Evaluation of open Digital Elevation Models: estimation of topographic indices relevant to erosion risk in the Wadi M’Goun watershed, Morocco

Maryam Khal; Abdellah Algouti; Ahmed Algouti; Nadia Akdim; Sergey A. Stankevich; Massimo Menenti; Maryam Khal; Abdellah Algouti; Ahmed Algouti; Nadia Akdim; Sergey A. Stankevich; Massimo Menenti

doi:10.3934/geosci.2020014

AIMS Geosciences

2020, Volume 6, Issue 2: 231-257. doi: 10.3934/geosci.2020014

Previous Article Next Article

Research article

Evaluation of open Digital Elevation Models: estimation of topographic indices relevant to erosion risk in the Wadi M’Goun watershed, Morocco

1.
Cadi Ayyad University, Faculty of Sciences, Department of Geology, Laboratory of Geosciences, Geotourism, Natural Hazards and Remote Sensing, Bd. BO 2390, 40000 Marrakech, Morocco
2.
Department of Geology, Faculty of Sciences, Chouaib Doukkali University, Route Ben Maachou, B.P 20, 24000, El Jadida, Morocco
3.
Scientific Centre for Aerospace Research of the Earth, NAS of Ukraine, 55-B O. Gonchar str., Kiev, 01054, Ukraine
4.
Geosciences and Remote Sensing Department, Delft University of Technology, Stevinweg 12628 CN Delft, The Netherlands
5.
State Key Laboratory of Remote Sensing Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100101, China

Received: 09 April 2020 Accepted: 11 June 2020 Published: 22 June 2020

Various Global Digital Elevation Models (DEM) are available freely on the Web. The main objective of this work is to evaluate the latest digital elevation models towards the estimation of morphological and topographic erosion parameters in the Wadi M’Goun watershed. We have evaluated multiple DEMs: SRTM (3-arcsec resolution, 90 m), ASTER GDEM (1-arcsec resolution, 30 m), SRTMGL1 V003 (30 m), and ALOS-PALSAR (12.5 m). We have applied for this purpose open source GIS software. To compare and evaluate each DEM, different processing methods have been applied to estimate the Wadi M’Goun watershed characteristics, namely Hypsometry, topographic slope extraction, retrieval of Slope Length and Steepness factor (LS-factor) and topographic wetness index. The accuracy of the ALOS-PALSAR and SRTMGL1 V003 (30 m) DEMs met the requirements applying to the required morphometric parameters. DEMs vertical accuracy has been evaluated by applying the root mean square error (RMSE) metric to DEM elevations vs. actual heights of 353 sample points extracted from an accurate survey-based map (toposheet). The RMSE was 1718 mm for ALOS-PALSAR, 1736 for SRTM 1-arcsec, 1958 for ASTER GDEM 1-arcsec and, 3189 for SRTM 3-arcsec. These results indicate that best accuracy is achieved with the high-resolution of the ALOS PALSAR DEM. This study suggests potential uncertainties in the open-source DEMs, which should be taken into account when estimating topographical and morphometric parameters related to erosion risk in the Wadi M’Goun watershed.

Keywords:

Citation: Maryam Khal, Abdellah Algouti, Ahmed Algouti, Nadia Akdim, Sergey A. Stankevich, Massimo Menenti. Evaluation of open Digital Elevation Models: estimation of topographic indices relevant to erosion risk in the Wadi M’Goun watershed, Morocco[J]. AIMS Geosciences, 2020, 6(2): 231-257. doi: 10.3934/geosci.2020014

Related Papers:

[1]	Umar Ishtiaq, Fahad Jahangeer, Doha A. Kattan, Manuel De la Sen . Generalized common best proximity point results in fuzzy multiplicative metric spaces. AIMS Mathematics, 2023, 8(11): 25454-25476. doi: 10.3934/math.20231299
[2]	Xiangxing Tao, Jiahui Wang . Commutators of multilinear $\theta$ -type generalized fractional integrals on non-homogeneous metric measure spaces. AIMS Mathematics, 2022, 7(6): 9627-9647. doi: 10.3934/math.2022535
[3]	Aftab Hussain . Fractional convex type contraction with solution of fractional differential equation. AIMS Mathematics, 2020, 5(5): 5364-5380. doi: 10.3934/math.2020344
[4]	Monairah Alansari, Mohammed Shehu Shagari, Akbar Azam, Nawab Hussain . Admissible multivalued hybrid $\mathcal{Z}$ -contractions with applications. AIMS Mathematics, 2021, 6(1): 420-441. doi: 10.3934/math.2021026
[5]	Md Hasanuzzaman, Mohammad Imdad . Relation theoretic metrical fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$ -contraction with an application. AIMS Mathematics, 2020, 5(3): 2071-2087. doi: 10.3934/math.2020137
[6]	Nizar Souayah, Nabil Mlaiki, Salma Haque, Doaa Rizk, Amani S. Baazeem, Wasfi Shatanawi . A new type of three dimensional metric spaces with applications to fractional differential equations. AIMS Mathematics, 2022, 7(10): 17802-17814. doi: 10.3934/math.2022980
[7]	Tatjana Došenović, Dušan Rakić, Stojan Radenović, Biljana Carić . Ćirić type nonunique fixed point theorems in the frame of fuzzy metric spaces. AIMS Mathematics, 2023, 8(1): 2154-2167. doi: 10.3934/math.2023111
[8]	Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for $\psi$ -Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244
[9]	Zijie Qin, Lin Chen . Characterization of ternary derivation of strongly double triangle subspace lattice algebras. AIMS Mathematics, 2023, 8(12): 29368-29381. doi: 10.3934/math.20231503
[10]	Muhammad Nazam, Aftab Hussain, Asim Asiri . On a common fixed point theorem in vector-valued $b$ -metric spaces: Its consequences and application. AIMS Mathematics, 2023, 8(11): 26021-26044. doi: 10.3934/math.20231326

Abstract

1. Introduction

Fractional derivatives, which have attracted considerable attention during the last few decades, can be defined according to their type. These include the Caputo ^{[1,2,3,4,5,6,8,7,9,10,11,12,13]}, Riemann-Liouville ^{[14,15,16,17,18,19,20,21]}, Riesz ^{[22,23,24,25,26]}, and Caputo-Fabrizio (CF) ^{[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]} types. Especially for the CF derivative, there were many related reports based on some discussions of different aspects, see the above references, and the Refs. ^[43,44,45]. In order to better grasp the fractional order problem, one can refer to the related works ^[47,48] and other fractional books.

Based on these fractional derivatives, numerous models have been developed. However, these models are difficult to solve directly by applying the general analytical methods because of the existence of fractional derivatives. This problem has inspired scholars to develop numerical algorithms to derive numerical solutions efficiently. In ^[5,49,50,51], a few high-order approximation formulas for the Riemann-Liouville, Caputo, and Riesz fractional derivatives were proposed and developed using different techniques or ideas. Recently, high-order discrete formulas for the CF fractional derivatives were designed and discussed in Refs. ^{[32,33,34,35,36,37,38]}.

Another difficulty for simulating the models with fractional derivatives is the non-locality which greatly reduces the efficiency of the algorithm and requires much more memory storage compared with the traditional local models. Specifically, to obtain the approximation solutions $\{U^k\}_{k = 1}^M$ with $M$ a positive integer, for the fractional models its computing complexity is $O(M^2)$ , and the memory storage is $O(M)$ , in contrast to the local models with $O(M)$ and $O(1)$ , respectively. For fast algorithms aimed at the Riemann-Liouville, Caputo, and Riesz fractional derivatives, see Refs. ^{[14,52,53,54,55]}. However, few scholars studied the fast algorithm for the CF fractional derivative. To the best of our knowledge, authors in ^[39] proposed numerically a fast method for the CF fractional derivative without further analysing the error accuracy.

In this study, our aim is to construct a novel efficient approximation formula for the following CF fractional derivative ^[31]

$\begin{equation} _0^{CF}\partial_t^\alpha u(t) = \frac{1}{1-\alpha} \int^{t}_0 u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t-s)\bigg]ds,\; 0 \lt \alpha \lt 1, \end{equation}$

(1.1)

where $\; t\in[0, T]$ , $0 < T < \infty$ . Our contributions in this study mainly focus on

$\bullet$ Propose a novel second-order approximation formula for the CF fractional derivative with detailed theoretical analysis for the truncation error.

