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A High Order Formula to Approximate the Caputo Fractional Derivative

Abstract

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order \(\alpha\) for \(0<\alpha<\)1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is \(4-\alpha ,\) while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The effectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial differential equations is presented by constructing a finite difference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

Introduction

Nowadays, to provide a better approach for describing complex phenomenon in nature, the fractional calculus has been developed in both theory and application. By entering the fractional derivative into the engineering and science problems, dozens of fractional problems such as the fractional diffusion equation, the fractional advection–diffusion equation, the fractional Fokker–Planck equation, the fractional cable equation and many other equations have appeared in recent years. Among various definitions of the fractional derivative, the Caputo fractional derivative is one of the most important and applicable tools in time-dependent problems. In fact in almost all of the time-fractional partial differential equations, the time-fractional derivative is usually considered in the sense of Caputo. A concise and somewhat complete literature survey can be found in [18].

Due to the complexity of computing of some special functions, such as the Fox H function and the hyperbolic geometry function [23], and the difficulties of finding exact solutions for most fractional problems, many researchers are working on extracting useful numerical algorithms. Some successful methods are the finite difference method (FDM) [2, 6, 21], the finite element method [24], the method of discontinuous Galerkin [8, 9, 26, 27], the meshless method [28], and the spectral approximation [19]; among them, the FDM is especially favored thanks to its simplicity in both calculation and analysis.

Obviously, a suitable numerical differentiation formula is necessary for numerically solving some fractional differential equations involving fractional derivatives. To the best of our knowledge, the Grünwald–Letnikov (GL) formula and the L1 formula are two more important numerical differentiation formulae for discretizing fractional derivatives. For the fractional derivative of order \(\alpha\), the accuracy of the GL formula is of order one [7, 23], while the L1 formula can achieve an improved accuracy of order \(2-\alpha\) [16]. In fact, the L1 formula is established by a piecewise linear interpolation approximation for the integrand function on each small interval. By applying a higher-order interpolant instead of the linear interpolant to improve the numerical accuracy, Cao et al. [5], Gao et al. [12], and Lv and Xu [20] proposed some \((3-\alpha )\)-order formulae for the \(\alpha\)-order Caputo fractional derivative. A three-point L1 approximation of the Caputo derivative with order \(3-\alpha\) has been also presented in [11]. Recently, Cao et al. [4] have presented a high-order algorithm with convergence order \(4-\alpha\) for the Caputo derivative, in which the coefficients and truncation errors of the proposed scheme are not discussed in detail. Moreover, the proposed method was used to present a higher-order numerical scheme to solve a time fractional advection–diffusion equation and its stability analysis was discussed by using the Fourier method. We also independently introduce a \(4-\alpha\) order algorithm for the Caputo derivative with a slightly different approach, where the implementation of the proposed scheme is distinctive, the weight coefficients are exactly examined, and the truncation error in our work is precisely inspected. For this purpose, three distinct numerical examples for testing the truncation error of the proposed formula and two different numerical examples of the time-fractional partial differential equations with the Caputo fractional derivative are carried out.

The objective of this paper is to establish an explicit high-order numerical differentiation formula for the Caputo fractional derivative and apply it to some schemes for numerically solving some fractional differential equations. Here after presenting a brief discussion about previous fractional numerical differentiation formulae for Caputo fractional of order \(\alpha\), i.e., L1 and L1-2 [12], we introduce L1-2-3 formula which has a better numerical accuracy compared to the L1 and L1-2 formulae. After that, we use the new L1-2-3 formula in various examples and make a comparison between the new and older formulae. Then, we construct a finite difference scheme for solving a time-fractional diffusion equation involving the Caputo derivative and provide some examples. Moreover, this new scheme is applied to the image denoising. Finally, we conclude the paper with a brief conclusion.

New Formula, Construction and Properties

In this section, to fully understand the proposed process, after representing both the L1 formula and the L1-2 formula, we give a new process of deriving the fractional numerical differentiation formula called L1-2-3. For this purpose, we denote \(\Delta t\) as the temporal step length and for the integer \(k\ge 0\), we set \(t_{k}=k \Delta t\), \(t_{k+1/2} = (t_{k+1}+t_{k})/2\), and

$$\begin{aligned} \delta _{t}\,f_{k-\frac{1}{2}}& = \frac{f(t_{k})-f(t_{k-1})}{\Delta t},\ \delta _{t}^{2}f_{k} = \frac{1}{\Delta t}(\delta _{t}\,f_{k+\frac{1}{2}} - \delta _{t}\,f_{k-\frac{1}{2}}),\\&\quad \delta _{t}^{3}f_{k-\frac{1}{2}} = \frac{1}{\Delta t^2}(\delta _{t}\,f_{k+\frac{1}{2}} - 2\delta _{t}f_{k-\frac{1}{2}} + \delta _{t}\,f_{k-\frac{3}{2}}). \end{aligned}$$

Denoting linear Lagrange interpolant of f on each small interval \([t_{j-1}, t_{j}]\, (1\le j \le k)\) as \(\Pi _{1,j}\,f\), we get

$$\begin{aligned} \Pi _{1,j}\,f(t) = f(t_{j-1}) \frac{t_{j}-t}{\Delta t} + f(t_{j}) \frac{t-t_{j-1}}{\Delta t}, \end{aligned}$$

which follows from the linear interpolation theory that

$$\begin{aligned}&f(t) - \Pi _{1,j}\,f(t) = \frac{f^{''}(\xi _{j})}{2} (t-t_{j-1})(t-t_{j}), \\&\quad t\in [t_{j-1},t_{j}],\, \xi _{j}\in (t_{j-1},t_{j}), 1\le j \le k. \end{aligned}$$

In a similar way, by constructing the second-degree Lagrange interpolating polynomial \(\Pi _{2,j}\,f\) of f using three points \((t_{j-2}, f(t_{j-2})), (t_{j-1}, f(t_{j-1}))\), and \((t_{j}, f(t_{j}))\) and taking a constraint of the result onto small interval \([t_{j-1}, t_{j}]\, (2\le j \le k)\), we have

$$\begin{aligned} \Pi _{2,j}\,f(t)& = f(t_{j-2}) \frac{(t-t_{j-1})(t-t_{j})}{2 \Delta t^{2}} \\& \quad + \, f(t_{j-1}) \frac{(t-t_{j-2})(t_{j}-t)}{\Delta t^{2}} + f(t_{j}) \frac{(t-t_{j-1})(t-t_{j-2})}{2 \Delta t^{2}}, \end{aligned}$$

and

$$\begin{aligned} & f(t) - \Pi _{2,j}\,f(t) = \frac{f^{'''}(\vartheta _{j})}{6} (t-t_{j-2})(t-t_{j-1})(t-t_{j}),\\&\quad t\in [t_{j-1},t_{j}],\, \vartheta _{j}\in (t_{j-2},t_{j}), 2\le j \le k. \end{aligned}$$

For \(j\ge 3\), constructing the third degree Lagrange interpolating polynomial \(\Pi _{3,j}\,f\) of f using four points \((t_{j-n}, f(t_{j-n})), n=3, 2, 1, 0\) and taking a constraint of the result onto the small interval \([t_{j-1}, t_{j}]\, (2\le j \le k)\), we get

$$\begin{aligned} \Pi _{3,j}\,f(t)& = f(t_{j-3}) {\frac{(t_{j}-t)(t-t_{j-1})(t-t_{j-2})}{6\, \Delta t^{3}}} + f(t_{j-2}) {\frac{(t-t_{j})(t-t_{j-1})(t-t_{j-3})}{2\, \Delta t^{3}}} \\& \quad + \, f(t_{j-1}) {\frac{(t_{j}-t)(t-t_{j-3})(t-t_{j-2})}{2\, \Delta t^{3}}} + f(t_{j}) {\frac{(t-t_{j-3})(t-t_{j-1})(t-t_{j-2})}{6\, \Delta t^{3}}} \\& = f(t_{j-3}) {\frac{(t_{j}-t)(t-t_{j-1})(t-t_{j-2})}{6\, \Delta t^{3}}} + f(t_{j-2}) {\frac{(t-t_{j})(t-t_{j-1})}{2\, \Delta t^{2}}} \left(1+\frac{t-t_{j-2}}{\Delta t}\right) \\& \quad + \, f(t_{j-1}) {\frac{(t_{j}-t)(t-t_{j-2})}{\Delta t^{2}}} \left(1+\frac{t-t_{j-1}}{2\, \Delta t}\right) + f(t_{j}) {\frac{(t-t_{j-1})(t-t_{j-2})}{2\, \Delta t^{2}}} \left(1+\frac{t-t_{j}}{3\, \Delta t}\right)\\& = \Pi _{2,j}\,f(t) + \frac{1}{6}(t-t_{j})(t-t_{j-1})(t-t_{j-2})\, \delta ^{3}_{t}f_{j-\frac{3}{2}}, \end{aligned}$$

and immediately

$$\begin{aligned} (\Pi _{3,j}\,f(t))^{'}& = \delta _{t}\,f_{j-\frac{1}{2}} + (\delta ^{2}_{t}f_{j-1})\,(t - t_{j-\frac{1}{2}}) + \frac{1}{6}\, \Phi _{j}(t)\, \delta ^{3}_{t}f_{j-\frac{3}{2}}, \end{aligned}$$

where \(\Phi _{j}(t) = 3t^{2} - 2t(t_{j-2}+t_{j-1}+t_{j}) +t_{j}t_{j-1}+t_{j}t_{j-2}+t_{j-2}t_{j-1}\). Moreover, we know that

$$\begin{aligned} &f(t) - \Pi _{3,j}\,f(t) = {\frac{f^{(4)}(\zeta _{j})}{4!}} {\prod _{i=0}^{3}}(t-t_{j-i}), \\&\quad t \in [t_{j-1},t_{j}], \, \, \zeta _{j}\in (t_{j-3},t_{j}), \, \, 3\le j \le k. \end{aligned}$$

For further details about Lagrange interpolants, we refer to [3]. Letting \(f\in C^{1}[0, t_{k}]\), the Caputo fractional derivative of order \(\alpha\) is defined as [19]

$$\begin{aligned} _{0}^{C}\mathcal{D}_{t}^{\alpha }f(t) \bigg \vert_{t=t_{k}}& = \, \frac{1}{\Gamma (1-\alpha )}\int _{0}^{t_{k}}\frac{f^{'}(s)}{(t_{k}-s)^{\alpha }}\, \mathrm{d}s \nonumber \\& = \, \frac{1}{\Gamma (1-\alpha )}\sum _{j=1}^{k} \int _{t_{j-1}}^{t_{j}}\frac{f^{'}(s)}{(t_{k}-s)^{\alpha }}\, \mathrm{d}s. \end{aligned}$$
(1)

