Finite Element Methods Based on Two Families of Second-Order Numerical Formulas for the Fractional Cable Model with Smooth Solutions

Approximation methods for the distributed order calculus using the convolution quadrature

“…For fractional calculus equations, one generally can not get the solution in closed form, thus different numerical methods have been proposed to efficiently obtain the approximate solution, see [2,5,7,8,10,20,21,24,25,31,38,41,43,49]. As is well known that the solution of fractional differential equations shows some singularity at the initial node, different techniques were employed to restore the optimal convergence rate, see [15,23,26,27,40,42,44,45,47]. For equations with distributed order calculus, things seem worse as the distributed order calculus is a natural generalization of the fractional calculus and hence is more complicated.…”

“…The starting part h α s j=1 w n,j f (jh) in (5) plays an essential role to improve the error accuracy of the formula for those f with initial singularity, and hence can be omitted provided f is smooth. For the calculation of the starting weights w n,j , we refer interested readers to [27,44]. Some well studied CQ methods with (α, ξ) satisfying ( 7) and ( 8) include (see [27,46]),…”

“…when conducting their numerical experiments (see [6,11,34]), we first explore the technique of adding starting part for solutions having the structure as (44), and for simplicity, we further assume…”

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

“…Zhao et al [47] presented a fully discrete L1 finite element method for multiterm time fractional diffusion equation with constant diffusivity, and a superconvergence result for H 1 -norm estimate was obtained. Yin et al [42] presented two families of novel fractional θ -methods to solve the fractional cable model, and an optimal convergence result with O(τ 2 + h k+1 ) for smooth solutions was obtained. Syed et al [26] proposed a homotopy analysis method for the space-time fractional Korteweg-de Vries (KdV) equation.…”

A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids

“…Fractional partial differential equations (FPDEs) have received extensive attentions by more and more scholars, and have been widely applied in many fields of science and engineering [1,2]. Many practical problems can be portrayed very well by the some FPDEs, such as fractional (reaction) diffusion equations [3][4][5][6][7][8][9][10][11][12], fractional Allen-Cahn equations [13][14][15], fractional Cable equations [16][17][18], and fractional mobile/immobile transport equations [19][20][21]. In the past few decades, a large number of numerical methods [22][23][24] have been proposed and used to solve the FPDEs, which have achieved excellent theoretical and numerical results.…”