$\bullet$ Develop a fast algorithm based on the novel discretization technique which reduces the computing complexity from $O(M^2)$ to $O(M)$ and the memory storage from $O(M)$ to $O(1)$ . Moreover, we theoretically show that the fast algorithm maintains the optimal convergence rate.

The remainder of this paper is structured as follows. In Section $2$ , we derive a novel approximation formula with second-order convergence rate for the CF fractional derivative. In Section $3$ , we develop a fast algorithm by splitting the CF fractional derivative into two parts, the history part and local part, and then rewrite the history part by a recursive formula. Further we prove the truncation error for the fast algorithm. In Section $4$ , two numerical examples are provided to verify the approximation results and the efficiency of our fast algorithm. In Section $5$ , we provide a conclusion and offer suggestions for future studies.

Throughout this article, we denote $C$ as a positive constant, which is free of the step size $\Delta_t$ .

2. Novel approximation formula for the Caputo-Fabrizio fractional derivative

To derive a novel approximation formula, we choose a uniform time step size $\Delta_{t} = \frac{T}{M} = t_{k}-t_{k-1}$ with nodes $t_{k} = k\Delta_{t}, k = 0, 1, \cdots, M$ , where $M$ is a positive constant. We denote $u^{k} = u(t_{k})$ on $[0, T]$ .

We next give a discrete approximation of CF fractional derivative $_0^{CF}\partial_t^\alpha u(t)$ at $t_{k+\frac{1}{2}}(k\geq1)$

$\begin{equation} \begin{split} &_0^{CF}\partial_t^\alpha u(t_{k+\frac{1}{2}})\\ = &\frac{1}{1-\alpha}\sum\limits_{j = 1}^{k}\int^{t_{j+\frac{1}{2}}}_{t_{j-\frac{1}{2}}}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds +\frac{1}{1-\alpha}\int^{t_{\frac{1}{2}}}_{t_{0}}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ = &\frac{1}{1-\alpha}\sum\limits_{j = 1}^{k}\int^{t_{j+\frac{1}{2}}}_{t_{j-\frac{1}{2}}}\bigg[u'(t_{j})+u''(t_j)(s-t_{j})+\frac{1}{2}u'''(\xi_j)(s-t_{j})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ &+\frac{1}{1-\alpha}\int^{t_{\frac{1}{2}}}_{t_{0}}\bigg[u'(t_{\frac{1}{2}})+u''(t_{\frac{1}{2}})(s-t_{\frac{1}{2}})+\frac{1}{2}u'''(\xi_0)(s-t_\frac{1}{2})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ = &\frac{1}{\alpha}\sum\limits_{j = 1}^{k}\frac{u^{j+1}-u^{j-1}}{2\Delta_{t}}\bigg(M^{k+\frac{1}{2}}_{j+\frac{1}{2}}-M^{k+\frac{1}{2}}_{j-\frac{1}{2}}\bigg) +\frac{1}{\alpha}\frac{u^1-u^0}{\Delta_{t}}\bigg(M^{k+\frac{1}{2}}_{\frac{1}{2}}-M^{k+\frac{1}{2}}_{0}\bigg)+R_{1}^{k+\frac{1}{2}}+R_{2}^{k+\frac{1}{2}}\\ \triangleq&_0^{CF}D_t^\alpha u(t_{k+\frac{1}{2}})+R^{k+\frac{1}{2}}, \end{split} \end{equation}$

(2.1)

where $\xi_j$ generally depending on $s$ satisfies $\xi_j\in (t_{j-\frac{1}{2}}, t_{j+\frac{1}{2}})$ for $j\geq 1$ and $\xi_0 \in (t_0, t_{\frac{1}{2}})$ . The coefficients $M_j^k$ and error $R^{k+\frac{1}{2}}$ are defined as follows

$\begin{equation} M^k_j = \exp\bigg[-\frac{\alpha}{1-\alpha}(t_k-t_j)\bigg], \ \ \ \ \ R^{k+\frac{1}{2}} = R_{1}^{k+\frac{1}{2}}+R_{2}^{k+\frac{1}{2}}, \end{equation}$

(2.2)

$\begin{equation} \begin{split} R_{1}^{k+\frac{1}{2}} = &\frac{1}{1-\alpha}\sum\limits_{j = 1}^{k}\int^{t_{j+\frac{1}{2}}}_{t_{j-\frac{1}{2}}}\bigg[u''(t_j)(s-t_{j})+\frac{1}{2}u'''(\xi_j)(s-t_{j})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ &+\frac{1}{1-\alpha}\int^{t_{\frac{1}{2}}}_{t_{0}}\bigg[u''(t_{\frac{1}{2}})(s-t_{\frac{1}{2}})+\frac{1}{2}u'''(\xi_0)(s-t_\frac{1}{2})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds,\\ R_{2}^{k+\frac{1}{2}} = &\frac{1}{1-\alpha}O(\Delta_{t}^{2})\int^{t_{k+\frac{1}{2}}}_{t_{0}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds. \end{split} \end{equation}$

(2.3)

From (2.1), we obtain the following approximation formula for $_0^{CF}\partial_t^\alpha u(t_{k+\frac{1}{2}})$ with $k\geq1$ .

$\begin{equation} \begin{split} _0^{CF}D_t^\alpha u(t_{k+\frac{1}{2}}) = \frac{1}{\alpha}\sum\limits_{j = 1}^{k}\frac{u^{j+1}-u^{j-1}}{2\Delta_{t}}\bigg(M^{k+\frac{1}{2}}_{j+\frac{1}{2}}-M^{k+\frac{1}{2}}_{j-\frac{1}{2}}\bigg)+\frac{1}{\alpha}\frac{u^1-u^0}{\Delta_{t}}\bigg(M^{k+\frac{1}{2}}_{\frac{1}{2}}-M^{k+\frac{1}{2}}_{0}\bigg). \end{split} \end{equation}$

(2.4)

Based on this discussion, we obtain the novel approximation formula (2.4). We next discuss the truncation error of the novel approximation formula.

Theorem 1. For $u(t)\in C^3[0, T]$ , the truncation error $R^{k+\frac{1}{2}}(k\geq0)$ satisfies the following estimate

$\begin{equation} |R^{k+\frac{1}{2}}|\leq C\Delta_t^2, \end{equation}$

(2.5)

where the constant $C$ is independent of $k$ and $\Delta_t$ .

Proof. According to formula (2.3), we can obtain:

$\begin{equation} \begin{split} \big|R^{k+\frac{1}{2}}\big|\leq&\big|R_{1}^{k+\frac{1}{2}}\big|+\big|R_{2}^{k+\frac{1}{2}}\big|\\ \leq&\bigg|\frac{1}{1-\alpha}\sum\limits_{j = 1}^{k}u''(t_j)\int^{t_{j+\frac{1}{2}}}_{t_{j-\frac{1}{2}}}(s-t_{j})\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\bigg|\\ &+\bigg|\frac{1}{1-\alpha}u''(t_{\frac{1}{2}})\int^{t_{\frac{1}{2}}}_{t_{0}}(s-t_{\frac{1}{2}})\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\bigg|\\ &+\bigg|\frac{1}{2(1-\alpha)}\sum\limits_{j = 1}^{k}\int^{t_{j+\frac{1}{2}}}_{t_{j-\frac{1}{2}}}u'''(\xi_j)(s-t_{j})^{2}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\bigg|\\ &+\bigg|\frac{1}{2(1-\alpha)}\int^{t_{\frac{1}{2}}}_{t_{0}}u'''(\xi_0)(s-t_\frac{1}{2})^{2}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\bigg|\\ &+\bigg|\frac{1}{1-\alpha}\int^{t_{k+\frac{1}{2}}}_{t_{0}}O(\Delta_{t}^{2})\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\bigg|\\ = &I_{1}+I_{2}+I_{3}+I_{4}+I_{5}. \end{split} \end{equation}$

(2.6)

For the term $I_{1}$ , using integration by parts, we can arrive at:

$\begin{equation} \begin{split} I_{1} \leq&\max\limits_{t\in[0,T]}|u''(t)|\frac{\Delta_{t}^{2}}{8(1-\alpha)}\sum\limits_{j = 1}^{k}\bigg\{\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-t_{j+\frac{1}{2}})\bigg]-\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-t_{j-\frac{1}{2}})\bigg]\bigg\}\\ &+\max\limits_{t\in[0,T]}|u''(t)|\frac{\alpha\Delta_{t}^{2}}{8(1-\alpha)^{2}}\int^{t_{k+\frac{1}{2}}}_{t_{\frac{1}{2}}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ \leq&\max\limits_{t\in[0,T]}|u''(t)|\frac{\Delta_{t}^{2}}{4(1-\alpha)}\bigg(1-M_{\frac{1}{2}}^{k+\frac{1}{2}}\bigg)\\ \leq&C\Delta_{t}^{2}. \end{split} \end{equation}$

(2.7)

Next, for the term $I_{2}$ , by using the mean value theorem of integrals, we obtain:

$\begin{equation} \begin{split} I_{2} = &\bigg|\frac{\Delta_t}{2(1-\alpha)}u''(t_{\frac{1}{2}})(t_{\varepsilon}-t_{\frac{1}{2}})\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-t_{\varepsilon})\bigg]\bigg|\\ \leq&\max\limits_{t\in[0,T]}|u''(t)|\frac{\Delta_{t}^{2}}{4(1-\alpha)}\\ \leq&C\Delta_{t}^{2}, \end{split} \end{equation}$

(2.8)

where $t_{0}\leq t_{\varepsilon}\leq t_{\frac{1}{2}}$ .

