Finite difference/spectral approximations for the time-fractional diffusion equation

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].…”

“…Lin et al [28] and Sun et al [47] independently proved that the truncation error of the L1 scheme is O(τ 2−α ) for a sufficiently smooth function u.…”

“…Hence there exists a constant C which depends only on θ and α such that, see Jin et al [21, (2.3)], (z α I + A) −1 ≤ C|z| −α , ∀ z ∈ Σ θ = {z = 0 : | arg z| < θ}. (1.3) Under the assumptions that the solutions of (1.1) are sufficiently smooth, for example u ∈ C 2 [0, T ], there are many time discretisation schemes in the literature, for example, Lubich's fractional multistep methods [7], [30], [55], [65], [61], [62], [64], the L1 scheme and its modification [28], [26], [18], [29], [47], the spectral method [58] [63], [4], [68], [57], nonpolynomial collocation method [14], discontinuous Galerkin method [39], etc.…”

Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method

“…A time-fractional diffusion equation occurs when replacing the standard time derivative with a time fractional derivative and can be applied in modeling of some problems in porous flows, rheology and mechanical systems, models of a variety of biological processes, control and robotics, transport in fusion plasmas, and many other areas of applications. The direct problems corresponding to the time-fractional diffusion equations have been studied extensively in recent years, including uniqueness and existence results [2], some analytical or numerical solutions [13,7,31], and numerical methods such as finite element methods or finite difference methods [12,14]. Here, we focus on an interesting inverse problem defined to the fractional inverse problem pioneered by Murio [18,16,17].…”

“…where b i = (i + 1) 1−α − i 1−α and γ n is the truncation error with the estimate [14] γ n ≤ C(∆t) 2−α .…”