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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

${D}_{*}^{{\alpha }_{p}}y\left(t\right)+\sum _{j=0}^{p-1}{a}_{j}\left(t\right){D}_{*}^{{\alpha }_{j}}y\left(t\right)=f\left(t\right),\phantom{\rule{1.em}{0ex}}{y}^{\left(j\right)}\left(0\right)={y}_{0}^{\left(j\right)}\phantom{\rule{1.em}{0ex}}j=0,1,...,⌈{\alpha }_{p}⌉-1,$

where ${D}_{*}^{\alpha }$ is a Caputo differential operator of order $\alpha$, $0\le {\alpha }_{0}<{\alpha }_{1}<\cdots <{\alpha }_{p}$ and the functions ${a}_{j}$ and $f$ are smooth on some interval $\left(0,b\right]$ 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.

##### MSC:
 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 65L05 Initial value problems for ODE (numerical methods) 65R20 Integral equations (numerical methods) 34A08 Fractional differential equations 34A30 Linear ODE and systems, general 65L20 Stability and convergence of numerical methods for ODE 65L70 Error bounds (numerical methods for ODE)