For the term $I_{3}$ , we can easily obtain:

$\begin{equation} \begin{split} I_{3} \leq&\max\limits_{t\in[0,T]}|u'''(t)|\frac{\Delta_{t}^{2}}{8(1-\alpha)}\int^{t_{k+\frac{1}{2}}}_{t_{\frac{1}{2}}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ \leq&C\Delta_{t}^{2}. \end{split} \end{equation}$

(2.9)

Similarly, we can estimate the term $I_{4}$ as follows:

$\begin{equation} \begin{split} I_{4} \leq&\max\limits_{t\in[0,T]}|u'''(t)|\frac{\Delta_{t}^{2}}{8(1-\alpha)}\int^{t_{\frac{1}{2}}}_{t_{0}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\\ \leq&C\Delta_{t}^{2}. \end{split} \end{equation}$

(2.10)

Finally, for the term $I_{5}$ , we can derive:

$\begin{equation} \begin{split} I_{5} \leq\frac{1}{1-\alpha}|O(\Delta_{t}^{2})|\bigg(1-M^{k+\frac{1}{2}}_{0}\bigg)\leq C\Delta_{t}^{2}. \end{split} \end{equation}$

(2.11)

Based on the aforementioned estimates for the terms $I_1$ , $\cdots, I_5$ , we can complete the proof of the Theorem.

3. Fast algorithm

It is obvious that the approximation formula (2.4) is nonlocal since the value at node $t_{k+\frac{1}{2}}$ for the CF fractional derivative is concerned with all the values of $u^j$ , $j = 0, 1, \cdots, k, k+1$ , which means the computing complexity when apply the formula (2.4) to ODEs is of $O(M^2)$ and the memory requirement is $O(M)$ . In the following analysis, inspired by the work ^[14], we develop a fast algorithm based on the new discretization technique used in this paper, with which the computing complexity is reduced from $O(M^2)$ to $O(M)$ and the memory requirement is $O(1)$ instead of $O(M)$ .

We split the derivative $_0^{CF}D_t^\alpha u(t_{k+\frac{1}{2}})$ for $k\geq 1$ into two parts: the history part denoted by $C_h(t_{k+\frac{1}{2}})$ and the local part denoted by $C_l(t_{k+\frac{1}{2}})$ , respectively, as follows

$\begin{equation} \begin{split} _0^{CF}\partial_t^\alpha u(t_{k+\frac{1}{2}})& = C_h(t_{k+\frac{1}{2}})+C_l(t_{k+\frac{1}{2}}) \\& = \frac{1}{1-\alpha}\int^{t_{k-\frac{1}{2}}}_{t_0}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\quad+\frac{1}{1-\alpha}\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds. \end{split} \end{equation}$

(3.1)

For the local part $C_l(t_{k+\frac{1}{2}})$ , we have

$\begin{equation} \begin{split} C_l(t_{k+\frac{1}{2}}) = \frac{u^{k+1}-u^{k-1}}{2\alpha\Delta_{t}}\bigg(1-M^{k+\frac{1}{2}}_{k-\frac{1}{2}}\bigg)+R^{k+\frac{1}{2}}_l, \end{split} \end{equation}$

(3.2)

where $M^k_j$ is defined by (2.2), and the truncation error $R^{k+\frac{1}{2}}_l$ is

$\begin{equation} \begin{split} R^{k+\frac{1}{2}}_l& = \frac{1}{1-\alpha}\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}\bigg[u''(t_k)(s-t_{k})+\frac{1}{2}u'''(\xi_k)(s-t_{k})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\quad+ \frac{1}{1-\alpha}\bigg[u'(t_k)-\frac{u^{k+1}-u^{k-1}}{2\Delta_t}\bigg]\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds,\quad k\geq 1. \end{split} \end{equation}$

(3.3)

For the history part $C_h(t_{k+\frac{1}{2}})$ , we rewrite it into a recursive formula when $k\geq 2$ in the following way

$\begin{equation} \begin{split} C_h(t_{k+\frac{1}{2}})& = \frac{1}{1-\alpha}\int^{t_{k-\frac{3}{2}}}_{t_0}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\quad+ \frac{1}{1-\alpha}\int^{t_{k-\frac{1}{2}}}_{t_{k-\frac{3}{2}}}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\triangleq C_h^{(1)}(t_{k+\frac{1}{2}})+C_h^{(2)}(t_{k+\frac{1}{2}}), \end{split} \end{equation}$

(3.4)

and when $k = 1$ ,

$\begin{equation} \begin{split} C_h(t_{\frac{3}{2}})& = \frac{1}{1-\alpha}\int^{t_{\frac{1}{2}}}_{t_0}u'(s)\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{\frac{3}{2}}-s)\bigg]ds \\&\triangleq C_h^{(2)}(t_{\frac{3}{2}}). \end{split} \end{equation}$

(3.5)

Careful calculations show that

$\begin{equation} \begin{split} C_h^{(1)}(t_{k+\frac{1}{2}}) = \exp\bigg(\frac{\alpha \Delta_t}{\alpha-1}\bigg)C_h(t_{k-\frac{1}{2}}), \quad k\geq 2. \end{split} \end{equation}$

(3.6)

For the term $C_h^{(2)}(t_{k+\frac{1}{2}})$ , by similar analysis for the Theorem 1, we have

$\begin{equation} C_h^{(2)}(t_{k+\frac{1}{2}}) = \begin{cases} \frac{u^k-u^{k-2}}{2\alpha\Delta_t}\bigg(M^{k+\frac{1}{2}}_{k-\frac{1}{2}}-M^{k+\frac{1}{2}}_{k-\frac{3}{2}}\bigg)+R_h^{k+\frac{1}{2}}, & \mbox{if } k\geq2 \\ \frac{u^1-u^0}{\alpha\Delta_t}\bigg(M^{\frac{3}{2}}_{\frac{1}{2}}-M^{\frac{3}{2}}_{0}\bigg)+R_h^{\frac{3}{2}}, & \mbox{if } k = 1, \end{cases} \end{equation}$

(3.7)

where, for $k\geq 2$ ,

$\begin{equation} \begin{split} R_h^{k+\frac{1}{2}}& = \frac{1}{1-\alpha}\int^{t_{k-\frac{1}{2}}}_{t_{k-\frac{3}{2}}}\bigg[u''(t_{k-1})(s-t_{k-1})+\frac{1}{2}u'''(\xi_{k-1})(s-t_{k-1})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\quad+ \frac{1}{1-\alpha}\bigg[u'(t_{k-1})-\frac{u^{k}-u^{k-2}}{2\Delta_t}\bigg]\int^{t_{k-\frac{1}{2}}}_{t_{k-\frac{3}{2}}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds, \end{split} \end{equation}$

(3.8)

and, for $k = 1$ ,

$\begin{equation} \begin{split} R_h^{\frac{3}{2}}& = \frac{1}{1-\alpha}\int^{t_{\frac{1}{2}}}_{t_{0}}\bigg[u''(t_{\frac{1}{2}})(s-t_{\frac{1}{2}})+\frac{1}{2}u'''(\xi_0)(s-t_\frac{1}{2})^{2}\bigg]\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\quad+ \frac{1}{1-\alpha}\bigg[u'(t_{\frac{1}{2}})-\frac{u^{1}-u^{0}}{\Delta_t}\bigg]\int^{t_{\frac{1}{2}}}_{t_{0}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{\frac{3}{2}}-s)\bigg]ds. \end{split} \end{equation}$

(3.9)

For the truncation error $R_l^{k+\frac{1}{2}}$ and $R_h^{k+\frac{1}{2}}$ defined respectively by (3.3) and (3.8)-(3.9), we have the estimates that

Lemma 1. Suppose that $u(t)\in C^3[0, T]$ , then for any $k\geq 1$ , $R_l^{k+\frac{1}{2}}$ and $R_h^{k+\frac{1}{2}}$ satisfy

$\begin{equation} \begin{split} |R^{k+\frac{1}{2}}_l|\leq C\Delta_t^3,\quad |R^{k+\frac{1}{2}}_h|\leq C\Delta_t^3, \end{split} \end{equation}$

(3.10)

where the constant $C$ is free of $k$ and $\Delta_t$ .