In this paper, we assume that \(0<\alpha <1\). In (1), we use \(\Pi _{1,j}\,f\) to approximate f on the first small interval \([t_{0},t_{1}]\), \(\Pi _{2,j}\,f\) to approximate f on the second small interval \([t_{1},t_{2}]\) and for \(j\ge 3\), \(\Pi _{3,j}\,f\) to approximate f on the interval \([t_{j-1},t_{j}]\). Noticing

$$\begin{aligned} \int _{t_{j-1}}^{t_{j}} \Phi _{j}(s) (t_{k}-s)^{-\alpha }\, \mathrm{d}s = \frac{6\, \Delta t ^{3-\alpha }}{1-\alpha }\beta _{k-j}^{(\alpha )}, \end{aligned}$$

where

$$\begin{aligned}&\beta _{j}^{(\alpha )} = -\left( \, \frac{1}{6}((j+1)^{1-\alpha }+2\, j^{1-\alpha }) + \frac{1}{2-\alpha } j^{2-\alpha } - \frac{1}{(2-\alpha )(3-\alpha )}((j+1)^{3-\alpha }- j^{3-\alpha })\, \right) , \, j\ge 0, \end{aligned}$$

we can obtain an improved numerical approximation of the Caputo fractional derivative of order \(\alpha\) for function f in the following form:

$$\begin{aligned} _{0}^{C}\mathcal{D}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}}& = { \frac{1}{\Gamma (1-\alpha )}} {\sum _{j=1}^{k} \int _{t_{j-1}}^{t_{j}}}\frac{f^{'}(s)}{(t_{k}-s)^{\alpha }}\, \mathrm{d}s\\ & \approx {\frac{1}{\Gamma (1-\alpha )}}\bigg ({\int _{t_{0}}^{t_{1}}}(t_{k}-s)^{-\alpha }\, (\Pi _{1,1}f(t))^{'}\, \mathrm{d}s + {\int _{t_{1}}^{t_{2}}}(t_{k}-s)^{-\alpha }\, (\Pi _{2,2}f(t))^{'} \, \mathrm{d}s \, \\& \quad +{\sum _{j=3}^{k} \int _{t_{j-1}}^{t_{j}}}(t_{k}-s)^{-\alpha }\, (\Pi _{3,j}f(t)) ^{'} \, \mathrm{d}s\bigg ) \\& = {\frac{1}{\Gamma (1-\alpha )}} \bigg (\delta _{t}f_{\frac{1}{2}} {\int _{t_{0}}^{t_{1}}}(t_{k}-s)^{-\alpha }\, \mathrm{d}s + {\int _{t_{1}}^{t_{2}}}(t_{k}-s)^{-\alpha }\, (\delta _{t}f_{\frac{3}{2}} +\delta _{t}^{2}f_{1} (s-t_{\frac{3}{2}}) ) \, \mathrm{d}s \\& \quad + {\sum _{j=3}^{k} \int _{t_{j-1}}^{t_{j}}}(t_{k}-s)^{-\alpha }\, (\delta _{t}f_{j-\frac{1}{2}} + \delta _{t}^{2}f_{j-1} (s-t_{j-\frac{1}{2}})+ \frac{1}{6}\, \Phi _{j}(s) \, \delta ^{3}_{t} f_{j-\frac{3}{2}})\,\mathrm{d}s\bigg ) \\& = {\frac{1}{\Gamma (1-\alpha )}} \bigg ({\sum ^{k}_{j=1}} (\delta _{t}f_{j-\frac{1}{2}})\,{\int _{t_{j-1}}^{t_j}} (t_k -s)^{-\alpha } \mathrm{d}s + {\sum ^{k}_{j=2}} (\delta ^{2} _{t}f_{j-1})\, {\int _{t_{j-1}}^{t_j}}(t_k -s)^{-\alpha } (s-t_{j-\frac{1}{2}}) \mathrm{d}s \\& \quad + {\sum _{j=3}^{k}}\frac{1}{6}\,(\delta ^{3}_{t} f_{j-\frac{3}{2}})\, { \int _{t_{j-1}}^{t_{j}}}(t_{k}-s)^{-\alpha }\, \Phi _{j}(s)\, \mathrm{d}s \bigg ). \end{aligned}$$

According to [12], \(D^{\alpha }_{t}f\) is the classical L1 operator which is derived from a piecewise linear interpolation approximation of f on each small interval \([t_{j-1}, t_{j}]\, (1\le j \le k)\) defined by

$$\begin{aligned} D_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}}& = { \frac{\, \Delta t ^{1-\alpha }}{\Gamma (2-\alpha )}}\, {\sum _{j=1}^{k}} a_{k-j}^{(\alpha )}\delta _{t}\, f_{j-\frac{1}{2}} \nonumber \\& = {\frac{\, \Delta t ^{-\alpha }}{\Gamma (2-\alpha )}}\, \left (f(t_{k}) - {\sum _{j=1}^{k-1}}(a_{k-j-1}^{(\alpha )} - a_{k-j}^{(\alpha )}) f(t_{j}) - a_{k-1}^{(\alpha )} f(t_{0})\right), \end{aligned}$$
(2)

with \(a_{j}^{(\alpha )} = (j+1)^{1-\alpha } - j^{1-\alpha },\, 0\le j \le k-1\), and \({\mathbb {D}}^{\alpha }_{t}f\) is the L1-2 operator which is derived from a second Lagrange interpolation approximation of f on each small interval \([t_{j-1}, t_{j}]\, (1\le j \le k)\) defined by

$$\begin{aligned} {\mathbb {D}}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}} = D_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}} + \frac{\, \Delta t ^{2-\alpha }}{\Gamma (2-\alpha )}\, \displaystyle {\sum _{j=2}^{k}}b_{k-j}^{(\alpha )} \delta _{t}^{2}f_{j-1}, \end{aligned}$$
(3)

with \(b_{j}^{(\alpha )} = [(\,j+1)^{2-\alpha } - j^{2-\alpha }]/(2-\alpha ) - [(\,j+1)^{1-\alpha } + j^{1-\alpha }]/2,\, j\ge 0\). Now, we define

$$\begin{aligned} \widetilde{{\mathbb {D}}}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}} = {\mathbb {D}}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}} + \frac{\, \Delta t ^{3-\alpha }}{\Gamma (2-\alpha )}\, \displaystyle {\sum _{j=3}^{k}}\beta _{k-j}^{(\alpha )} \delta _{t}^{3}f_{j-\frac{3}{2}}. \end{aligned}$$
(4)

The operator \(\widetilde{{\mathbb {D}}}_{t}^{\alpha }\) is the improved fractional numerical differentiation operator for the Caputo fractional derivative \(_{0}^{C}\mathcal{D}_{t}^{\alpha }f\) and we call it L1-2-3 operator. Looking at (3) and (4), we understand that at \(t=t_{k}\) the new L1-2-3 operator \(\widetilde{{\mathbb {D}}}_{t}^{\alpha }f\) is actually a modification of the L1-2 operator \({\mathbb {D}}_{t}^{\alpha }f\) by adding a correction term \(\frac{\, \Delta t ^{3-\alpha }}{\Gamma (2-\alpha )}\, \sum\limits_{j=3}^{k}\beta _{k-j}^{(\alpha )} \delta _{t}^{3}f_{j-\frac{3}{2}}\) for \(k\ge 3\). The following lemma remarks the properties of coefficient \(\{\beta _{j}^{(\alpha )}\}\) in the correction term.

Lemma 2.1

For any\(\alpha\), we have

  1. (i)

    \(\beta _{j}^{(\alpha )}>0,\, j\ge 0\);

  2. (ii)

    \(\beta _{0}^{(\alpha )} \ge \beta _{1}^{(\alpha )} \ge \beta _{2}^{(\alpha )}\ge \cdots\), i.e., \(\{\beta _{j}^{(\alpha )} \}\)is strictly monotonically decreasing with respect toj.

Proof

The only root of \(\Phi _{j}\) in \([t_{j-1}, t_{j}]\) is \(t_{j}^{*}=t_{j-1}+\Delta t/\sqrt{3}\). \(\Phi _{j}\) is an increasing function on \([t_{j-1}, t_{j}]\). Moreover, \(\int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, \mathrm{d}s = \int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, \mathrm{d}s = 2 \sqrt{3} \Delta t^3/9 = A\), and \((t_{k}-s)^{-\alpha }\) is a continuous, positive and increasing function of s on \([t_{j-1}, t_{j}]\). Using the weighted mean value theorem for integrals [3], we have

  1. (a)

    there is \(c_{1}\in [t_{j-1}, t_{j}^{*}]\) such that

    $$\begin{aligned}&0< \int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, (t_{k}-s)^{-\alpha }\, \mathrm{d}s\, = (t_{k}-c_{1})^{-\alpha }\, \int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, \mathrm{d}s\, = A\, (t_{k}-c_{1})^{-\alpha } < A\,(t_{k}-t_{j}^{*})^{-\alpha }, \end{aligned}$$
  2. (b)

    there is \(c_{2}\in [t_{j}^{*}, t_{j}]\) such that

    $$\begin{aligned}&\int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, (t_{k}-s)^{-\alpha }\, \mathrm{d}s\, = (t_{k}-c_{2})^{-\alpha }\, \int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, \mathrm{d}s\, = A\, (t_{k}-c_{2})^{-\alpha } > A\, (t_{k}-t_{j}^{*})^{-\alpha }. \end{aligned}$$

Therefore,

$$\begin{aligned}0&<\int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, (t_{k}-s)^{-\alpha }\, \mathrm{d}s\, < \int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, (t_{k}-s)^{-\alpha }\, \mathrm{d}s, \end{aligned}$$

and it means that \(\beta _{j}^{(\alpha )}>0\). To prove that \(\{\beta _{j}^{(\alpha )} \}\) is strictly monotonically decreasing with respect to j, we must show

$$\begin{aligned}0&<\int _{t_{j-1}}^{t_{j}}\Phi _{j}(s)\, (t_{k}-s)^{-\alpha }\, \mathrm{d}s\, < \int _{t_{j}}^{t_{j+1}}\Phi _{j+1}(s)\, (t_{k}-s)^{-\alpha }\, \mathrm{d}s. \end{aligned}$$
(5)

