Two-step collocation methods for fractional differential equations

Abstract

We propose two-step collocation methods for the numerical solution of fractional differential equations. These methods increase the order of convergence of one-step collocation methods, with the same number of collocation points. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. We describe the derivation of two-step collocation methods and analyse convergence. Some numerical experiments confirm theoretical expectations.

Jacobian-dependent vs Jacobian-free discretizations for nonlinear differential problems

“…In future developments of this research, we aim to provide alternative formulation of the Jacobian-dependent discretizations that avoid the matrix inversion along the integration process, but take advantage from the structure of the involved matrices, especially when the ODE belongs to the discretization of PDEs in space. Moreover, it is worth assessing the effectiveness of the approach also to other kind of operators, such as problems with memory (Burrage et al 2017;Cardone et al 2009Cardone et al , 2018Conte and Califano 2018;Conte et al 2017) and stochastic problems (Cardone et al 2019;Chen et al 2020;D'Ambrosio et al 2018b), and other families of methods, such as multivalue methods (Conte et al 2013;D'Ambrosio et al 2014a, b;D'Ambrosio and Hairer 2014;D'Ambrosio and Paternoster 2015). In addition, we will consider the effectiveness of this approach on the parallel solution of high-dimensional problems , for which a CPU time comparison is a further measure of efficiency.…”

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

“…Thanks to the structure of the coefficient matrix, those methods can be easily parallelized, so the computational effort can be reduced. In the future we aim to construct such types of methods for different operators such as stochastic differential equations [ 15 , 21 , 31 ], fractional differential equations [ 2 , 10 , 13 , 16 ], partial differential equations [ 1 , 11 , 14 , 20 , 30 , 32 , 35 , 38 , 40 ], Volterra integral equations [ 8 , 9 , 12 , 17 , 23 ], second order problems [ 26 , 37 ], oscillatory problems [ 19 , 22 , 24 , 33 , 36 , 53 ], as well as to the development of algebraically stable high order collocation based multivalue methods [ 18 , 29 ].…”

User-Friendly Expressions of the Coefficients of Some Exponentially Fitted Methods

“…Further developments of this research will be oriented to the reformulation, through C and S functions, of existing methods for ordinary differential equations [ 2 , 17 , 20 , 25 , 26 , 28 , 37 – 39 , 41 , 42 , 44 , 48 , 51 , 53 , 56 , 77 , 80 ], integral equations [ 5 – 8 , 10 , 11 , 24 , 29 , 32 , 34 , 55 , 71 ], stochastic problems [ 9 , 12 , 13 , 18 , 19 , 29 , 47 ], fractional equations [ 12 , 13 , 21 , 22 , 36 ], partial differential equations [ 57 – 59 , 66 , 74 ].…”