Stability of collocation methods for fractional differential equations

Collocation methods for fractional differential equations have been introduced by Blank [1] and later on rigorously analyzed by Pedas and Tamme [3,4]. Recently Cardone, Conte and Paternoster [2] introduced two step collocation methods, which raise the order of convergence, by using additional information from the past, without increasing the computational cost. Here we study the stability of both classes of methods, in order to find methods with unbounded stability regions. This is a joint work with D. CONTE and B. PATERNOSTER from UNIVERSITY OF SALERNO. [1] L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World, 4 (1997), 473-491. [2] A. Cardone, D. Conte, B. Paternoster, Two-step collocation methods for fractional differential equations, to appear in Discrete Cont.Dyn.-B. [3] A. Pedas and E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl. Math., 235 (2011), 3502-3514. [4] A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math., 255 (2014), 216-230.