Again, using the weighted mean value theorem for integrals, we can write

  1. (a)

    there is \(c_{1}\in [t_{j-1}, t_{j}^{*}]\) such that

    $$\begin{aligned}0&< \int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, g(s)\, \mathrm{d}s\,= g(c_{1})\, \int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, \mathrm{d}s\, = A\, g(c_{1}), \end{aligned}$$
  2. (b)

    there is \(c_{2}\in [t_{j}^{*}, t_{j}]\) such that

    $$\begin{aligned}0&< \int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, g(s)\, \mathrm{d}s\, = g(c_{2})\, \int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, \mathrm{d}s\, = A\, g(c_{2}), \end{aligned}$$

where \(g(s) = (t_{k} - s - \Delta t)^{-\alpha } - (t_{k} - s)^{-\alpha }\) which is an increasing function of s on \([t_{j-1}, t_{j}]\) since \(g^{'}(s) = \alpha \, ( (t_{k} - s - \Delta t)^{-(\alpha +1)} - (t_{k} - s)^{-(\alpha +1)})>0\). Therefore, \(A\, g(c_{1}) < A\, g(c_{2})\), and it means that

$$\begin{aligned}&0< \int _{t_{j-1}}^{t_{j}^{*}}-\Phi _{j}(s)\, g(s)\, \mathrm{d}s\, < \int _{t_{j}^{*}}^{t_{j}}\Phi _{j}(s)\, g(s)\, \mathrm{d}s. \end{aligned}$$

Then,

$$\begin{aligned}&0 < \int _{t_{j-1}}^{t_{j}}\Phi _{j}(s)\, g(s)\, \mathrm{d}s\, = \int _{t_{j-1}}^{t_{j}} \Phi _{j}(s)\, (t_{k} - s - \Delta t)^{-\alpha } - \Phi _{j}(s)\, (t_{k} - s)^{-\alpha } \, \mathrm{d}s. \end{aligned}$$

Knowing \(\Phi _{j}(s) =\Phi _{j}(s+\Delta t), \, s \in [t_{j-1}, t_{j}]\), we have

$$\begin{aligned} 0 < \int _{t_{j}}^{t_{j+1}} \Phi _{j+1}(s )\, (t_{k} - s)^{-\alpha } - \int _{t_{j-1}}^{t_{j}}\Phi _{j}(s)\, (t_{k} - s)^{-\alpha } \, \mathrm{d}s. \end{aligned}$$

Therefore, (5) holds, and so the proof is complete.

The new, improved fractional numerical differentiation formula (4) can be rewritten as

$$\begin{aligned} \widetilde{{\mathbb {D}}}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}}& = {\frac{\Delta t ^{1-\alpha }}{\Gamma (2-\alpha )}} \bigg ( {\sum ^{k}_{j=1}} a_{k-j}^{(\alpha )} (\delta _{t}\,f_{j-\frac{1}{2}}) + {\sum ^{k}_{j=2}}b_{k-j}^{(\alpha )} (\delta _{t}\,f_{j-\frac{1}{2}} - \delta _{t}\,f_{j-\frac{3}{2}}) \nonumber \\& \quad + \, {\sum _{j=3}^{k}}\beta _{k-j}^{(\alpha )} (\delta _{t}\,f_{j-\frac{1}{2}} - 2\,\delta _{t}\,f_{j-\frac{3}{2}} + \delta _{t}\,f_{j-\frac{5}{2}}) \bigg )\nonumber \\& = {\frac{\Delta t ^{1-\alpha }}{\Gamma (2-\alpha )}} \bigg ({\sum ^{k}_{j=1}} a_{k-j}^{(\alpha )} (\delta _{t}\,f_{j-\frac{1}{2}}) + {\sum ^{k}_{j=2}}b_{k-j}^{(\alpha )} \delta _{t}\,f_{j-\frac{1}{2}} - {\sum ^{k-1}_{j=1}}b_{k-j-1}^{(\alpha )} \delta _{t}\,f_{j-\frac{1}{2}}\nonumber \\& \quad + \, {\sum _{j=3}^{k}}\beta _{k-j}^{(\alpha )} \delta _{t}\,f_{j-\frac{1}{2}} -2 {\sum _{j=2}^{k-1}}\beta _{k-j-1}^{(\alpha )} \delta _{t}\,f_{j-\frac{1}{2}} + {\sum _{j=1}^{k-2}}\beta _{k-j-2}^{(\alpha )} \delta _{t}\,f_{j-\frac{1}{2}}\bigg )\nonumber \\& = {\frac{\Delta t ^{1-\alpha }}{\Gamma (2-\alpha )} } {\sum _{j=1}^{k}} \gamma _{k-j}^{(\alpha )}\delta _{t}\,f_{j-\frac{1}{2}}\nonumber \\& = {\frac{\, \Delta t ^{-\alpha }}{\Gamma (2-\alpha )}} \bigg (\gamma _{0}^{(\alpha )}\, f_{k} - {\sum _{j=1}^{k-1}}(\gamma _{k-j-1}^{(\alpha )} - \gamma _{k-j}^{(\alpha )})\, f_{j} - \gamma _{k-1}^{(\alpha )}\, f_{0}\bigg ), \end{aligned}$$
(6)

where for \(k=1\), \(\gamma _{0}^{(\alpha )} = 1\); for \(k=2\), \(\gamma _{0}^{(\alpha )}=a_{0}^{(\alpha )} + b_{0}^{(\alpha )}, \gamma _{1}^{(\alpha )}=a_{1}^{(\alpha )} - b_{0}^{(\alpha )}\); for \(k= 3\),

$$\begin{aligned} \begin{array}{ll} \gamma _{l}^{(\alpha )} & = \left\{ \begin{array}{lr} a_{l}^{(\alpha )} + b_{l}^{(\alpha )} + \beta _{l}^{(\alpha )}, & l=0,\\ a_{l}^{(\alpha )} + b_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} -2\beta _{l-1}^{(\alpha )}, & l=1,\\ a_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} + \beta _{l-2}^{(\alpha )}, & l=2,\\ \end{array} \right. \end{array} \end{aligned}$$

and for \(k\ge 4\),

$$\begin{aligned} \begin{array}{rl} \gamma _{l}^{(\alpha )} & = \left\{ \begin{array}{lr} a_{l}^{(\alpha )} + b_{l}^{(\alpha )} + \beta _{l}^{(\alpha )}, & l=0,\\ a_{l}^{(\alpha )} + b_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} + \beta _{l}^{(\alpha )} -2\beta _{l-1}^{(\alpha )}, & l=1,\\ a_{l}^{(\alpha )} + b_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} + \beta _{l}^{(\alpha )} -2\beta _{l-1}^{(\alpha )} + \beta _{l-2}^{(\alpha )}, & 2\le l \le k-3,\\ a_{l}^{(\alpha )} + b_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} -2\beta _{l-1}^{(\alpha )} + \beta _{l-2}^{(\alpha )}, & l=k-2,\\ a_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} + \beta _{l-2}^{(\alpha )}, & l=k-1. \end{array} \right. \end{array} \end{aligned}$$
(7)

For \(k=1\), (4) is the same as (2); for \(k=2\), (4) is the same as (3); for \(k=3\),

$$\begin{aligned} \gamma _{0}^{(\alpha )}& = \, \frac{1}{3}+\frac{1}{2-\alpha }+\frac{1}{(2-\alpha )\,(3-\alpha )} \in \left( 1,\frac{11}{6}\right) , \\ \\ \gamma _{1}^{(\alpha )}= \,& 2^{-\alpha }+\frac{2^{2-\alpha }-2}{2-\alpha } -\frac{2}{3} -\frac{2}{(2-\alpha )\, (3-\alpha )}\in \left( -\frac{7}{6},1\right) , \\ \\ \gamma _{2}^{(\alpha )}& = \, 3^{1-\alpha }-2^{-\alpha }-\frac{2^{2-\alpha }-1}{2-\alpha }+\frac{1}{3}+\frac{1}{(2-\alpha )\, (3-\alpha )} \in (0,1). \end{aligned}$$

Lemma 2.2

For any\(\alpha\) and \(\gamma _{j}^{(\alpha )} (0\le j \le k-1, k\ge 4)\)defined in  (7), we have

  1. (i)

    \(\gamma _{0}^{(\alpha )} > \vert \gamma _{1}^{(\alpha )}\vert\);

  2. (ii)

    \(\gamma _{j}^{(\alpha )} >0,\ j\ne 1\);

  3. (iii)

    \(\gamma _{2}^{(\alpha )} \ge \gamma _{3}^{(\alpha )} \ge \cdots \ge \gamma _{k-3}^{(\alpha )}\)\(\ge \gamma _{k-2}^{(\alpha )}\ge \gamma _{k-1}^{(\alpha )}\);

  4. (iv)

    \(\gamma _{0}^{(\alpha )} > \gamma _{2}^{(\alpha )}\);

  5. (v)

    \(\sum\limits_{j=0}^{k-1} \gamma _{j}^{(\alpha )} = k^{1-\alpha }\).