Proof. To avoid repetition we just prove the estimate for $R^{k+\frac{1}{2}}_l$ , since the estimate for $R^{k+\frac{1}{2}}_h$ can be derived similarly. By the definition (3.3), we have

$\begin{equation} \begin{split} \big|R^{k+\frac{1}{2}}_l\big| &\leq \frac{1}{1-\alpha}\max\limits_{t\in[0,T]}|u''(t)|\bigg|\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}(s-t_{k})\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds\bigg| \\&\quad+\frac{1}{2(1-\alpha)}\max\limits_{t\in[0,T]}|u'''(t)|\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}(s-t_{k})^{2}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\quad+ \frac{C\Delta_t^2}{1-\alpha}\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds. \\&\triangleq L_1+L_2+L_3. \end{split} \end{equation}$

(3.11)

Then, for the term $L_1$ , using integration by parts and the Taylor expansion for $\exp(t)$ at zero, we have

$\begin{equation} \begin{split} L_1 &\leq C \Delta_t^2\bigg(M^{k+\frac{1}{2}}_{k+\frac{1}{2}}-M^{k+\frac{1}{2}}_{k-\frac{1}{2}}\bigg)+C\int^{t_{k+\frac{1}{2}}}_{t_{k-\frac{1}{2}}}(s-t_k)^2\exp\bigg[-\frac{\alpha}{1-\alpha}(t_{k+\frac{1}{2}}-s)\bigg]ds \\&\leq C\Delta_t^2\bigg[1-\bigg(1-\frac{\alpha\Delta_t}{1-\alpha}-|O(\Delta_t^2)|\bigg)\bigg]+C\Delta_t^3 \\&\leq C\Delta_t^3. \end{split} \end{equation}$

(3.12)

For the terms $L_2$ and $L_3$ , by the mean value theorem of integrals we can easily get $L_2 \leq C\Delta_t^3$ and $L_3 \leq C\Delta_t^3$ . Hence, we have proved the estimate for $R^{k+\frac{1}{2}}_l$ .

Now, based on the above analysis, and for a better presentation, we can introduce an operator ${}^{CF}_0 \mathcal{F}_t^{\alpha}$ for the fast algorithm defined by

$\begin{equation} \begin{split} {}^{CF}_0 \mathcal{F}_t^{\alpha}u(t_{k+\frac{1}{2}}) = \frac{u^{k+1}-u^{k-1}}{2\alpha\Delta_t}\bigg(1-M^{k+\frac{1}{2}}_{k-\frac{1}{2}}\bigg)+\mathcal{F}_h(t_{k+\frac{1}{2}}), \quad k\geq 1, \end{split} \end{equation}$

(3.13)

where the history part $\mathcal{F}_h(t_{k+\frac{1}{2}})$ satisfies

$\begin{equation} \begin{split} \mathcal{F}_h(t_{k+\frac{1}{2}}) = \begin{cases} \exp\bigg( \frac{\alpha \Delta_t}{\alpha-1}\bigg)\mathcal{F}_h(t_{k-\frac{1}{2}})+ \frac{u^k-u^{k-2}}{2\alpha\Delta_t}\bigg(M^{k+\frac{1}{2}}_{k-\frac{1}{2}}-M^{k+\frac{1}{2}}_{k-\frac{3}{2}}\bigg), & \mbox{if } k\geq 2 \\ \frac{u^1-u^0}{\alpha\Delta_t}\bigg(M^{\frac{3}{2}}_{\frac{1}{2}}-M^{\frac{3}{2}}_{0}\bigg), & \mbox{if } k = 1. \end{cases} \end{split} \end{equation}$

(3.14)

We note that with (3.13) and (3.14), $u^{k+1}$ only depends on $u^k$ , $u^{k-1}$ and $u^{k-2}$ , which reduces the algorithm complexity from $O(M^2)$ to $O(M)$ and the memory requirement from $O(M)$ to $O(1)$ .

The following theorem confirms the efficiency of the operator ${}^{CF}_0 \mathcal{F}_t^{\alpha}$ , with which we can still obtain the second-order convergence rate.

Theorem 2. Assume $u(t) \in C^3[0, T]$ and the operator ${}^{CF}_0 \mathcal{F}_t^{\alpha}$ is defined by (3.13). Then

$\begin{equation} \begin{split} \big|{}_0^{CF}\partial_t^\alpha u(t_{k+\frac{1}{2}})-{}^{CF}_0 \mathcal{F}_t^{\alpha}u(t_{k+\frac{1}{2}})\big| \leq C\Delta_t^2, \end{split} \end{equation}$

(3.15)

where the constant $C$ is independent of $k$ and $\Delta_t$ .

Proof. Combining (3.1), (3.2), (3.4)-(3.5) with (3.13), (3.14), we can get

$\begin{equation} \begin{split} \big|{}_0^{CF}\partial_t^\alpha u(t_{k+\frac{1}{2}})-{}^{CF}_0 \mathcal{F}_t^{\alpha}u(t_{k+\frac{1}{2}})\big| \leq \big|C_h(t_{k+\frac{1}{2}})-\mathcal{F}_h(t_{k+\frac{1}{2}})\big|+\big|R_l^{k+\frac{1}{2}}\big|, \quad k\geq 2. \end{split} \end{equation}$

(3.16)

Then, next we mainly analyse the estimate for $\big|C_h(t_{k+\frac{1}{2}})-\mathcal{F}_h(t_{k+\frac{1}{2}})\big|$ . Actually, by definitions we obtain

$\begin{equation} \begin{split} C_h(t_{k+\frac{1}{2}})-\mathcal{F}_h(t_{k+\frac{1}{2}}) = \exp\bigg(\frac{\alpha \Delta_t}{\alpha-1}\bigg)\big[C_h(t_{k-\frac{1}{2}})-\mathcal{F}_h(t_{k-\frac{1}{2}})\big]+R_h^{k+\frac{1}{2}}. \end{split} \end{equation}$

(3.17)

We introduce some notations to simplify the presentation. Let

$\begin{equation} \begin{split} T_{k+1} = C_h(t_{k+\frac{1}{2}})-\mathcal{F}_h(t_{k+\frac{1}{2}}), \quad L = \exp\bigg(\frac{\alpha \Delta_t}{\alpha-1}\bigg). \end{split} \end{equation}$

(3.18)

Then, the recursive formula (3.17) reads that

$\begin{equation} \begin{split} T_{k+1} = L^{k-1}T_2+\mathcal{R}_h^{k+1}, \end{split} \end{equation}$

(3.19)

where the term $\mathcal{R}_h^{k+1}$ is defined by

$\begin{equation} \begin{split} \mathcal{R}_h^{k+1} = L^{k-2}R_h^{2+\frac{1}{2}}+L^{k-3}R_h^{3+\frac{1}{2}}+\cdots+R_h^{k+\frac{1}{2}}. \end{split} \end{equation}$

(3.20)

Now, by (3.7) and (3.14) as well as the Lemma 1, we can get

$\begin{equation} \begin{split} |T_2| = |R_h^{\frac{3}{2}}|\leq C\Delta_t^3, \end{split} \end{equation}$

(3.21)

and

$\begin{equation} \begin{split} \big|\mathcal{R}_h^{k+1}\big|\leq C\Delta_t^3\big(L^{k-2}+L^{k-3}+\cdots+1\big) = C\Delta_t^3\frac{1-L^{k-1}}{1-L}. \end{split} \end{equation}$

(3.22)

Noting here that $L\in (0, 1)$ we have

$\begin{equation} \begin{split} 1-L = 1-\exp\bigg(\frac{\alpha \Delta_t}{\alpha-1}\bigg) \sim\frac{\alpha \Delta_t}{1-\alpha}\geq \frac{\alpha \Delta_t}{2(1-\alpha)}. \end{split} \end{equation}$

(3.23)

Combining (3.19), (3.21)-(3.23), we obtain that

$\begin{equation} \begin{split} \big|T_{k+1}\big|\leq C\Delta_t^2. \end{split} \end{equation}$

(3.24)

Now, with (3.16), (3.17), (3.24) and the Lemma 1, we complete the proof for the theorem.

