A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

Abstract

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for $L_{log,2-1_{\sigma }}$ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp $L^2$-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

Introduction

In the last few decades, a great deal of efforts have been done in the study of fractional calculus and fractional differential equations (FDEs) [1], [2], [3]. Until now, many researchers have carried on the thorough study to several kinds of fractional integrals and derivatives, such as Riemann–Liouville calculus, Caputo calculus, etc. Although Hadamard fractional derivative was first proposed by Hadamard in 1892 [4], there are few studies on this kind of fractional problem besides [5], [6], [7], [8]. In practice, the Hadamard derivative is worth being further investigated as well. It has been found that the Hadamard derivative and Hadamard-type fractional differential equations are of value in mechanics, engineering and so on for the past few years [8]. In addition, compared with the Riemann–Liouville derivative, there are two characteristics of Hadamard derivative [9]. For one thing, in Hadamard derivatives, the integral is kernel to the power of $logt-logs$, while that in the Riemann–Liouville one is $t-s$. For another, the Hadamard derivative is regarded as a generalization of the operator ${\left(t\frac{\mathrm{d}}{\mathrm{d}t}\right)}^n$, while the Riemann–Liouville one is considered as an extension of the classical operator ${\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)}^n$. Furthermore, the Hadamard derivative begins at the initial moment $a$ that is greater than zero, whereas the Riemann–Liouville one tends to start at the origin or any other real number.

Particularly, Gohar et al. [10] first investigated the smoothness properties of the solution to the initial value problems with Caputo–Hadamard derivative. In [11], the existence and uniqueness of the solution to the fractional ordinary differential equations with Caputo–Hadamard derivative were studied. Li et al. [9] studied the regularity and logarithmic decay for Caputo–Hadamard fractional sub-diffusion equations. In addition, the finite difference scheme with nonuniform meshes was employed to approximate the time fractional derivative and the local discontinuous Galerkin method was used to approximate the spacial derivative. However, the temporary accuracy of its derived scheme is $O\left(N^{-min\{r\alpha ,2-\alpha \}}\right)$, where $N$ denotes the numbers of grids in temporal direction and $\alpha \in (0,1)$ is the fractional order. In this paper, we study the high order finite difference scheme for the following Caputo–Hadamard fractional sub-diffusion equations:

$${}_{CH}{D}_{a,t}^{\alpha }u(x,t)-\Delta u(x,t)=f(x,t),0<x<L,a<t\le T,0<\alpha <1,$$$$u(x,a)=\varphi \left(x\right),0\le x\le L,$$$$u(0,t)=u(L,t)=0,a<t\le T,$$

where $\Delta$ is the Laplacian, $f(x,t),\varphi \left(x\right)$ are given functions, and the symbol ${}_{CH}{D}_{a,t}^{\alpha }$ is Caputo–Hadamard derivative with order $\alpha$:

$${}_{CH}{D}_{a,t}^{\alpha }u(x,t)={\int }_a^t{\omega }_{1-\alpha }(logt-logs)\delta u(x,s)\frac{\mathrm{d}s}{s},0<a<t,$$

where ${\omega }_{\beta }\left(t\right)≔\frac{t^{\beta -1}}{\Gamma \left(\beta \right)}$, $\delta u(x,s)=\left(s\frac{\partial }{\partial s}\right)u(x,s),0<\alpha <1$.

In the literature, several numerical Caputo formulas have a high-order accuracy, such as the $L$1-2 formula [12] and the $L2-1_{\sigma }$ formula [13]. But they achieved second-order accuracy for adequately smooth solutions. It is worth mentioning that the solution of time-fractional differential equations typically displays a weak singularity near the initial time, that is, for more general given data, the high-order time partial derivatives of the solution may not be bounded in the whole closed time domain especially at the initial point, which leads to the loss of time accuracy for many related high-order numerical methods. Recently, more and more researchers have begun to take the initial singularity into account in time fractional models, see [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33] and their references. The nonuniform grids technique [14], [17], [19], [22], [27], [28], [29], [30], [31], [32] works efficiently for time fractional partial/ordinary differential equations with non-smooth solutions. This motivates us to select the nonuniform time meshes $t_k=a{\left(\frac{T}{a}\right)}^{{(k/N)}^r}$, where the grading parameter $r\ge 1$ is adopted to represent the strength of the singularity. And its efficiency is demonstrated by the theoretical and numerical results that will be provided later.

Liao et al. [14] derived a second-order Crank–Nicolson-like scheme with nonuniform meshes for linear reaction–subdiffusion problems with non-smooth solutions by $L2-1_{\sigma }$ approximation formula. By virtue of the ideas derived in [14], we consider the $L_{log,2-1_{\sigma }}$ interpolation for the Caputo–Hadamard derivative. The proposed method in this paper is analyzed under the following regularity assumptions [9]:

$$|\frac{{\partial }^{k}u(x,t)}{\partial x^k}|\le C,k=0,1,2,3,4,$$$$\left|{\delta }^{k}u(x,t)\right|\le C\left(1+{\left(log\frac{t}{a}\right)}^{\alpha -k}\right),k=0,1,2,3,$$

for all $(x,t)\in [0,L]\times (a,T]$ and $C$ is a positive constant. Throughout this paper, the generic constant $C$ is a positive constant and may take different values at different places.

The structure of the paper is as follows. In Section 2, the approximation to the Caputo–Hadamard fractional derivative and the Crank–Nicolson-like scheme are proposed. In Section 3, the properties of discrete kernels $A_{n-k}^{\left(n\right)}$ and global consistency error are investigated. The stability and convergence of the resulting discrete scheme are analyzed by the energy method in Section 4. Eventually, the numerical example presenting the validity of theoretical results is given in Section 5 and some conclusions are drawn in Section 6.