Proof

For any \(\alpha\) and \(k\ge 4\), noticing \(\gamma _{k}^{(\alpha )}\) in (7), we have

$$\begin{aligned} \begin{array}{c} \gamma _{0}^{(\alpha )}= {\frac{1}{3}+\frac{1}{2-\alpha }+\frac{1}{(2-\alpha )\,(3-\alpha )} \in \left( 1,\frac{11}{6}\right) }, \qquad \gamma _{1}^{(\alpha )} = \displaystyle {\frac{2^{1-\alpha }}{3} + \frac{2^{2-\alpha }-3}{2-\alpha } + \frac{2^{3-\alpha }-3}{(2-\alpha )(3-\alpha )} -1}. \end{array} \end{aligned}$$

Let \(g(x) = \frac{2^{1-x}}{3} + \frac{2^{2-x}-3}{2-x} + \frac{2^{3-x}-3}{(2-x)(3-x)} -1\). Then, \(g^{'}(x)<0\) for \(x\in (0, 1)\). Therefore, g is monotonically decreasing on (0, 1), and then \(-\frac{7}{6} = \gamma _{1}^{(1)}<\gamma _{1}^{(\alpha )}<\gamma _{1}^{(0)}=1\); thus, \(\gamma _{0}^{(\alpha )} > \vert \gamma _{1}^{(\alpha )}\vert\), i.e., (i) holds. Moreover, the root of g on (0, 1) is about \(x_{0}=0.532\,49\), thus, \(\gamma _{1}^{(\alpha )}>0\) for \(\alpha <x_{0}\), and \(\gamma _{1}^{(\alpha )}<0\) for \(\alpha >x_{0}\). For \(2\le j \le k-3,\)

$$\begin{aligned} \gamma _{j}^{(\alpha )}& = \, a_{l}^{(\alpha )} + b_{l}^{(\alpha )} - b_{l-1}^{(\alpha )} + \beta _{l}^{(\alpha )} -2\beta _{l-1}^{(\alpha )} + \beta _{l-2}^{(\alpha )}\\& = \displaystyle { \bigg (\frac{1}{3}((j+1)^{1-\alpha }-2j^{1-\alpha }+(j-1)^{1-\alpha })\, + \, \frac{1}{2-\alpha }((j+1)^{2-\alpha }-2j^{2-\alpha }+(j-1)^{2-\alpha })}\, \\& \quad +\displaystyle {\, \frac{1}{(2-\alpha )(3-\alpha )}((j+1)^{3-\alpha }-2j^{3-\alpha }+(j-1)^{3-\alpha }) \bigg ) }\\& \quad - \displaystyle { \bigg (\frac{1}{3}(j^{1-\alpha }-2(j-1)^{1-\alpha }+(j-2)^{1-\alpha })\, + \, \frac{1}{2-\alpha }(j^{1-\alpha }-2(j-1)^{1-\alpha }+(j-2)^{1-\alpha })}\, \\& \quad +\, \displaystyle {\frac{1}{(2-\alpha )(3-\alpha )}(j^{1-\alpha }-2(j-1)^{1-\alpha }+(j-2)^{1-\alpha }) \bigg ) }\\& = \, I_{j} - I_{j-1}, \end{aligned}$$

where

$$\begin{aligned} I_{j}& = \displaystyle { \frac{1}{3}((j+1)^{1-\alpha }-2j^{1-\alpha }+(j-1)^{1-\alpha })\, + \, \frac{1}{2-\alpha }((j+1)^{2-\alpha }-2j^{2-\alpha }+(j-1)^{2-\alpha })}\, \\& \quad +\displaystyle {\, \frac{1}{(2-\alpha )(3-\alpha )}((j+1)^{3-\alpha }-2j^{3-\alpha }+(j-1)^{3-\alpha }) }\\& = \displaystyle { \bigg ( \frac{1}{3} (j+1)^{1-\alpha } + \frac{1}{2-\alpha }(j+1)^{2-\alpha } + \frac{1}{(2-\alpha )(3-\alpha )}(j+1)^{3-\alpha }\bigg )}\\& \quad -2\, \displaystyle {\bigg (\frac{1}{3} j^{1-\alpha } + \frac{1}{2-\alpha }j^{2-\alpha } + \frac{1}{(2-\alpha )(3-\alpha )}j^{3-\alpha }\bigg )}\\& \quad +\, \displaystyle {\bigg (\frac{1}{3} (j-1)^{1-\alpha } + \frac{1}{2-\alpha }(j-1)^{2-\alpha } + \frac{1}{(2-\alpha )(3-\alpha )}(j-1)^{3-\alpha }\bigg )},\quad j\ge 1. \end{aligned}$$

Let \(h(x) = \frac{1}{3} x^{1-\alpha } + \frac{1}{2-\alpha }x^{2-\alpha } + \frac{1}{(2-\alpha )(3-\alpha )}x^{3-\alpha }, \, x\ge 1\). Then \(I_{j} = h(j+1) - 2 h(j) + h(j-1)\). For \(x\ge 1\), we have \(h^{'}(x)>0,\,h^{''}(x)>0,\,h^{'''}(x)>0\), and \(h^{(4)}(x)<0\). Therefore,

$$\begin{aligned} \gamma _{j}^{(\alpha )}& = I_{j}-I_{j-1}=h(j+1)-3\, h(j)+3\, h(j-1)-h(j-2)=h^{'''}(\zeta _{j})>0,\\ \\ \gamma _{j}^{(\alpha )}-\gamma _{j+1}^{(\alpha )}& = -h(j+2)+4\, h(j+1)-6\, h(j)+4\, h(j-1)-h(j-2)=-h^{(4)}(\sigma _{j})>0, \end{aligned}$$

where \(j-2<\zeta _{j} < j+1\) and \(j-2<\sigma _{j} < j+2\). That is to say that \(\gamma _{2}^{(\alpha )} \ge \gamma _{3}^{(\alpha )} \ge \cdots \ge \gamma _{k-3}^{(\alpha )}\). In addition,

$$\begin{aligned} \gamma _{k-3}^{(\alpha )} - \gamma _{k-2}^{(\alpha )} = -h(k-1) +3\, h(k-2) - 6\,h(k-3) + 4\, h(k-4) - h(k-5) + \frac{(k-1)^{3-\alpha }}{(2-\alpha )(3-\alpha )} \end{aligned}$$

is continuous for all \(\alpha \in [0,1]\), then it attains its maximum and minimum values on [0, 1]. Because the minimum of this function on [0, 1] is zero, \(\gamma _{k-3}^{(\alpha )} - \gamma _{k-2}^{(\alpha )}\ge 0\), so \(\gamma _{k-3}^{(\alpha )} \ge \gamma _{k-2}^{(\alpha )}\). Similarly,

$$\begin{aligned} \gamma _{k-2}^{(\alpha )} - \gamma _{k-1}^{(\alpha )} = -3\, h(k-2) + 4\, h(k-3) - h(k-4) - k ^{1-\alpha } + (k-1)^{1-\alpha } + \frac{2\,(k-1)^{2-\alpha }}{2-\alpha } \end{aligned}$$

is continuous for all \(\alpha \in [0,1]\), then due to attaining its minimum values on [0, 1], which is zero, \(\gamma _{k-2}^{(\alpha )} - \gamma _{k-1}^{(\alpha )}\ge 0\), then \(\gamma _{k-2}^{(\alpha )} \ge \gamma _{k-1}^{(\alpha )}\). Moreover,

$$\begin{aligned} \begin{array}{cl} \gamma _{k-2}^{(\alpha )}&= -2\,h(k-2) + 3\, h(k-3) - h(k-4) + \frac{1}{2}\,(k-1)^{1-\alpha } + \frac{(k-1)^{2-\alpha }}{2-\alpha }:= p(\alpha ) \end{array} \end{aligned}$$

is continuous for all \(\alpha \in [0, 1]\). Because \(p^{'}(\alpha )<0\) on [0, 1], this function is decreasing on [0, 1], and then \(0 = p(1) < p(\alpha )\) for all \(\alpha \in (0,1)\); thus, \(\gamma _{k-2}^{(\alpha )}>0\). In a similar way,

$$\begin{aligned} \begin{array}{cl} \gamma _{k-1}^{(\alpha )}&= h(k-2) - h(k-3) + k^{1-\alpha } - \frac{1}{2}\,(k-1)^{1-\alpha } - \frac{(k-1)^{2-\alpha }}{2-\alpha }:= q(\alpha ) \end{array} \end{aligned}$$

is continuous, thus for all \(\alpha \in [0, 1]\). Because \(q^{'}(\alpha )<0\) on [0, 1], \(0 = q(1) < q(\alpha )\) for all \(\alpha \in (0,1)\), \(\gamma _{k-1}^{(\alpha )}>0\). That is to say, (ii) and (iii) hold. Due to the fact that \(\gamma _{0}^{(\alpha )} - \gamma _{2}^{(\alpha )} = 3\, h(2) - 2\, h(1) - h(3)\) is positive, continuous, and increasing on (0, 1), we have \(\gamma _{0}^{(\alpha )} > \gamma _{2}^{(\alpha )}\), so (iv) holds. The validity of (v) can be directly derived from the definition (7) of \(\gamma _{j}^{(\alpha )}\). The proof is completed.

The truncation errors of the new L1-2-3 formula (6) are illustrated in the following theorem.

Theorem 2.1

Suppose\(f\in C^{4}[0,t_{k}]\). For any\(\alpha\), \(\widetilde{{\mathbb {D}}}_{t}^{\alpha }f(t) \vert_{t=t_{k}}\) is defined in (6). Denote\(\vert R(\,f(t_{k})) \vert = _{0}^{C}\mathcal{D}_{t}^{\alpha }f(t) \vert_{t=t_{k}} - \widetilde{{\mathbb {D}}}_{t}^{\alpha }f(t) \vert_{t=t_{k}}\). Then, we have

$$\begin{aligned} \begin{array}{rcl} \vert R(\,f(t_{1})) \vert \le & \frac{\alpha }{2\, \Gamma (3-\alpha )}& \Delta t ^{2-\alpha } \, \displaystyle {\max _{t_{0}\le t \le t_{1}}} f^{''}(t) ,\\ \vert R(\,f(t_{2})) \vert \le & \frac{\alpha }{ \Gamma (1-\alpha )} &\left( \, \frac{1}{12}\displaystyle {\max _{t_{0}\le t \le t_{1}}}\vert\,f^{''}(t)\vert \, (t_{2}-t_{1})^{-\alpha -1}\,\Delta t ^{3}\right. \\ & & \left. + \frac{1}{3(1-\alpha )(2-\alpha )} \left( \frac{1 }{2} + \frac{1}{3-\alpha }\right) \, \displaystyle {\max _{t_{0}\le t \le t_{2}}}\vert\, f^{'''}(t) \vert \, \Delta t ^{3-\alpha }\, \right) ,\\ \vert R(\,f(t_{k}))\vert \le & \frac{\alpha }{\Gamma (1-\alpha )} & \left( \, 12\, (t_{k}-t_{1})^{-\alpha -1}\displaystyle {\max _{t_{0}\le t\le t_{1}}}\vert\,f^{''}(t)\vert \, \Delta t ^{3} +\frac{1}{8}(t_{k}-t_{2})^{-\alpha -1}\,\displaystyle {\max _{t_{0}\le t\le t_{2}}} \vert\,f^{'''}(t)\vert \Delta t^{4} \right. \\ & & \left. + \left( \frac{1}{2} + \frac{1}{12}\,\frac{27-10\alpha + \alpha ^{2}}{\prod _{i=1}^{4}(\alpha -i)}\right) \displaystyle { \max _{t_{0}\le t\le t_{k}} }\vert \,f^{(4)}(t)\vert \Delta t^{4-\alpha }\, \right) ,\qquad k\ge 3. \end{array} \end{aligned}$$
(8)