4. Numerical tests

To check the second-order convergence rate and the efficiency of the fast algorithm for the novel approximation formula, we choose two fractional ordinary differential equation models with the domain $I = (0, T]$ . Let $U^k$ be the numerical solution for the chosen models at $t_k$ , and define $U^0 = u(0)$ . Define the error as $Err(\Delta_t) = \max_{1\leq k \leq M}|U^k-u^k|$ . For the sufficiently smooth function $u(t)$ , we have the approximation formulas for $u(t_{k+\frac{1}{2}})$ and its first derivative $\frac{\mathrm{d}u}{\mathrm{d}t}\big|_{t = t_{k+\frac{1}{2}}}$ :

$\begin{equation} \begin{split} u(t_{k+\frac{1}{2}})& = \frac{1}{2}(u^k+u^{k+1})+O(\Delta_t^2), \\ \frac{\mathrm{d}u}{\mathrm{d}t}\bigg|_{t = t_{k+\frac{1}{2}}}& = \frac{u^{k+1}-u^k}{\Delta_t}+O(\Delta_t^2). \end{split} \end{equation}$

(4.1)

Then, combined with results (2.5) and (3.15), the second-order convergence rate in the following tests is expected.

4.1. Example 1

First, we consider the following fractional ordinary differential equation with an initial value:

$\begin{equation} \begin{split} \left\{ \begin{aligned} &_{0}^{CF}\partial_{t}^{\alpha}u(t)+u(t) = g_{1}(t),\ t\in \bar{I}, \\ &u(0) = \varphi_{0}. \end{aligned} \right. \end{split} \end{equation}$

(4.2)

Next, by taking the exact solution $u(t) = t^2$ and the initial value $\varphi_0 = 0$ , we derive the source function as follows:

$\begin{equation} \begin{split} g_{1}(t) = \frac{2t}{\alpha}+t^2-\frac{2(1-\alpha)}{\alpha^{2}}\bigg[1-\exp(-\frac{\alpha}{1-\alpha}t)\bigg]. \end{split} \end{equation}$

(4.3)

Direct scheme: Based on the novel approximation formula (2.4), we derive the following discrete system at $t_{k+\frac{1}{2}}$ :

Case $k = 0$

$\begin{equation} \begin{split} \bigg(\frac{1}{2}+\frac{1-M_{0}^{\frac{1}{2}}}{\alpha\Delta_{t}}\bigg)U^{1} = \bigg(-\frac{1}{2}+\frac{1-M_{0}^{\frac{1}{2}}}{\alpha\Delta_{t}}\bigg)U^{0}+g_{1}(t_{\frac{1}{2}}), \end{split} \end{equation}$

(4.4)

Case $k\geq 1$

$\begin{equation} \begin{split} \bigg(\frac{1}{2}+\frac{1-M_{k-\frac{1}{2}}^{k+\frac{1}{2}}}{2\alpha\Delta_{t}}\bigg)U^{k+1} = &\frac{1-M_{k-\frac{1}{2}}^{k+\frac{1}{2}}}{2\alpha\Delta_{t}}U^{k-1}-\frac{1}{2}U^{k} -\frac{M_{\frac{1}{2}}^{k+\frac{1}{2}}-M_{0}^{k+\frac{1}{2}}}{\alpha\Delta_{t}}(U^{1}-U^{0})\\ &-\frac{1}{2\alpha\Delta_{t}}\sum\limits_{j = 1}^{k-1}(U^{j+1}-U^{j-1})\big(M_{j+\frac{1}{2}}^{k+\frac{1}{2}}-M_{j-\frac{1}{2}}^{k+\frac{1}{2}}\big)+g_{1}(t_{k+\frac{1}{2}}). \end{split} \end{equation}$

(4.5)

Fast scheme: Applying the fast algorithm to the equation (4.2), we can get, for $k\geq 1$ :

(4.6)

where $\mathcal{F}_h(t_{k+\frac{1}{2}})$ is defined by (3.14). For the case $k = 0$ , the formula (4.4) is used to derive $U^1$ .

Let $T = 1$ . By calculating based on the direct scheme (4.4)–(4.5) and the fast scheme (4.6), we obtain the error results by choosing changed mesh sizes time step $\Delta_{t} = 2^{-10}, 2^{-11}, 2^{-12}, 2^{-13}, 2^{-14}$ for different fractional parameters $\alpha = 0.1, 0.5, 0.9$ , respectively, in Table 1. From the computed results, we can see that the convergence rate for both of the schemes is close to 2, which is in agreement with our theoretical result.

Table 1. Convergence results of Example 1.

$\alpha$	$\Delta_t$	Direct scheme			Fast scheme
$\alpha$	$\Delta_t$	$Err(\Delta_t)$	Rate	CPU(s)	$Err(\Delta_t)$	Rate	CPU(s)
0.1	$2^{-10}$	4.76834890E-07	—	0.0625	4.76834890E-07	—	0.0072
	$2^{-11}$	1.19209015E-07	1.999996	0.2508	1.19209014E-07	1.999996	0.0067
	$2^{-12}$	2.98022864E-08	1.999998	0.8776	2.98022864E-08	1.999998	0.0088
	$2^{-13}$	7.45057174E-09	2.000000	3.3688	7.45057174E-09	2.000000	0.0100
	$2^{-14}$	1.86264512E-09	1.999998	13.2991	1.86264512E-09	1.999998	0.0140
0.5	$2^{-10}$	4.76811290E-07	—	0.0716	4.76811290E-07	—	0.0067
	$2^{-11}$	1.19206056E-07	1.999961	0.2768	1.19206056E-07	1.999961	0.0083
	$2^{-12}$	2.98019183E-08	1.999980	0.9433	2.98019183E-08	1.999980	0.0091
	$2^{-13}$	7.45052997E-09	1.999990	3.6143	7.45052997E-09	1.999990	0.0098
	$2^{-14}$	1.86263893E-09	1.999995	14.2937	1.86263893E-09	1.999995	0.0135
0.9	$2^{-10}$	8.88400608E-07	—	0.0834	8.88400608E-07	—	0.0071
	$2^{-11}$	2.22067159E-07	2.000214	0.2935	2.22067163E-07	2.000214	0.0074
	$2^{-12}$	5.55126489E-08	2.000108	1.1452	5.55126587E-08	2.000107	0.0083
	$2^{-13}$	1.38776781E-08	2.000050	4.3128	1.38775857E-08	2.000060	0.0099
	$2^{-14}$	3.46934370E-09	2.000032	17.1595	3.46943940E-09	1.999982	0.0199

| Show Table

DownLoad: CSV

Moreover, we manifest the efficiency of our fast scheme in two aspects: (i) by comparing with a published second-order scheme ^[35] which is denoted as Scheme I and (ii) with the direct scheme (4.5). In , we take $T = 10$ and plot the CPU time consumed for Scheme I and our fast scheme under the condition $|Err(\Delta_t)|\leq 10^{-7}$ for each $\alpha = 0.1, 0.2, \cdots, 0.9$ . It is evident that our fast scheme is much more efficient. Further, to check the computing complexity of our direct and fast schemes, we depict in the CPU time in seconds needed with $\alpha = 0.1$ in the $\log$ - $\log$ coordinate system, by taking $T = 1$ , $M = 10^3\times 2^m$ , $m = 1, 2, \cdots, 6$ . One can see that the fast scheme has reduced the computing complexity from $O(M^2)$ to $O(M)$ .

Figure 1. Comparison of CPU time between our fast method and the Scheme I with the error satisfying

$|Err(\Delta_t)|\leq 10^{-7}$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. CPU time for Example

$1$ with

$\alpha = 0.1$ .

