, Yu., Algorithms for the fractional calculus: A selection of numerical methods, Comput. meth. appl. mech. eng., 194, 743-773, (2005) · Zbl 1119.65352
[31] Samko, G.; Kilbas, A.A.; Marichev, O.I., ()
[32] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[33] Magin, R.L., ()
[34] ()
[35] Luchko, Yu.; Gorenflo, R., An operational method for solving fractional differential equation with the Caputo derivatives, Acta math. vietnamica, 24, 2, 207-233, (1999) · Zbl 0931.44003
[36] Achar, N.; Lorenzo, C.F.; Hartley, T.T., ()
[37] Ortigueira, M.D.; Coito, F.J., Initial conditions: what are we talking about?, ()
[38] J. Sabatier, R. Malti, M. Merveillaut and A. Oustaloup: How to impose physically coherent initial conditions to a fractional system?, Commun. Nonlin. Sci. Numer. Simul., to appear 2009. · Zbl 1221.34019
[39] Craiem, D.; Magin, R.L., Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics, Phys. biol., 7, 1, 013001, (2010)
[40] Jeon, J.H.; Metzler, R., Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries, Phys. rev. E, 81, 2, 021103, (2010)
[41] Baleanu, D., Fractional variational principles in action, Physica scripta, 7136, 014006, (2009)
[42] Machado, J.A.T.; Silva, M.F.; Barbosa, R.S.; Jesus, J.; Reis, C.M.; Marcos, M.G.; Galhano, A.F., Some applications of fractional calculus in engineering, Math. probl. eng., 639801, (2010) · Zbl 1191.26004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.