Proof

For \(k=1\), (6) is just the L1 formula, and we have

$$\begin{aligned} R(\,f(t_{1})) = \frac{1}{\Gamma (1-\alpha )} \displaystyle {\int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))]^{'}(t_{1}-s)^{-\alpha }\, \mathrm{d}s = \frac{\alpha }{2\, \Gamma (3-\alpha )} f^{''}(\eta _{1}) \Delta t ^{2-\alpha }, \end{aligned}$$

where \(\eta _{1}\in (t_{0}, t_{1})\). For \(k=2\), (6) is just the L1-2 formula, and we have

$$\begin{aligned} R(\,f(t_{2}))& = \frac{1}{\Gamma (1-\alpha )}\Bigg \{\displaystyle {\int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(f(s))]^{'}(t_{2}-s)^{-\alpha }\, \mathrm{d}s + \displaystyle {\int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))]^{'}(t_{2}-s)^{-\alpha }\, \mathrm{d}s\Bigg \} \\& = \frac{-\alpha }{\Gamma (1-\alpha )}\Bigg \{ \displaystyle { \int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))](t_{2}-s)^{-\alpha -1}\, \mathrm{d}s + \displaystyle { \int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))](t_{2}-s)^{-\alpha -1}\, \mathrm{d}s\Bigg \}, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} \bigg \vert \displaystyle { \int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))](t_{2}-s)^{-\alpha -1}\, \mathrm{d}s \bigg \vert & = \bigg \vert \displaystyle { \int _{t_{0}}^{t_{1}}} \frac{f^{''}(\xi _{1})}{2}\, (s-t_{0})(s-t_{1})(t_{2}-s)^{-\alpha -1}\, \mathrm{d}s \bigg \vert \\ & = \bigg \vert \frac{1}{2} f^{''}(\eta _{1})\, \displaystyle { \int _{t_{0}}^{t_{1}}} (s-t_{0})(s-t_{1})(t_{2}-s)^{-\alpha -1}\, \mathrm{d}s \bigg \vert \\ & \le \frac{1}{12}\, (t_{2}-t_{1})^{-\alpha -1} \, \Delta t ^{3}\displaystyle {\max _{t_{0}\le t \le t_{1}}} \vert \,f^{''}(t) \vert , \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{rl} \displaystyle { \int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))](t_{2}-s)^{-\alpha -1}\, \mathrm{d}s & = \displaystyle { \int _{t_{1}}^{t_{2}}} \frac{f^{'''}(\xi _{2})}{6}\, (s-t_{0})(s-t_{1})(t_{2}-s)^{-\alpha }\, \mathrm{d}s \\ & = \frac{1}{6} f^{'''}(\eta _{2})\, \displaystyle { \int _{t_{1}}^{t_{2}}} (s-t_{0})(s-t_{1})(t_{2}-s)^{-\alpha }\, \mathrm{d}s \\ & = -\frac{1}{6} \vert\, f^{'''}(\eta _{2}) \vert \, \Delta t ^{3-\alpha }\, \frac{2}{(1-\alpha )(2-\alpha )} \left( \frac{1 }{2} + \frac{1}{3-\alpha}\right) , \end{array} \end{aligned}$$

where \(\eta _{2}\in (t_{1}, t_{2})\). For \(k\ge 3\), we get

$$\begin{aligned} R(\,f(t_{k}))& = \frac{1}{\Gamma (1-\alpha )} \Bigg \{ \displaystyle {\int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))]^{'}(t_{k}-s)^{-\alpha }\, \mathrm{d}s + \displaystyle {\int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))]^{'}(t_{k}-s)^{-\alpha }\, \mathrm{d}s \\& \quad +\displaystyle {\sum _{j=3}^{k}}\, \displaystyle {\int _{t_{j-1}}^{t_{j}}} [\,f(s) - \Pi _{3,j}(\,f(s))]^{'}(t_{k}-s)^{-\alpha }\, \mathrm{d}s\, \Bigg \}\\& = \frac{1}{\Gamma (1-\alpha )} \Bigg \{\, [\,f(s) - \Pi _{1,1}(\,f(s))](t_{k}-s)^{-\alpha }\bigg \vert_{t_{0}}^{t_{1}} - \alpha \,\displaystyle {\int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \\& \quad + [\,f(s) - \Pi _{2,2}(\,f(s))](t_{k}-s)^{-\alpha }\bigg \vert_{t_{1}}^{t_{2}} - \alpha \,\displaystyle {\int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \\& \quad +\displaystyle {\sum _{j=3}^{k}}\Bigg\{\, [\,f(s) - \Pi _{3,j}(\,f(s))](t_{k}-s)^{-\alpha }\bigg \vert_{t_{j-1}}^{t_{j}} \\& \quad - \alpha \,\displaystyle {\int _{t_{j-1}}^{t_{j}}} [\,f(s) - \Pi _{3,j}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s\Bigg \} \, \Bigg \}\\& = \frac{-\alpha }{\Gamma (1-\alpha )} \Bigg \{\, \displaystyle {\int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \\& \quad + \displaystyle {\int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \\& \quad +\displaystyle {\sum _{j=3}^{k}} \Bigg\{\displaystyle {\int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \Bigg\}\, \Bigg \}, \end{aligned}$$

where

$$\begin{aligned}\left[\,f(s)-\Pi_{3,k}(\,f(s))\right](t_{k}-s)^{-\alpha }\bigg\vert_{t_{k-1}}^{t_{k}} = -\frac{1}{4!}\, f^{(4)}(\vartheta _{k})\, (s-t_{k-3})(s-t_{k-2})(s-t_{k-1})(t_{k}-s)^{1-\alpha }\, \bigg\vert_{t_{k-1}}^{t_{k}} = 0 \end{aligned}$$

is used. Moreover,

$$\begin{aligned} \left \vert {\int _{t_{0}}^{t_{1}}} [\,f(s) - \Pi _{1,1}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \right \vert \, & = \left \vert {\int _{t_{0}}^{t_{1}}} \frac{1}{2}f^{''}(\xi _{1})\, (s-t_{0})(s-t_{1})(t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \right \vert \\ & = \left \vert \frac{1}{2}f^{''}(\eta _{1})\, {\int _{t_{0}}^{t_{1}}} (s-t_{0})(s-t_{1})(t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \right \vert \\ & \le \frac{1}{8} \vert\,f^{''}(\eta _{1}) \vert \, \Delta t ^{2}\, { \int _{t_{0}}^{t_{1}}}(t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \\ & \le 12 \vert\,f^{''}(\eta _{1}) \vert (t_{k}-t_{1})^{-\alpha -1} \Delta t^{3}, \end{aligned}$$

where \(\eta _{1}\in (t_{0}, t_{1})\),

$$\begin{aligned} \begin{array}{rl} \bigg \vert \displaystyle {\int _{t_{1}}^{t_{2}}} [\,f(s) - \Pi _{2,2}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \bigg \vert \, & = \bigg \vert \displaystyle {\int _{t_{1}}^{t_{2}}} \frac{1}{6}f^{'''}(\xi _{2})\, (s-t_{0})(s-t_{1})(s-t_{2})(t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \bigg \vert \\ & = \bigg \vert \frac{1}{6}f^{'''}(\eta _{2})\, \displaystyle {\int _{t_{1}}^{t_{2}}} (s-t_{0})(s-t_{1})(s-t_{2})(t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \bigg \vert \\ & \le \frac{1}{12} \vert\, f^{'''}(\eta _{2}) \vert \Delta t^{3}\, \displaystyle {\int _{t_{1}}^{t_{2}}}(t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \\ & \le \frac{1}{8} \vert\,f^{'''}(\eta _{2}) \vert \, (t_{k}-t_{2})^{-\alpha -1} \Delta t^{4},\\ \end{array} \end{aligned}$$

where \(\eta _{2}\in (t_{1}, t_{2})\), and

$$\begin{aligned} \begin{array}{rl} \bigg \vert \displaystyle {\sum _{j=3}^{k-1} }\displaystyle {\int _{t_{j-1}}^{t_{j}}} [\,f(s) - \Pi _{3,j}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \bigg \vert \, & = \bigg \vert \displaystyle {\sum _{j=3}^{k-1} }\displaystyle {\int _{t_{j-1}}^{t_{j}}} \frac{1}{4!}f^{(4)}(\xi _{j})\, \prod _{i=0}^{3} (s-t_{j-i}) (t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \bigg \vert \\ & = \frac{1}{4!} \vert\,f^{(4)}(\eta ) \vert \, \displaystyle {\sum _{j=3}^{k-1} }\displaystyle {\int _{t_{j-1}}^{t_{j}}} \prod _{i=0}^{3} (s-t_{j-i})(t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \\ & \le \frac{1}{4!} \vert\,f^{(4)}(\eta ) \vert \Delta t^{4}\, \displaystyle {\int _{t_{2}}^{t_{k-1}}}(t_{k}-s)^{-\alpha -1}\, \mathrm{d}s\\ & \le \frac{1}{2} \vert\,f^{(4)}(\eta ) \vert \Delta t^{4-\alpha }, \end{array} \end{aligned}$$

where \(\eta _{j}\in (t_{j-3}, t_{j}),\ 3\le j \le k-1\), \(\eta \in (t_{0}, t_{k-1})\). In addition,

$$\begin{aligned} \begin{array}{rl} \displaystyle {\int _{t_{k-1}}^{t_{k}}} [\,f(s) - \Pi _{3,k}(\,f(s))](t_{k}-s)^{-\alpha -1}\, \mathrm{d}s \, & = \displaystyle {\int _{t_{k-1}}^{t_{k}}} \frac{1}{4!}\, f^{(4)}(\nu _{k})\, \prod _{i=0}^{3} (s-t_{k-i}) (t_{k}-s)^{-\alpha -1} \, \mathrm{d}s \\ & = -\frac{1}{4!}\, f^{(4)}(\eta _{4}) \, \displaystyle {\int _{t_{k-1}}^{t_{k}}} \prod _{i=1}^{3} (s-t_{k-i})(t_{k}-s)^{-\alpha } \, \mathrm{d}s \\ & = -\frac{1}{4!}\, f^{(4)}(\eta _{4})\, \Delta t^{4-\alpha }\, \left( \frac{2(27-10\alpha + \alpha ^{2})}{\prod _{i=1}^{4}(\alpha -i)}\right) , \end{array} \end{aligned}$$

where \(\eta _{k}\in (t_{k-3}, t_{k})\). Hence, (8) holds.