DownLoad: Full-Size Img PowerPoint

4.2. Example 2

We next consider another initial value problem of the fractional ordinary differential equation:

$\begin{equation} \begin{split} \left\{ \begin{aligned} &\frac{du(t)}{dt}+_{0}^{CF}\partial_{t}^{\alpha}u(t) = g_{2}(t),\ t\in \bar{I},\\ &u(0) = \psi_{0}, \end{aligned} \right. \end{split} \end{equation}$

(4.7)

where the exact solution is $u(t) = \exp(2t)$ , the initial value is $\psi_0 = 1$ , and the source function is:

$\begin{equation} \begin{split} g_{2}(t) = \frac{2}{2-\alpha}\bigg[\exp(2t)-\exp\bigg(-\frac{\alpha}{1-\alpha}t\bigg)\bigg]+2\exp(2t). \end{split} \end{equation}$

(4.8)

Direct scheme: For the model (4.7), we formulate the Crank-Nicolson scheme based on the new approximation formula (2.4) at $t_{k+\frac{1}{2}}$ as follows:

Case $k = 0$

$\begin{equation} \begin{split} \bigg(\frac{1}{\Delta_{t}}+\frac{1-M_{0}^{\frac{1}{2}}}{\alpha\Delta_{t}}\bigg)U^{1} = \bigg(\frac{1}{\Delta_{t}}+\frac{1-M_{0}^{\frac{1}{2}}}{\alpha\Delta_{t}}\bigg)U^{0}+g_{2}(t_{\frac{1}{2}}), \end{split} \end{equation}$

(4.9)

Case $k\geq 1$

$\begin{equation} \begin{split} \bigg(\frac{1}{\Delta_{t}}+\frac{1-M_{k-\frac{1}{2}}^{k+\frac{1}{2}}}{2\alpha\Delta_{t}}\bigg)U^{k+1} = &\frac{1-M_{k-\frac{1}{2}}^{k+\frac{1}{2}}}{2\alpha\Delta_{t}}U^{k-1}+\frac{1}{\Delta_{t}}U^{k}-\frac{M_{\frac{1}{2}}^{k+\frac{1}{2}}-M_{0}^{k+\frac{1}{2}}}{\alpha\Delta_{t}}(U^{1}-U^{0})\\ &-\frac{1}{2\alpha\Delta_{t}}\sum\limits_{j = 1}^{k-1}(U^{j+1}-U^{j-1})(M_{j+\frac{1}{2}}^{k+\frac{1}{2}}-M_{j-\frac{1}{2}}^{k+\frac{1}{2}}) +g_{2}(t_{k+\frac{1}{2}}). \end{split} \end{equation}$

(4.10)

Fast scheme: Applying the fast algorithm to the model (4.7), we have, for $k\geq 1$ :

$\begin{equation} \begin{split} \bigg(\frac{1}{\Delta_t}+\frac{1-M_{k-\frac{1}{2}}^{k+\frac{1}{2}}}{2\alpha\Delta_{t}}\bigg)U^{k+1} = &\frac{1-M_{k-\frac{1}{2}}^{k+\frac{1}{2}}}{2\alpha\Delta_{t}}U^{k-1}+\frac{U^{k}}{\Delta_t}+g_{1}(t_{k+\frac{1}{2}})-\mathcal{F}_h(t_{k+\frac{1}{2}}). \end{split} \end{equation}$

(4.11)

Similarly, we also compute and list the convergence data in Table 2 to show further the effectiveness of the novel approximation and the fast algorithm.

Table 2. Convergence results of Example 2.

$\alpha$	$\Delta_t$	Direct scheme			Fast scheme
$\alpha$	$\Delta_t$	$Err(\Delta_t)$	Rate	CPU(s)	$Err(\Delta_t)$	Rate	CPU(s)
0.2	$2^{-10}$	1.85604636E-06	—	0.0620	1.85604607E-06	—	0.0066
	$2^{-11}$	4.64007204E-07	2.000014	0.2317	4.64006418E-07	2.000016	0.0075
	$2^{-12}$	1.16000631E-07	2.000015	0.8691	1.16003294E-07	1.999979	0.0091
	$2^{-13}$	2.90011650E-08	1.999950	3.3423	2.89965145E-08	2.000214	0.0107
	$2^{-14}$	7.24794447E-09	2.000467	13.4227	7.25755100E-09	1.998325	0.0154
0.4	$2^{-10}$	1.78086742E-06	—	0.0638	1.78086759E-06	—	0.0071
	$2^{-11}$	4.45204337E-07	2.000041	0.2516	4.45203661E-07	2.000043	0.0080
	$2^{-12}$	1.11298823E-07	2.000029	0.9043	1.11298892E-07	2.000026	0.0095
	$2^{-13}$	2.78237513E-08	2.000049	3.5048	2.78246395E-08	2.000004	0.0110
	$2^{-14}$	6.95804836E-09	1.999562	14.8677	6.95847024E-09	1.999521	0.0143
0.8	$2^{-10}$	3.89820152E-07	—	0.0785	3.89820181E-07	—	0.0115
	$2^{-11}$	9.73901404E-08	2.000961	0.2962	9.73902248E-08	2.000960	0.0079
	$2^{-12}$	2.43394851E-08	2.000477	1.0595	2.43393918E-08	2.000484	0.0095
	$2^{-13}$	6.08529405E-09	1.999900	4.1052	6.08559336E-09	1.999823	0.0116
	$2^{-14}$	1.51925406E-09	2.001964	16.5226	1.51877799E-09	2.002487	0.0156

| Show Table

DownLoad: CSV

From the computed data summarized in , both of the schemes have a second-order convergence rate, and the fast scheme indeed improves the efficiency of the novel approximation formula without losing too much precision. Similarly as the Example $1$ , we compare in the times for both of the methods under different $M = 10^2 \times 2^m$ , for $\alpha = 0.9$ and $m = 1, 2, \cdots, 6$ in the $\log$ - $\log$ coordinate system. One can see clearly that the computing complexity for the direct scheme is $O(M^2)$ , and for the fast scheme it is $O(M)$ .

Figure 3. CPU time for Example

$2$ with

$\alpha = 0.9$ .

DownLoad: Full-Size Img PowerPoint

5. Conclusion and suggestions for future study

In this study, we constructed a novel discrete formula for approximating the CF fractional derivative and proved the second-order convergence rate for the novel approximation formula. To overcome the nonlocal property of the derivative, we proposed a fast algorithm that tremendously improves the efficiency of the approximation formula. Moreover, we demonstrated the fast algorithm maintains the second-order convergence rate. In future works, this novel approximation formula and fast algorithm can be applied with the finite element, finite difference, or other numerical methods to specific fractional differential equation models with Caputo-Fabrizio derivatives.

