Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution

Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

“…Moreover, the choice of collocation basis makes the numerics more adapted to the problem, with meaningful improvements when qualitative behaviors of the solution are merged in the numerical scheme. Adapted functional basis are relevant, for instance, in the case of oscillatory problems [62][63][64][65]. Further developments of this research will be oriented to the establishment of a theory of collocation methods for stochastic problems (see, for instance, refs.…”

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

“…For example, recently two-step collocation methods have been proposed for fractional differential equations [44], and further developments may be achieved for other fractional models, as time fractional differential equations [45]. Further issues of this research will focus on oscillatory problems [46,47] and in particular on the application of multistep collocation methods to periodic integral equations [48,49]. Moreover, it seems reasonable to consider the possibility of employing collocation spaces based on functions other than polynomials, as in [50][51][52] and similarly as in the case of oscillatory problems [53], and merge into the numerical scheme as many known qualitative properties of the continuous problem as possible, in a structure-preserving perspective [54].…”

“…. , c m , conditions (49) and (50) lead to the following non linear system of (r + m) 2 equations, where the (r + m) 2 unknowns are the coefficients of the polynomials ϕ k (s) and ψ j (s):…”

Exponentially fitted methods with a local energy conservation law

“…However, when unknown, the frequency can be estimated by using one of the many approaches suggested in literature [18,42,43]. Exponential fitting techniques have been successfully used to solve problems of very different nature, such as fractional differential equations [7], quadrature [14,16,24], interpolation [21], time and space integrators for ODEs [12,15,41] and PDEs [8,13,19], integral equations [9,10], boundary value problems [32].…”