Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations

Abstract

In this article, we introduce two families of novel fractional θ-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator I α with a second order convergence rate. A new fractional BT-θ method connects the fractional BDF2 (when θ = 0) with fractional trapezoidal rule (when θ = 1/2), and another novel fractional BN-θ method joins the fractional BDF2 (when θ = 0) with the second order fractional Newton-Gregory formula (when θ = 1/2). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different θ-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional θ-methods are A(ϑ)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis.

Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions

“…exactly hold for i = 1, • • • , s (see [2] and [3]), where we have assumed that the solution of (1) can be expanded at initial time with the expression…”

“…In the following discussions we mainly analyse two families of novel fractional θ-methods applied to the equation (1), which, from the aspect of generating function, can be stated as (see [3]), FBT-θ method:…”

“…Lemma 2 (See [3]) The convolution weights ω k which are defined as the coefficients of the generating function for the FBN-θ method can be derived by the recursive formula…”

“…As is well known that the solutions of fractional PDEs show some singularity at initial value, some methods or techniques were developed to cope with such difficulty, see [2,32,33]. In this paper, we apply two families of novel fractional θ-methods, i.e., the fractional BT-θ (FBT-θ) method and fractional BN-θ (FBN-θ) method (see the generating functions for both of these two methods defined by ( 6)), developed by authors in [3], to the fractional Cable model,      u t = RL D 1−γ 0,t ∆u − µ 2 RL D 1−κ 0,t u + f (z, t), (z, t) ∈ Ω × (0, T ], u(z, t) = 0, (z, t) ∈ ∂Ω × [0, T ], u(z, 0) = u 0 (z), z ∈ Ω,…”

The unified theory of shifted convolution quadrature for fractional calculus

“…The fractional calculus has drawn much attention in recent years for its wide applications and theoretical interests, see [9,10,[22][23][24][25][26][27][28][29][30][31]33]. In this paper, we are particularly concerned about the Riemann-Liouville calculus operator I α which is defined by…”