Acknowledgments

The authors are grateful to the three anonymous referees and editors for their valuable comments and good suggestions which greatly improved the presentation of the paper. This work is supported by the National Natural Science Fund (11661058, 11761053), the Natural Science Fund of Inner Mongolia Autonomous Region (2017MS0107), the program for Young Talents of Science, and Technology in Universities of the Inner Mongolia Autonomous Region (NJYT-17-A07).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. No. OSU/DGSS-355, Ohio State University.
[2]	Müller-wohlfeil DI, lahmer W, krysanova V, et al. (1996) Topography-based hydrological modeling in the Elbe River drainage basin. In: Third International Conference/Workshop on Integrating GIS and Environmental Modeling, National Center for Geographic Information and Analysis, C.A, Santa Fe.
[3]	Mark DM, Smith B (2004) A science of topography: from qualitative ontology to digital representations. In: Bishop MP, Shroder JF (Eds.), Geographic Information Science and Mountain Geomorphology, Springer-Praxis, Chichester, England, 75-97.
[4]	Khal M, Algouti Ab, Algouti A (2018) Modeling of Water Erosion in the M'Goun Watershed Using OpenGIS Software. In: World Academy of Science, Engineering and Technology International Journal of Computer and Systems Engineering, 12: 1102-1106.
[5]	Mcluckie D, NFRAC (2008) Flood risk management in Australia. Aust J Emerg Manag 23: 21-27.
[6]	Ait Mlouk M, Algouti Ab, Algouti Ah, et al. (2018) Assessment of river bank erosion in semi-arid climate regions using remote sensing and GIS data: a case study of Rdat River, Marrakech, Morocco. Estud Geol 74: 81. doi: 10.3989/egeol.43217.493
[7]	Williams J (2009) Weather Forecasting. The AMS Weather Book: The Ultimate Guide to America's Weather. American Meteorological Society, Boston, MA. doi: 10.1007/978-1-935704-55-3
[8]	Da ros D, Borga M (1997) Use of digital elevation model data for the derivation of the geomorphological instantaneous unit hydrograph. Hydrol Process 11: 13-33. doi: 10.1002/(SICI)1099-1085(199701)11:1<13::AID-HYP400>3.0.CO;2-M
[9]	Tesfa TK, Tarboton DG, Watson DW, et al. (2011) Extraction of hydrological proximity measures from DEMs using parallel processing. Environ Model Softw 26: 1696-1709. doi: 10.1016/j.envsoft.2011.07.018
[10]	Jobin T, Prasannakumar V (2015) Comparison of basin morphometry derived from topographic maps, ASTER and SRTM DEMs: an example from Kerala, India. Geocarto Int 30: 346-364. doi: 10.1080/10106049.2014.955063
[11]	Kishan SR, Anil KM, Vinay KS, et al. (2012) Comparative evaluation of horizontal accuracy of elevations of selected ground control points from ASTER and SRTM DEM with respect to CARTOSAT-1 DEM: a case study of Shahjahanpur district, Uttar Pradesh, India. Geocarto Int 28: 439-452.
[12]	Tian Y, Lei S, Bian Z, et al. (2018) Improving the Accuracy of Open Source Digital Elevation Models with Multi-Scale Fusion and a Slope Position-Based Linear Regression Method. Remote Sens 10: 1861. doi: 10.3390/rs10121861
[13]	Cuartero A, Felicsimo AM, Ariza FJ (2004) Accuracy of DEM generation from TERRA-ASTER stereo data. Int Arch Photogramm Remote Sens 35: 559-563.
[14]	Day T, Muller J (1988) Quality assessment of digital elevation models produced by automatic stereo-matchers from SPOT image pairs. Photogramm Rec 12: 797-808. doi: 10.1111/j.1477-9730.1988.tb00630.x
[15]	Fujisada H (1994) Overview of ASTER instrument on EOS-AM1 platform. In: Proceedings of SPIE, 2268: 14-36. doi: 10.1117/12.185838
[16]	Toutin T (2008) ASTER DEMs for geomatic and geoscientific applications. Int J Remote Sens 29: 1855-1875. doi: 10.1080/01431160701408477
[17]	Bolstad PV, Stowe T (1994) An evaluation of DEM accuracy: elevation, slope, and aspect. Photogramm Eng Remote Sens 60: 1327-1332.
[18]	Blöschl G, Sivapalan M (1995) Scale issues in hydrological modelling: A review. Hydrol Process 9: 251-290. doi: 10.1002/hyp.3360090305
[19]	Vijith H, Seling LW, Dodge-Wan D (2015) Comparison and Suitability of SRTM and ASTER Digital Elevation Data for Terrain Analysis and Geomorphometric Parameters: Case Study of Sungai PatahSubwatershed (Baram River, Sarawak, Malaysia). Environ Res Eng Manag 71: 23-35. doi: 10.5755/j01.erem.71.3.12566
[20]	Beven KJ, Moore ID (1993) Terrain analysis and distributed modelling in hydrology. New York: Wiley.
[21]	Wang XH, Yin ZY (1998) A comparison of drainage networks derived from digital elevation models at two scales. J Hydrol 210: 221-241. doi: 10.1016/S0022-1694(98)00189-9
[22]	Wang W, Yang X, Yao T (2012) Evaluation of ASTER GDEM and SRTM and their suitability in hydraulic modelling of a glacial lake outburst flood in southeast Tibet. Hydrol Process 26: 213-225. doi: 10.1002/hyp.8127
[23]	Nikolakopoulos KG, Kamaratakis EK, Chrysoulakis N (2006) SRTM vs ASTER elevation products. Comparison for two regions in Crete, Greece. Int J Remote Sens 27: 4819-4838. doi: 10.1080/01431160600835853
[24]	Pryde JK, Osorio J, Wolfe ML, et al. (2007) USGS. An ASABE Meeting Presentation Paper Number: 072093, Minneapolis Convention Center Minneapolis, Minnesota, June; 072093.
[25]	Jing C, Shortridge A, Lin S, et al. (2014) Comparison and validation of SRTM and ASTER GDEM for a subtropical landscape in Southeastern China. Int J Digit Earth 7: 969-992. doi: 10.1080/17538947.2013.807307
[26]	Dewitt JD, Warner TA, Conley JF (2015) Comparison of DEMS derived from USGS DLG, SRTM, a statewide photogrammetry program, ASTER GDEM and LiDAR: implications for change detection. GIScience Remote Sens 52: 179-197. doi: 10.1080/15481603.2015.1019708
[27]	Moudrý V, Lecours V, Gdulová K, et al. (2018) On the use of global DEMs in ecological modelling and the accuracy of new bare-earth DEMs. Ecol Modell 383: 3-9. doi: 10.1016/j.ecolmodel.2018.05.006
[28]	Zhang K, Gann D, Ross M, et al. (2019) Comparison of TanDEM-X DEM with LiDAR Data for Accuracy Assessment in a Coastal Urban Area. Remote Sens 11: 876. doi: 10.3390/rs11070876
[29]	Kinsey-Henderson AE, Wilkinson SN (2012) Evaluating Shuttle radar and interpolated DEMs for slope gradient and soil erosion estimation in low relief terrain. Environ Modell Softw 40: 128-139. doi: 10.1016/j.envsoft.2012.08.010
[30]	Lin S, Jing C, Coles NA, et al. (2013) Evaluating DEM source and resolution uncertainties in the Soil and Water Assessment Tool. Stoch Environ Res Risk Assess 27: 209-221. doi: 10.1007/s00477-012-0577-x
[31]	Williams JR, Berndt HD (1977) Sediment yield prediction based on watershed hydrology. Transactions of the American Society of Agricultural and Biological Engineers. Trans ASAE 20: 1100-1104. doi: 10.13031/2013.35710
[32]	Rexer M, Hirt C (2014) Comparison of free high-resolution digital elevation data sets (ASTER GDEM2, SRTM v2.1/v4.1) and validation against accurate heights from the Australian National Gravity Database. Aust J Earth Sci 61: 213-226.
[33]	Renard KG, Foster GR, Weesies GA, et al. (1997) Predicting soil erosion by water: a guide to conservation planning with the revised universal soil loss equation (RUSLE). Agriculture Handbook, U.S. Department of Agriculture, No 703, 404.
[34]	Prasuhn V, Liniger H, Gisler S, et al. (2013) A high-resolution soil erosion risk map of Switzerland as strategic policy support system. Land Use Policy 32: 281-291. doi: 10.1016/j.landusepol.2012.11.006
[35]	Mondal A, Khare D, Kundu S, et al. (2016) Uncertainty of soil erosion modelling using open source high resolution and aggregated DEMs. Geosci Front 8: 425-436. doi: 10.1016/j.gsf.2016.03.004
[36]	Mondal A, Khare D, Kundu S (2017) Uncertainty analysis of soil erosion modelling using different resolution of open-source DEMs. Geocarto Int 32: 334-349. doi: 10.1080/10106049.2016.1140822
[37]	Uhlemann S, Thieken AH, Merz B (2014) A quality assessment framework for natural hazard event documentation: application to trans-basin flood reports in Germany. Nat Hazards Earth Syst Sci 14: 189-208. doi: 10.5194/nhess-14-189-2014
[38]	USGS (2006) Earth Resources Observation and Science. Available from: https://www.usgs.gov/centers/eros.
[39]	Wang L, Liu H (2006) An efficient method for identifying and filling surface depressions in digital elevation models for hydrologic analysis and modelling. Int J Geogr Inf Sci 20: 193-213. doi: 10.1080/13658810500433453
[40]	Das A, Agrawala R, Mohan S (2015) Topographic correction of ALOS-PALSAR images using InSAR-derived DEM. Geocarto Int 30: 145-153.
[41]	Jäger R, Kaminskis J, Balodis J (2012) Determination of Quasi-geoid as Height Component of the Geodetic Infrastructure for GNSS-Positioning Services in the Baltic States. Latv J Phys Tech Sci 49: 2.
[42]	Ghilani CD, Wolf PR (2006) Adjustment Computations: Spatial Data Analysis, 4th Edition, John Wiley & Sons, Hoboken.
[43]	Al-Fugara A (2015) Comparison and Validation of the Recent Freely Available DEMs over Parts of the Earth's Lowest Elevation Area: Dead Sea, Jordan. Int J Geosci 6: 1221-1232. doi: 10.4236/ijg.2015.611096
[44]	Shaw EM (1988) Van Nostrand Reinhold International, London, United Kingdom. Hydrology in practice.
[45]	Strahler AN (1964) Quantitative geomorphology of drainage basin and channel network. In Chow VT (ed), Handbook of Applied Hydrology, McGrawHill, NewYork, NY, USA.
[46]	Wanielista MP, Kersten R, Eaglin R (1997) Hydrology: Water Quantity and Quality Control, Wiley, New York.
[47]	Musy A (2001) Ecole Polytechnique Fédérale, Lausanne, Suisse, e-drologie.
[48]	Roche M (1963) Hydrologie de Surface. Gauthier-Villars, Paris, 140: 659.
[49]	Horton RE (1945) Erosional development of streams and their drainage basins: hydro physical approach to quantitative morphology. Geol Soc Am Bull 56: 275-370. doi: 10.1130/0016-7606(1945)56[275:EDOSAT]2.0.CO;2
[50]	Strahler AN (1952) Hypsometric analysis of erosional topography. Bull Geol Soc Am 63: 1117-1142. doi: 10.1130/0016-7606(1952)63[1117:HAAOET]2.0.CO;2
[51]	Schumm SA (1956) Evolution of drainage systems and slopes in badlands at perth amboy, new jersey. Geol Soc Am Bull 67: 597-646. doi: 10.1130/0016-7606(1956)67[597:EODSAS]2.0.CO;2
[52]	Beven KJ, Kirkby MJ (1979) A physically based, variable contributing area model of basin hydrology. Hydrol Sci Bull 24: 43-69. doi: 10.1080/02626667909491834
[53]	Pandey A, Chowdary VM, Mal BC (2007) Identification of critical erosion prone areas in the small agricultural watershed using USLE, GIS and remote sensing. Water Resour Manage 21: 729-746. doi: 10.1007/s11269-006-9061-z
[54]	Freeman TG (1991) Calculating Catchment Area With Divergent Flow Based on a Regular Grid. Comput Geosci 17: 413-422. doi: 10.1016/0098-3004(91)90048-I
[55]	Kamp U, Bolch T, Olsenholler J (2005) Geomorphometry of Cerro Sillajhuay (Andes, Chile/Bolivia): Comparison of digital elevation models (DEMs) from ASTER remote sensing data and contour maps. Geocarto Int 20: 23-33. doi: 10.1080/10106040508542333
[56]	Datta PS, Schack-Kirchner H (2010) Erosion Relevant Topographical Parameters Derived from Different DEMs-A Comparative Study from the Indian Lesser Himalayas. Remote Sens 2: 1941-1961. doi: 10.3390/rs2081941
[57]	Luo W (1998) Hypsometric analysis with a geographic information system. Comput Geosci 24: 815-821. doi: 10.1016/S0098-3004(98)00076-4
[58]	Vaze J, Teng J, Spencer G (2010) Impact of DEM accuracy and resolution on topographic indices. Environ Modell Softw 25: 1086-1098. doi: 10.1016/j.envsoft.2010.03.014
[59]	Holmes KW, Chadwick OA, Kyriankidis PC (2000) Error in USGS 30-meter digital elevation model and its impact on terrain modelling. J Hydrol 233: 154-173. doi: 10.1016/S0022-1694(00)00229-8
[60]	Huggel C, Schneider D, Miranda PJ, et al. (2008) Evaluation of ASTER and SRTM DEM data for lahar modelling: A case study on lahars from Popocatépetl volcano, Mexico. J Volcanol Geotherm Res 170: 99-110. doi: 10.1016/j.jvolgeores.2007.09.005
[61]	De Vente J, Poesen J, Govers G, et al. (2009) The implications of data selection for regional erosion and sediment yield modelling. Earth Surf Process Landf 34: 1994-2007. doi: 10.1002/esp.1884
[62]	Nitheshnirmal S, Thilagaraj P, Abdul Rahaman S, et al. (2019) Erosion risk assessment through morphometric indices for prioritisation of Arjuna watershed using ALOS-PALSAR DEM. Model Earth Syst Environ 5: 907-924. doi: 10.1007/s40808-019-00578-y
[63]	Bhakar R, Srivastav SK, Punia M (2010) Assessment of the relative accuracy of aster and SRTM digital elevation models along irrigation channel banks of Indira Gandhi Canal. J Water Land-use Manage 10: 1-11.
[64]	Hasan A, Pilesjo P, Persson A (2011) The use of LIDAR as a data source for digital elevation models-a study of the relationship between the accuracy of digital elevation models and topographical attributes in northern peatlands. Hydrol Earth Syst Sci Discuss 8: 5497-5522. doi: 10.5194/hessd-8-5497-2011

