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Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations. (English) Zbl 1166.65066
Summary: We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

MSC:
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
65L05 Numerical methods for initial value problems
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[1] Bagley, R.L.; Torvik, P.J., On the appearance of the fractional derivative in the behaviour of real materials, J. appl. mech., 51, 294-298, (1984) · Zbl 1203.74022
[2] J.T. Chern, Finite element modelling of viscoelastic materials on the theory of fractional calculus, Ph.D. Thesis, Pennsylvania State University, 1993
[3] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Elec. transact. numer. anal., 5, 1-6, (1997) · Zbl 0890.65071
[4] Diethelm, K., Efficient solution of multi-term fractional differential equations using \(P(E C)^m E\) methods, Computing, 71, 305-319, (2003) · Zbl 1035.65066
[5] Diethelm, K.; Ford, J.M.; Ford, N.J.; Weilbeer, M., Pitfalls in fast numerical solution of fractional differential equations, J. comput. appl. math., 186, 482-503, (2006) · Zbl 1078.65550
[6] Diethelm, K.; Ford, N.J., Numerical solution methods for distributed order differential equations, Fract. calc. appl. anal., 4, 531-542, (2001) · Zbl 1032.65070
[7] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[8] Diethelm, K.; Ford, N.J., Numerical solution of the bagley-torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067
[9] Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Appl. math. comp., 154, 621-640, (2004) · Zbl 1060.65070
[10] Diethelm, K.; Ford, N.J.; Freed, A.D., Detailed error analysis for a fractional Adams method, Numerical algorithms, 36, 31-52, (2004) · Zbl 1055.65098
[11] Diethelm, K.; Ford, N.J.; Freed, A.D.; Luchko, Y., Algorithms for the fractional calculus: A selection of numerical methods, Comput. methods appl. mech. engrg., 194, 743-773, (2005) · Zbl 1119.65352
[12] K. Diethelm, N.J. Ford, Numerical analysis for distributed order differential equations, (2007) (submitted for publication) · Zbl 1159.65103
[13] Diethelm, K.; Luchko, Y., Numerical solution of linear multi-term initial value problems of fractional order, J. comput. anal. appl., 6, 243-263, (2004) · Zbl 1083.65064
[14] Edwards, J.T.; Ford, N.J.; Simpson, A.C., The numerical solution of multi-term fractional differential equations: systems of equations, J. comput. appl. math., 148, 401-418, (2002) · Zbl 1019.65048
[15] Ford, N.J.; Connolly, J.A., Comparison of numerical methods for fractional differential equations, Comm. pure appl. anal., 5, 289-307, (2006) · Zbl 1133.65115
[16] Lubich, C., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. comp., 45, 463-469, (1985) · Zbl 0584.65090
[17] Lubich, C., Convolution quadrature and discretized operational calculus. I, Num. math., 52, 129-145, (1988) · Zbl 0637.65016
[18] Lubich, C., Convolution quadrature and discretized operational calculus. II, Num. math., 52, 413-425, (1988) · Zbl 0643.65094
[19] Lubich, C., A stability analysis of convolution quadratures for Abel-Volterra integral equations, IMA J. numer. anal., 6, 87-101, (1986) · Zbl 0587.65090
[20] Lubich, C., Convolution quadrature revisited, Bit, 44, 503-514, (2004) · Zbl 1083.65123
[21] Luchko, Y.; Gorenfl, R., An operational method for solving fractional differential equations with the Caputo derivative, Acta math. Vietnam., 24, 207-233, (1999) · Zbl 0931.44003
[22] A.R. Nkamnang, Diskretisierung von mehrgliedrigen Abelschen Integralgleichungen und gewöhnlichen Differentialgleichungen gebrochener Ordnung, Ph.D. Dissertation, Freie Universiät Berlin, 1998
[23] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
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