Numerical Examples

To verify our constructed numerical formula for the Caputo fractional derivative and show that L1-2-3 formula (6) can act better than the L1 and L1-2 formulae, we test the following three numerical examples. Although the first two examples (\(f(t)=t^{4+r}\) and \(f(t)=t^3\)) are polynomial-based examples with zero initial values, the convergence order of them is different. In addition, the third one (\(f(t)=\exp (2t)\)) is a non-polynomial example with non-zero initial value and its exact solution does not have a closed form. Actually, we aim to present these three examples to show that the convergence order of the formula depends on the function f. In fact, the convergence order of the following three examples are, respectively, \(4-\alpha\), 4 and 3.

For \(0\le k \le N,\) denote

$$\begin{aligned} \begin{array}{c} F^{k} = _{0}^{C}\!\!\mathcal{D}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}},\quad f_{1}^{k}= D_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}},\quad f_{2}^{k}={\mathbb {D}}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}},\quad f_{3}^{k}= \widetilde{{\mathbb {D}}}_{t}^{\alpha }f(t)\bigg \vert_{t=t_{k}} ,\\ \\ E_{j}^{N} = \vert F^{N} - f_{j}^{N} \vert ,\quad \mathop {\mathrm{order}}_{j} = \displaystyle {\log _{2}\frac{E_{j}^{N}(\Delta t)}{E_{j}^{N}(\Delta t / 2)}},\qquad j=1, 2, 3 \end{array} \end{aligned}$$

and assume that \(t_{N}=T=1\).

Example 1

Let \(f(t) = t^{4+r}\), where \(0\le r<1\). We compute the \(\alpha\)-order Caputo fractional derivative of f(T) numerically. The exact solution is given by

$$\begin{aligned} _{0}^{C}\mathcal{D}_{t}^{\alpha }f(t)\bigg \vert_{t=1}\, = \frac{ t^{4-\alpha +r}\,\Gamma (5+r)}{\Gamma (5-\alpha +r)}\bigg \vert_{t=1}\,=\frac{\Gamma (5+r)}{\Gamma (5-\alpha +r)}. \end{aligned}$$

We use three methods presented in Sect. 2, and list the computational errors and numerical convergence orders for different parameters \(\alpha = 0.9, 0.5, 0.1\) and different temporal step sizes in Table 1. Noticing Table 1, we find that the computed errors by the L1-2-3 formula (6) are obviously much smaller than that by the L1-2 and L1 formulae (3) and (2), and the attainable convergence order of the L1-2-3 formula (6) is \(4-\alpha\), which is also higher than the convergence order of \(3-\alpha\) and \(2-\alpha\) for the L1-2 and L1 formulae (3) and (2). From the above test example, we can draw the conclusion that the computational efficiency and numerical accuracy of the new formula (6) are superior to that of the L1-2 formula (3) and L1 formula (2), and the convergence order of (6) reaches \(4-\alpha\) if \(N\ne 1\) and N is relatively large.

Table 1 (Example 1) Computational errors and convergence orders

Example 2

Consider \(f(t) = t^3\). We compute the \(\alpha\)-order Caputo fractional derivative of f(T) numerically. The exact solution is given by

$$\begin{aligned} _{0}^{C}\mathcal{D}_{t}^{\alpha }f(t)\bigg \vert_{t=1}\, = \frac{6}{\Gamma (4-\alpha )}\, t^{3-\alpha }\bigg \vert_{t=1}\, =\frac{6}{\Gamma (4-\alpha )}. \end{aligned}$$

Taking the same varying temporal step sizes and parameters \(\alpha\) as in Example 1, we use formulae (2), (3), and (6), and list the computational errors and numerical convergence orders in Table 2. According to (8), for \(f(t)=t^3\), we have

$$\begin{aligned} \vert R(\,f(t_{k}))\vert \le \frac{\alpha }{\Gamma (1-\alpha )} ( \, 72\, (t_{k}-t_{1})^{-\alpha -1}+\frac{6}{8}(t_{k}-t_{2})^{-\alpha -1})\, \Delta t^4,\qquad k\ge 3, \end{aligned}$$

which shows that the convergence order of (6) for this example reaches 4 since \((t_{k}-t_{1})^{-\alpha -1}=O(1)\) and \((t_{k}-t_{2})^{-\alpha -1}=O(1)\). The results of Table 2 confirm this fact.

Table 2 (Example 2) Computational errors and convergence orders

Example 3

Suppose \(f(t) = \exp (2t)\). We compute the \(\alpha\)-order Caputo fractional derivative of f(T) numerically. The exact solution is given [10] by

$$\begin{aligned} _{0}^{C}\mathcal{D}_{t}^{\alpha }f(t)\bigg \vert_{t=1}\, = 2t^{1-\alpha }\, E_{1,2-\alpha }(2t)\bigg \vert_{t=1}\, =2\, E_{1,2-\alpha }(2), \end{aligned}$$

where \(E_{\mu , \nu }(z)\) is the Mittag-Leffler function with two parameters defined by

$$\begin{aligned} E_{\mu , \nu }(z) = \displaystyle {\sum _{n=0}^{\infty } \frac{z^{n}}{\Gamma (\mu n + \nu )}}. \end{aligned}$$

Due to the lack of closed form for the exact solution of this problem, the Mittag-Leffler function \(E_{\mu , \nu }(z)\) which depends on the precise values of the parameters \(\mu\) and \(\nu\) and the argument z has been numerically computed. Table 3 is reported to make a comparison with [4]. According to the results of Table 3, numerical convergence order of formula (6) reaches 3. Also, being determined by (8), for \(f(t)=\exp (2t)\), we have

$$\begin{aligned} \left| {R(\,f(t_{k} ))} \right| &\le \frac{\alpha }{{\Gamma (1 - \alpha )}}\left\{\vphantom{\left. { + 32k\left( {\frac{1}{2} + \frac{1}{{12}}\frac{{(27 - 10\alpha + \alpha ^{2} )}}{{\prod\limits_{{i = 1}}^{4} {(\alpha - i)} }}} \right)\Delta t^{{5 - \alpha }} + 4(t_{k} - t_{2} )^{{ - \alpha - 1}} \Delta t^{5} } \right\}} {48(t_{k} - t_{1} )^{{ - \alpha - 1}} \Delta t^{3} } \right. \hfill \\ &\quad+ 16\left( {\frac{1}{2} + \frac{1}{{12}}\frac{{27 - 10\alpha + \alpha ^{2} }}{{\prod\limits_{{i = 1}}^{4} {(\alpha - i)} }}} \right)\Delta t^{{4 - \alpha }} + (96(t_{k} - t_{1} )^{{ - \alpha - 1}} + (t_{k} - t_{2} )^{{ - \alpha - 1}} )\Delta t^{4} \hfill \\ &\quad \left. { + 32k\left( {\frac{1}{2} + \frac{1}{{12}}\frac{{27 - 10\alpha + \alpha ^{2} }}{{\prod\limits_{{i = 1}}^{4} {(\alpha - i)} }}} \right)\Delta t^{{5 - \alpha }} + 4(t_{k} - t_{2} )^{{ - \alpha - 1}} \Delta t^{5} } \right\},\quad k \ge 3, \hfill \\ \end{aligned}$$

which shows that the convergence order of (6) for this example must be 3,  since \((t_{k}-t_{1})^{-\alpha -1}=O(1)\) and \((t_{k}-t_{2})^{-\alpha -1}=O(1)\). Since truncation error of the new L1-2-3 formula has not been obtained correctly in [4], the authors claimed and showed that the convergence order of (6) for this example is \(4-\alpha ,\) which is not correct. Specifically, for \(\alpha =0.8\), when the temporal step size has decreased to 1/5 120, the convergence order has increased to 3.196 2, which seems to reach \(4-\alpha\). Nevertheless, for this value of \(\alpha\), the convergence order will steadily decrease and reach 3 when \(\Delta t\) decreases to less than 1/40 960. For these three numerical tests, Fig. 1 shows that the computational error curves with \(\alpha = 0.5\) and \(N=20\) using three different formulae, the L1 formula (2), L1-2 formula (3), and L1-2-3 formula (6). According to Fig. 1, it is clearly obvious that the computational results by (6) are better than the ones by (2) and (3).

Table 3 (Example 3) Computational errors and convergence orders
Fig. 1
figure1

The error curves of absolute errors with \(\alpha = 0.5, N=20\)