This article has been cited by:

1.	Saïd Abbas, Mouffak Benchohra, Juan J. Nieto, Caputo–Fabrizio fractional differential equations with non instantaneous impulses, 2021, 0009-725X, 10.1007/s12215-020-00591-6
2.	Zahir Shah, Rashid Jan, Poom Kumam, Wejdan Deebani, Meshal Shutaywi, Fractional Dynamics of HIV with Source Term for the Supply of New CD4+ T-Cells Depending on the Viral Load via Caputo–Fabrizio Derivative, 2021, 26, 1420-3049, 1806, 10.3390/molecules26061806
3.	Rajarama Mohan Jena, Snehashish Chakraverty, Dumitru Baleanu, Maysaa M. Alqurashi, New Aspects of ZZ Transform to Fractional Operators With Mittag-Leffler Kernel, 2020, 8, 2296-424X, 10.3389/fphy.2020.00352
4.	Fatima Si Bachir, Saïd Abbas, Maamar Benbachir, Mouffak Benchohra, Successive Approximations for Caputo-Fabrizio Fractional Differential Equations, 2022, 81, 1338-9750, 117, 10.2478/tmmp-2022-0009
5.	Zahir Shah, Ebenezer Bonyah, Ebraheem Alzahrani, Rashid Jan, Nasser Aedh Alreshidi, Mariya Gubareva, Chaotic Phenomena and Oscillations in Dynamical Behaviour of Financial System via Fractional Calculus, 2022, 2022, 1099-0526, 1, 10.1155/2022/8113760
6.	Saïd Abbas, Mouffak Benchohra, Juan J. Nieto, Caputo-Fabrizio fractional differential equations with instantaneous impulses, 2021, 6, 2473-6988, 2932, 10.3934/math.2021177
7.	Rashid Jan, Salah Boulaaras, Hussain Ahmad, Muhammad Jawad, Sulima Zubair, Mohamed Abdalla, A Robust Study of Tumor-Immune Cells Dynamics through Non-Integer Derivative, 2023, 7, 2504-3110, 164, 10.3390/fractalfract7020164
8.	Shorog Aljoudi, Existence and uniqueness results for coupled system of fractional differential equations with exponential kernel derivatives, 2022, 8, 2473-6988, 590, 10.3934/math.2023027
9.	Abdulrahman Obaid Alshammari, Imtiaz Ahmad, Rashid Jan, Sahar Ahmed Idris, Fractional-calculus analysis of the dynamics of $\text {CD4}^{+}$ T cells and human immunodeficiency viruses, 2024, 1951-6355, 10.1140/epjs/s11734-024-01192-5
10.	Tao-Qian Tang, Rashid Jan, Hassan Ahmad, Zahir Shah, Narcisa Vrinceanu, Mihaela Racheriu, A Fractional Perspective on the Dynamics of HIV, Considering the Interaction of Viruses and Immune System with the Effect of Antiretroviral Therapy, 2023, 30, 1776-0852, 1327, 10.1007/s44198-023-00133-5
11.	Salah Boulaaras, Rashid Jan, Amin Khan, Ali Allahem, Imtiaz Ahmad, Salma Bahramand, Modeling the dynamical behavior of the interaction of T-cells and human immunodeficiency virus with saturated incidence, 2024, 76, 0253-6102, 035001, 10.1088/1572-9494/ad2368
12.	Bin Fan, Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives, 2024, 9, 2473-6988, 7293, 10.3934/math.2024354
13.	Mubashara Wali, Sadia Arshad, Sayed M Eldin, Imran Siddique, Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations, 2023, 8, 2473-6988, 15129, 10.3934/math.2023772

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)