Least squares finite-element solution of a fractional order two-point boundary value problem. (English) Zbl 1069.65094

Fix, G. J.; Roof, J. P.

Comput. Math. Appl. 48, No. 7-8, 1017-1033 (2004).

This paper deals with approximation of fractional differential equations on finite domains using variational methods. A least squares variational form for two point boundary value problems containing a fractional differential operator is derived and the existence and uniqueness results are proven. Error estimates for the variational form for piecewise linear trial elements are presented. Two numerical results are given.

Reviewer: Pavol Chocholatý (Bratislava)

MSC:

65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10	Numerical solution of boundary value problems involving ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
65L70	Error bounds for numerical methods for ordinary differential equations

Keywords:

Least squares finite-element methods; Fractional differential operators; Fractional diffusion equations; error estimates; numerical results

References:

[1]	Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., The fractional order governing equations of levy motion, Water resour. res., 36, 1413-1423, (2000)
[2]	Chaves, A.S., A fractional diffusion equation to describe levy flights, Phys. lett. A, 239, 13-16, (1998) · Zbl 1026.82524
[3]	Compte, A., Stochastic foundations of fractional dynamics, Phys. rev. E, 53, 4, 4191-4193, (1996)
[4]	Metzler, R.; Klafter, J., The random Walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep., 339, 1-77, (2000) · Zbl 0984.82032
[5]	Zaslavsky, G.M., Chaos fractional kinetics, and anomalous transport, Phys. rep., 371, 461-580, (2002) · Zbl 0999.82053
[6]	Lu, S.; Molz, F.J.; Fix, G.J., Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media, Water resour. res., 38, 9, 1165-1171, (2002)
[7]	Podlubny, I., Fractional differential equations, (1999), Academic Press · Zbl 0918.34010
[8]	Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach New York · Zbl 0818.26003
[9]	Molz, F.J.; Fix, G.J.; Lu, S., A physical interpretation for the fractional derivative in Lévy diffusion, Appl. math. lett., 15, 7, 907-911, (2002) · Zbl 1043.76056
[10]	Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Wheatcraft, S.W., Eulerian derivation of the fractional advection-dispersion equation, J. cont. hyd., 48, 69-88, (2001)
[11]	Meerschaert, M.M.; Benson, D.A.; Baeumer, B., Multidimensional advection and fractional dispersion, Phys. rev. E, 59, 5, 5026-5028, (1999)
[12]	Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[13]	Grisvard, P., Singularities in boundary value problems, (1992), Springer-Verlag New York · Zbl 0766.35001
[14]	Fix, G.J.; Gunzburger, M.D.; Nicolades, R.A., On finite element methods of the least squares type, Computers math. applic., 6, 2, 87-98, (1979) · Zbl 0422.65064
[15]	Jiang, B., The least squares finite element method: theory and applications in computational fluid dynamics and electromagnetics, (1998), Springer-Verlag New York
[16]	Clément, P., Approximation by finite element functions using local regularization, RAIRO anal. numér., R-2, 77-84, (1975) · Zbl 0368.65008
[17]	Brenner, S.; Scott, L.R., The mathematical theory of finite element methods, (1994), Springer-Verlag New York · Zbl 0804.65101
[18]	Bedivan, D.M.; Fix, G.J., Least squares methods for Volterra equations and generalizations, Numer. meth. P.D.E., 14, 5, 679-693, (1998) · Zbl 0931.65131
[19]	Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley & Sons New York · Zbl 0789.26002