Time-Fractional Partial Differential Equations

We carry out in this section a series of numerical examples of time-fractional partial differential equations with the Caputo fractional derivatives and present some results to confirm our theoretical statements. The main purpose is to check the convergence behavior of the numerical solution with respect to the time step \(\Delta t\). Consider the time-fractional diffusion equation of the form

$$\begin{aligned} \begin{array}{lr} _{0}^{C}\mathcal{D}_{t}^{\alpha }\, u(x, t) = \displaystyle {\frac{\partial ^{2}u(x, t)}{\partial x^{2}}} + f(x,t),&x \in (0,1), t\in [0, T], \end{array} \end{aligned}$$
(9)

subject to the following boundary and initial conditions:

$$\begin{aligned} \begin{array}{lr} u(0, t)=\phi (t),\, \, u(1,t)=\varphi (t), & t \in (0, T],\\ u(x, 0) = u^{0}(x), & x \in [0,1], \end{array} \end{aligned}$$
(10)

where \(\alpha\) is the order of the Caputo fractional derivative, \(T>0\), and \(f, \phi , \varphi\), and \(u^{0}\) are given functions. Introducing a finite difference approximation to discretize the time-fractional derivative, let \(\Omega _{\Delta t}=\{ t_{k} \vert t_{k}= k\, \Delta t\,, k=0, 1, \ldots , N\},\) where \(\Delta t = T/ N\) is the time step size. By using (6), Eq. (9) is converted to

$$\begin{aligned} \begin{array}{c} \gamma _{0}^{(\alpha )}\, u(x, t_{k}) - \mu \, \displaystyle {\frac{\partial ^{2}u(x,t_{k})}{\partial x^{2}}} = \gamma _{k-1}^{(\alpha )}\, u(x,t_{0}) + \displaystyle {\sum _{i=1}^{k-1}}(\gamma _{i-1}^{(\alpha )}-\gamma _{i}^{(\alpha )})u(x,t_{k-i})\, +\, \mu f(x,t_{k}) + R_{k}, \end{array} \end{aligned}$$
(11)

where \(\mu = (\Delta \, t)^{\alpha } \Gamma (2-\alpha )\), and according to Theorem 2.1 for \(k\ge 3,\)

$$\begin{aligned} \vert R_{k}\vert& = \vert R(u(x,t_{k}))\vert \le \frac{\alpha }{\Gamma (1-\alpha )} \left\{\vphantom{\left. + \left( \frac{1}{2} + \frac{1}{12}\,\frac{(27-10\alpha + \alpha ^{2})}{\prod _{i=1}^{4}(\alpha -i)}\right) \displaystyle { \max _{t_{0}\le t\le t_{k}} }\vert \frac{\partial ^{4}u}{\partial t^{4}}(x,t)\vert \Delta t^{4-\alpha } \right \}}12\, (t_{k}-t_{1})^{-\alpha -1}\displaystyle {\max _{t_{0}\le t\le t_{1}}}\left\vert \frac{\partial ^{2}u}{\partial t^{2}}(x,t)\right\vert \, \Delta t ^{3} \right.\\& \quad +\frac{1}{8}(t_{k}-t_{2})^{-\alpha -1}\,\displaystyle {\max _{t_{0}\le t\le t_{2}}} \left\vert \frac{\partial ^{3}u}{\partial t^{3}}(x,t)\right\vert \Delta t^{4}\\&\left. \quad + \left( \frac{1}{2} + \frac{1}{12}\,\frac{27-10\alpha + \alpha ^{2}}{\prod\limits_{i=1}^{4}(\alpha -i)}\right) \displaystyle { \max _{t_{0}\le t\le t_{k}} }\left\vert \frac{\partial ^{4}u}{\partial t^{4}}(x,t)\right\vert \Delta t^{4-\alpha } \right \}. \end{aligned}$$

Let \(u^{k}=u^{k}(x)\) be the numerical approximation to \(u(x,t_{k})\), and \(f^{k}=f(x,t_{k})\). Then, Eq. (11) can be discretized as the following scheme:

$$\begin{aligned} \begin{array}{c} \gamma _{0}^{(\alpha )}\, u^{k} - \mu \, \displaystyle {\frac{\partial ^{2}u^{k}}{\partial x^{2}}} = \gamma _{k-1}^{(\alpha )}\, u^{0}+ \displaystyle {\sum _{i=1}^{k-1}}(\gamma _{i-1}^{(\alpha )}-\gamma _{i}^{(\alpha )})u^{k-i}\, +\, \mu f^{k}. \end{array} \end{aligned}$$
(12)

To discretize (11) in space, let \(\Omega _{\Delta x}=\{ x_{i} \vert x_{i}= i\, \Delta x, i=0, 1, \ldots , M\}\), where \(\Delta x = 1/ M\) is the space step size. Suppose \(u =\{u_{i}^{k} \vert 0\le i\le M, \, 0\le k\le N \}\) is a grid function on \(\Omega _{\Delta x}\times \Omega _{\Delta t}\), and introduce \(\partial _{x} u^{k}_{i-1/2} = (u^{k}_{i}-u^{k}_{i-1})/\Delta x\), where \(\partial _{x} u^{k}_{i-1/2}\) is the first-order difference quotient of u on the points \((x_{i},t_{k})\) and \((x_{i-1},t_{k})\), and \(\partial _{x}^{2} u^{k}_{i} = (\partial _{x}u^{k}_{i+1/2}-\partial _{x}u^{k}_{i-1/2})/\Delta x\), where \(\partial _{x}^{2} u^{k}_{i}\) is the second-order difference quotient of u on the points \((x_{i-1},t_{k})\), \((x_{i},t_{k})\) and \((x_{i+1},t_{k})\). The finite difference scheme which we consider for (9) and (10) is as follows:

$$\begin{aligned} \left\{ \begin{array}{lr} \widetilde{{\mathbb {D}}}_{t}^{\alpha }u_{i}^{k} = \partial _{x}^{2}u_{i}^{k} + f^{k}, & 1\le i \le M-1,\, 1\le k \le N,\\ u_{0}^{k}=\phi (t_{k}), \, u_{M}^{k} =\varphi (t_{k}),& \, 1\le k \le N,\\ u_{i}^{0}=u^{0} (x_{i}),& \, 0\le i \le M.\\ \end{array} \right. \end{aligned}$$
(13)

Remark 3.1

The truncation error of (13) is

$$\begin{aligned} \displaystyle {\vert T(x, t_{k})\vert }&\le \displaystyle {\frac{\alpha }{\Gamma (1-\alpha )}} \left \{ \vphantom{\left.+ \left( \frac{1}{2} + \frac{1}{12}\,\frac{(27-10\alpha + \alpha ^{2})}{\prod _{i=1}^{4}(\alpha -i)}\right) \displaystyle { \max _{t_{0}\le t\le t_{k}} }\vert \frac{\partial ^{4}u}{\partial t^{4}}(x,t)\vert \Delta t^{4-\alpha }\right\}}12\, (t_{k}-t_{1})^{-\alpha -1}\displaystyle {\max _{t_{0}\le t\le t_{1}}}\left\vert \frac{\partial ^{2}u}{\partial t^{2}}(x,t)\right\vert \, \Delta t ^{3} \right.\\&\quad+\frac{1}{8}(t_{k}-t_{2})^{-\alpha -1}\,\displaystyle {\max _{t_{0}\le t\le t_{2}}} \left\vert \frac{\partial ^{3}u}{\partial t^{3}}(x,t)\right\vert \Delta t^{4} \\&\quad\left.+ \left( \frac{1}{2} + \frac{1}{12}\,\frac{27-10\alpha + \alpha ^{2}}{\prod\limits_{i=1}^{4}(\alpha -i)}\right) \displaystyle { \max _{t_{0}\le t\le t_{k}} }\left\vert \frac{\partial ^{4}u}{\partial t^{4}}(x,t)\right\vert \Delta t^{4-\alpha }\right\} \\&\quad+ \frac{1}{12}\, \displaystyle {\max _{0< x <1}} \left\vert \frac{\partial ^{4}u}{\partial x^{4}}(x,t) \right\vert \, \Delta x^{2},\qquad \qquad k\ge 3. \end{aligned}$$

The corresponding difference schemes using the L1-2 formula [12] and L1 formula [25] for the Caputo derivative are, respectively,

$$\begin{aligned} \left\{ \begin{array}{lr} {\mathbb {D}}_{t}^{\alpha }u_{i}^{k} = \partial _{x}^{2}u_{i}^{k} + f^{k}, & 1\le i \le M-1,\, 1\le k \le N,\\ u_{0}^{k}=\phi (t_{k}), \, u_{M}^{k} =\varphi (t_{k}),& \, 1\le k \le N,\\ u_{i}^{0}=u^{0} (x_{i}),& \, 0\le i \le M, \end{array} \right. \end{aligned}$$
(14)
$$\begin{aligned} \left\{ \begin{array}{lr} D_{t}^{\alpha }u_{i}^{k} = \partial _{x}^{2}u_{i}^{k} + f^{k}, & 1\le i \le M-1,\, 1\le k \le N,\\ u_{0}^{k}=\phi (t_{k}), \, u_{M}^{k} =\varphi (t_{k}),& \, 1\le k \le N,\\ u_{i}^{0}=u^{0} (x_{i}),& \, 0\le i \le M. \end{array} \right. \end{aligned}$$
(15)

Letting \(\{U_{i}^{k}\}\), \(\{(u_{1})_{i}^{k}\}\), \(\{(u_{2})_{i}^{k}\}\), and \(\{(u_{3})_{i}^{k}\}\,\) for \(0\le i \le M\) and \(0\le k \le N\) be, respectively, solutions of problem (9) and (10), finite difference schemes (15), (14), and (13), we denote

$$\begin{aligned} \begin{array}{c} {{EU}}_{j}^{N}(\Delta t) = \displaystyle {\max _{0\le i\le M}}\bigg \vert U_{i}^{N} - (u_{j})_{i}^{N}\bigg \vert ,\ {\mathrm{and}}\qquad \mathop {\mathrm{order}} (u_{j}) = \log _{2}\frac{{{EU}}_{j}^{N}(\Delta t)}{{{EU}}_{j}^{N}(\Delta t / 2)}, \end{array} \end{aligned}$$

where \(j= 1, 2, 3\). To test the efficiency of the difference scheme (13), two numerical examples, which have different nature (in particular, in the time-dependent part of the solution and therefore in the initial condition which affect the convergence order of the formula) are examined.

Example 4

In (9) and (10), take \(f(x,t)=\text{e}^{x}t^{4}[\frac{\Gamma (5+\alpha )}{24}-t^\alpha ],\, \phi (t)=t^{4+\alpha },\, \varphi (t)=\text{e} t^{4+\alpha }\), and \(u^{0}(x) = 0\). The exact solution is \(u(x,t)=\text{e}^{x}t^{4+\alpha }\).

