Numerical Schemes for Fractional Ordinary Differential Equations

Efficient algorithms for solving the fractional ordinary differential equations

“…As mentioned above, how to compute the integral tn 0 (t n+1 − τ ) α−1 f (τ, x(τ ))dτ efficiently is the key of reducing the computation cost for obtaining the numerical approximation to x(t n+1 ). In the previous predictor-corrector scheme introduced in [10][11][12], as well as its improved version in [8,9], n steps' calculations are used to approximate the integral tn 0 · dτ . While, by noticing that (t n+1 − τ ) α−1 decays with power 1 − α (see figures below), we can actually select less mesh points, say 0 = τ 0,n < τ 1,n < · · · < τ mn,n = t n , (2.6) at which to approximate the term tn 0 · dτ by compounding two-point trapezoidal quadrature formula.…”

“…This method is improved in [8] (or see the review article [9]), where almost half of the computation cost is reduced and the computation accuracy is improved from O(h min{1+α,2} ) to O(h min{1+2α,2} ), but the computation expenditure is still O(h −2 ), which also means that the computation cost is proportional to t 2 . Along the direction of reducing the computation cost, some efforts have been made using the nested meshes which arebased on the fixed memory principle [19] and the short memory principle [14] of fractional operator when α ∈ (0, 1) in (1.1), and the short memory principle is apprehended from a new point of view in [7] where the range can be extended to α ∈ (0, 2).…”

Fast predictor-corrector approach for the tempered fractional differential equations

“…As mentioned above, how to compute the integral tn 0 e −λ(tn+1−τ ) (t n+1 −τ ) α−1 f (τ, x(τ ))dτ efficiently is the key of reducing the computation cost to obtain the numerical approximations to x(t n+1 ). In the previous predictor-corrector work introduced in [7,8,9], as well as its improved version in [5,6], n steps' calculations are used to approximate the integral tn 0 · dτ . While, by noticing that (t n+1 − τ ) α−1 decays with power 1 − α, and e −λ(tn+1−τ ) damps exponentially (see figures below), we can actually select less mesh points, say 0 = τ 0,n < τ 1,n < · · · < τ mn,n = t n , (2.6) at which to approximate the term tn 0 · dτ by compounding two-point trapezoidal quadrature formula.…”

Finite time stability of linear fractional order dynam-ical system with variable delays

“…Thus, there remain only geometric methods (Nyquist) which can be used for the stability check for bounded input bounded output. Different techniques have been proposed in the investigation of the stability for various fractional dynamical system, such as analytical approach [4], [27], fixed point theorem [9], [10], [51], the Lyapunov method [34], [35], linear matrix inequality [47], Gronwall inequality [12], [30]. Recently there have been advances in control theory of fractional order dynamical systems for different kinds of stability.…”