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Spline collocation methods for linear multi-term fractional differential equations. (English) Zbl 1229.65138
The authors consider the numerical solution of the linear fractional order multi-term initial value problem $$D_*^{\alpha_p} y(t) + \sum_{j=0}^{p-1} a_j(t) D_*^{\alpha_j} y(t) = f(t),\quad y^{(j)}(0) = y_0^{(j)}\quad j = 0, 1, \ldots, \lceil \alpha_p \rceil - 1,$$ where $D_*^\alpha$ is a Caputo differential operator of order $\alpha$, $0 \le \alpha_0 < \alpha_1 < \dots < \alpha_p$ and the functions $a_j$ and $f$ are smooth on some interval $(0,b]$ but may have unbounded derivatives of certain orders at the origin. The algorithms under investigation are generalized spline collocation methods, thus extending earlier work of the authors [J. Comput. Appl. Math. 235, No. 12, 3502--3514 (2011; Zbl 1217.65154)]. The first results are some regularity statements on the exact solution. Based on these regularity properties, the authors next construct a nonuniform partition of the basic interval that has the form of a graded mesh. The approximate solutions are then constructed as piecewise polynomials over this partition without any transition conditions for moving from a subinterval to its neighbor. The precise form of each polynomial on its subinterval is determined by prescribing collocation points and by requiring the differential equation to be fulfilled exactly at these points. Under suitable conditions on the collocation points, the authors then prove convergence of the algorithm and provide error bounds not only for the solution itself but also for its lower order derivatives. Superconvergence behaviour is shown to be present for certain choices of the collocation points. Regrettably, some of the most important results heavily rely on the linearity of the differential equations, and it is not immediately obvious how to extend the results to nonlinear problems.

65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L05Initial value problems for ODE (numerical methods)
65R20Integral equations (numerical methods)
34A08Fractional differential equations
34A30Linear ODE and systems, general
65L20Stability and convergence of numerical methods for ODE
65L70Error bounds (numerical methods for ODE)
Full Text: DOI
[1] Diethelm, K.: The analysis of fractional differential equations, Lecture notes in mathematics 2004 (2010)
[2] Brunner, H.; Pedas, A.; Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. anal. 39, 957-982 (2001) · Zbl 0998.65134 · doi:10.1137/S0036142900376560
[3] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006) · Zbl 1092.45003
[4] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[5] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives, theory and applications, (1993) · Zbl 0818.26003
[6] Diethelm, K.: Efficient solution of multi-term fractional differential equations using $P(EC)$mE methods, Computing 71, 305-319 (2003) · Zbl 1035.65066 · doi:10.1007/s00607-003-0033-3
[7] Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solution, Appl. math. Comput. 154, 621-640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2
[8] Edwards, J. T.; Ford, N. J.; Simpson, A. C.: The numerical solution of linear multi-term fractional differential equations: systems of equations, J. comput. Appl. math. 148, 401-418 (2002) · Zbl 1019.65048 · doi:10.1016/S0377-0427(02)00558-7
[9] El-Mesiry, A. E. M.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. math. Comput. 160, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026
[10] Ford, N. J.; Connoly, J. A.: Systems-based decomposition schemes for approximate solution of multi-term fractional differential equations, J. comput. Appl. math. 229, 382-391 (2009) · Zbl 1166.65066 · doi:10.1016/j.cam.2008.04.003
[11] Brunner, H.; Van Der Houven, P. J.: The numerical solution of Volterra equations, (1986) · Zbl 0611.65092
[12] Vainikko, G.: Multidimensional weakly singular integral equations, Lecture notes in mathematics 1549 (1993) · Zbl 0789.65097
[13] Brunner, H.: Collocation methods for Volterra integral and related functional equations, Cambridge monographs on applied and computational mathematics 15 (2004) · Zbl 1059.65122
[14] Pedas, A.; Tamme, E.: Spline collocation method for integro-differential equations with weakly singular kernels, J. comput. Appl. math. 197, 253-269 (2006) · Zbl 1104.65129 · doi:10.1016/j.cam.2005.07.035
[15] Pedas, A.; Tamme, E.: On the convergence of spline collocation methods for solving fractional differential equations, J. comput. Appl. math. 235, 3502-3514 (2011) · Zbl 1217.65154 · doi:10.1016/j.cam.2010.10.054
[16] Blank, L.: Numerical treatment of differential equations of fractional order, Nonlinear world 4, 473-491 (1997) · Zbl 0907.65066
[17] Dubois, F.; Menguë, S.: Mixed collocation for fractional differential equations, Numer. algorithms 34, 303-311 (2003) · Zbl 1038.65059 · doi:10.1023/B:NUMA.0000005367.21295.05
[18] Rawashdeh, E.: Numerical solution of semidifferential equations by collocation method, Appl. math. Comput. 174, 869-876 (2006) · Zbl 1090.65097 · doi:10.1016/j.amc.2005.05.029
[19] Hamarsheh, M. H.; Rawashdeh, E. A.: A numerical method for solution of semidifferential equations, Mat. vesnik 62, 117-126 (2010) · Zbl 1265.65146
[20] Lepik, Ü.: Solving fractional integral equations by the Haar wavelet method, Appl. math. Comput. 214, 468-478 (2009) · Zbl 1170.65106 · doi:10.1016/j.amc.2009.04.015
[21] Kangro, R.; Parts, I.: Superconvergence in the maximum norm of a class of piecewise polynomial collocation methods for solving linear weakly singular Volterra integro-differential equations, J. integral equations appl. 15, 403-427 (2003) · Zbl 1065.65148 · doi:10.1216/jiea/1181074984
[22] I. Parts, Piecewise polynomial collocation methods for solving weakly singular integro-differential equations, Ph.D. Thesis, Tartu, 2005. Available from Internet: http://dspace.utlib.ee/bitstream/10062/851/5/parts.pdf. · Zbl 1093.65126