In Table 4, the numerical results using difference schemes (15), (14), and (13) for \(T=1\) are computed. Due to verifying the temporal numerical accuracy, a fixed and sufficient small spatial step size \(\Delta x\)\((\Delta x=1/20\,000)\) is chosen. Noticing Table 4, we find that the computational errors of the difference scheme (13) are smaller, and its numerical accuracy is superior to the results of \(3-\alpha\) order of accuracy of (14) and \(2-\alpha\) order of accuracy of (15) in time. Moreover, for \(T=1\) and \(\alpha =0.1\) with \(\Delta x=1/1\,000\) and different temporal step sizes, the cell numbers N and the CPU time (on the same machine equipped with Intel 2.4-GHz Intel Core processor, Matlab software) are recorded when the absolute errors of three schemes are on the same orders. According to Fig. 2 showing the relationship between N, absolute errors, and CPU time for three schemes, the cell number N would be much smaller for the scheme (13) than for the schemes (14) and (15). It is clearly obvious that the larger temporal step size can be used for the scheme (13) than for the other two schemes. In addition, Fig. 2 shows that for the same absolute errors, the CPU time is much shorter by using the scheme (13) than that by using the schemes (14) and (15). To reach absolute error less than 5E−8 when \(T=1, \alpha =0.1\), and \(\Delta x=1/1\,000\), we must compute \(u_3^{(80)}, u_2^{(410)}\), and \(u_1^{(6\,800)}\) which are, respectively, the solution of scheme (13), (14), and (15). For obtaining \(u_3^{(80)}\), we need three arrays with dimension \(80\times 1\), \(79\times 1\), and \(78\times 1,\) respectively, to compute \(a_{j}^{(\alpha )}, b_{j}^{(\alpha )}\), and \(\beta _{j}^{(\alpha )}\); for obtaining \(u_2^{(410)}\), we need two arrays with dimension \(410\times 1\) and \(409\times 1,\) respectively, to compute \(a_{j}^{(\alpha )}\) and \(b_{j}^{(\alpha )}\); and for obtaining \(u_1^{(6\,800)}\), we need one array with dimension \(6\,800\times 1\) to compute \(a_{j}^{(\alpha )}\). Therefore, we can draw a conclusion that less memory for computing \(u_3^{(80)}\) (obtained by L1-2-3) is required.

Table 4 (Example 4) Numerical results of finite difference schemes with \(\Delta x=1/20\,000\)
Fig. 2
figure2

The absolute errors and CPU time versus cell number N with \(\alpha = 0.1, \Delta x=1/1\,000\)

Example 5

In (9) and (10), take \(T=1, \, f(x,t)=\text{e}^{x}[2t^{1-\alpha }E_{1,2-\alpha }(2t) - \text{e}^{2t}],\, \phi (t)=\text{e}^{2t},\, \varphi (t)=\text{e}^{1+2t}\), and \(u^{0}(x) = {\rm{e}}^{x}\). The exact solution is \(u(x,t)={\rm{e}}^{x+2t}\).

The computational results listed in Table 5 state the advantages of the difference scheme (13) over (14) and (15). Noticing this table, we find that the computational efficiency and numerical accuracy of the difference scheme (13) are better; moreover, its convergence order of (13) is superior.

Table 5 (Example 5) Numerical results of finite difference schemes with \(\Delta x=1/20\,000\)

To investigate the numerical stability of scheme (13), we only follow up some tests and report corresponding results in Tables 6 and 7. Results imply that even by setting large time and space step sizes, satisfactory results may be produced and there is no track of the instability. So it seems that the method is unconditionaly stable, but proving this fact needs a rigorous and strict theoretical analysis which we leave it to future works.

Table 6 Numerical stability analysis of finite difference scheme (13) (Part I)
Table 7 Numerical stability analysis of finite difference scheme (13) (Part II)

PDE-Based Image Denoising

Image noise removal constitutes a very important process; the most common noise results from the image acquisition system can be modeled as Gaussian random noise in most cases. Gaussian noise represents statistical noise having the probability density function equal to that of the normal distribution. Numerous image denoising models have been introduced in the past few decades. Although the conventional filters such as averaging filter, median filter, and the 2D Gaussian filter are efficient in smoothing the noise, they have the disadvantage of blurring image edges [13, 15]. PDE-based methods for image processing (denoising, restorations, inpainting, segmentation, etc.) have been largely studied in the literature [1] due to their remarkable advantages in both theory and computation. To preserve the image structures when removing the noise, Perona and Malik (PM) [22] proposed a nonlinear equation which replaced isotropic diffusion expressed through a linear heat equation with an anisotropic diffusion. While the backward diffusion of the PM equation results in enhancing the edges, it is an ill-posed process in the sense that it is very sensitive to perturbations in the initial noisy data. Since the work of Perona and Malik, a large number of nonlinear PDE-based anisotropic diffusion models have been proposed [14, 17].

In this section, to confirm the ability of the new L1-2-3 formula to solve fractional differential equations in two dimension and illustrate the applications of it in image denoising, the fractional-order PDE has been applied to the image processing and computer vision. We proposed a fractional-order equation for image denoising as

$$\begin{aligned} \begin{array}{lr} _{0}^{C}\mathcal{D}_{t}^{\alpha }\, u(\mathbf{x }, t) = \displaystyle {\Delta u(\mathbf{x }, t)},&(\mathbf{x }, t) \in \Omega \times [0, T], \end{array} \end{aligned}$$
(16)

where \(\Delta\) denotes the Laplacian operator, \(\Omega \subset {\mathbb {R}}^{2}\) represents the image domain, and \(_{0}^{C}\mathcal{D}_{t}^{\alpha }\) stands for the Caputo fractional time derivative of order \(\alpha\). Assume that an image has \((M_{x}+2)\times (M_{y}+2)\) pixels, and \(u^{k}_{i,j} \simeq u(i, j, t_{k})\) for \(i=0,1,\ldots , M_{x}+1\) and \(j=0,1,\ldots , M_{y}+1\). The following explicit finite difference scheme by using L1-2-3 formula (6) to approximate the time-fractional derivative and the second-order central difference quotient has been considered for (16):

$$\begin{aligned} \begin{array}{c} \gamma _{0}^{(\alpha )}\, u^{k}_{i,j} = \gamma _{k-1}^{(\alpha )}\, u^{0}_{i,j}+ \displaystyle {\sum _{r=1}^{k-1}}(\gamma _{r-1}^{(\alpha )}-\gamma _{r}^{(\alpha )})u^{k-r}_{i,j}\, +\, \mu \, (u^{k-1}_{i-1,j}+u^{k-1}_{i+1,j}+u^{k-1}_{i,j-1}+u^{k-1}_{i,j+1}-4u^{k-1}_{i,j}), \end{array} \end{aligned}$$
(17)

to treat the spatial derivative term. Here \(\mu = (\Delta t)^{\alpha }\Gamma (2-\alpha )\), \(u^{0}_{i,j}\) is an image affected by a noise, and the boundary conditions are zero. Similarly, the following explicit finite difference scheme by using L1-2 formula to approximate the time-fractional derivative term has been considered for (16)

$$\begin{aligned} \begin{array}{c} c _{0}^{(\alpha )}\, u^{k}_{i,j} = c _{k-1}^{(\alpha )}\, u^{0}_{i,j}+ \displaystyle {\sum _{r=1}^{k-1}}(c _{r-1}^{(\alpha )} - c _{r}^{(\alpha )})u^{k-r}_{i,j}\, +\, \mu \, (u^{k-1}_{i-1,j}+u^{k-1}_{i+1,j}+u^{k-1}_{i,j-1}+u^{k-1}_{i,j+1}-4u^{k-1}_{i,j}), \end{array} \end{aligned}$$
(18)

with \(c _{r}^{(\alpha )}\) introduced in [12]. To assess the performance levels of each image denoising technique, performance evaluation of the proposed methods are measured by the peak signal to noise ratio (PSNR)

$$\begin{aligned} \mathop {\mathrm{PSNR}} = \displaystyle {10 \log _{10}(255^2/\mathop {\mathrm{MSE}})},\qquad \mathop {\mathrm{MSE}} = \displaystyle { \frac{\sum\limits_{i=0}^{M_{x}+1}\sum\limits_{j=0}^{M_{y}+1}(u(i,j)-U(i,j))^2}{(M_{x}+1)\,(M_{y}+1)}}, \end{aligned}$$

where U is an initial image and u is a restored version. Due to the purpose of the paper, developing a fractional numerical differentiation formula called the L1-2-3 formula to approximate the Caputo fractional derivative of order \(\alpha\), the denoising performance of our method has not been compared with performances of other noise removal techniques. The images shown in Fig. 3a represents the original \([512\times 512]\) image of Baboon; (b) represents the degraded image by Gaussian noise characterized by parameters 0.1 (mean) and 0.01 (variance); (c, e) represent the image denoising result produced by the L1-2 model (18) with \(\alpha =0.4, N=25\) and \(\alpha =0.5, N=8\); (d, f) represent the image denoising result produced by the L1-2-3 model (17) with \(\alpha =0.4, N=25\) and \(\alpha =0.5, N=8\). Another two image denoising method comparison examples are displayed in Figs. 4 and 5. The same noise removal approaches are applied on the standard Lena and Peppers images corrupted by the same amount of Gaussian noise, their results being represented in Figs. 4c–f and 5c–f. According to Table 8, the values of PSNR for noise reduction approaches corresponding to Figs. 35, the L1-2-3 model (17) can smooth the noise better than the L1-2 model (18).

Fig. 3
figure3

Gaussian noise removal results produced by the L1-2 model (18) and the L1-2-3 model (17)

Fig. 4
figure4

Gaussian noise removal results produced by the L1-2 model (18) and the L1-2-3 model (17)

Fig. 5
figure5

Gaussian noise removal results produced by the L1-2 model (18) and the L1-2-3 model (17)

Table 8 Values of PSNR for noise reduction approaches

Conclusion

In this work, a new fractional numerical differentiation formula called the L1-2-3 formula, obtained by constructing a third degree Lagrange interpolation function for the integrand f, is established to approximate the Caputo fractional derivative of order \(\alpha\). After providing some analysis for the coefficient features and truncation errors of the L1-2-3 formula and considering two test examples which are conducted to effectively confirm its computational validity and numerical accuracy, we apply it in solving some time-fractional sub-diffusion equations on a bounded spatial domain. To illustrate the efficiency and advantages of the proposed scheme, we use it for smoothing the noise of some digital images. Some numerical tests imply that the scheme is unconditional stable. Providing a rigorous theoretical stability analysis of the proposed scheme can be considered as a future work.

In summary, in comparison with the previous methods such as the L1-2 and L1 formulae, the convergence order of the L1-2-3 formula is higher and its computational efficiency and numerical accuracy are superior. Moreover, this new formula can be used for both numerically solving other various fractional differential equations that involve the Caputo time-fractional derivatives and smoothing the noise. Experiments show improved performances in these areas.

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Correspondence to R. Mokhtari.

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Mokhtari, R., Mostajeran, F. A High Order Formula to Approximate the Caputo Fractional Derivative. Commun. Appl. Math. Comput. 2, 1–29 (2020). https://doi.org/10.1007/s42967-019-00023-y

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Keywords

  • Caputo fractional derivative
  • Fractional partial differential equation
  • Finite difference scheme

Mathematics Subject Classification

  • 65M15
  • 65M06