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Research article

Broiler chicken carcasses and their associated abattoirs as a source of enterotoxigenic Clostridium perfringens: Prevalence and critical steps for contamination

  • Received: 22 March 2018 Accepted: 31 May 2018 Published: 13 June 2018
  • Clostridium perfringens ranks among the three most frequent bacterial pathogens causing human foodborne diseases in Canada, and poultry meat products are identified as a source of infection for humans. The objective of the current study was to estimate the proportion of broiler chicken flocks, carcasses and various environmental samples from critical locations of the slaughter plant positive for the presence of C. perfringens enterotoxin encoding gene (cpe). From the 16 visits conducted, 25% of the 79 flocks sampled, 10% of the 379 carcasses sampled and 5% of the 217 environmental samples collected were found positive for cpe. The proportion of cpe-positive carcasses was statistically different between surveyed plants, with 17.0% for one abattoir and 2.2% for the other. For the most contaminated plant, cpe-positive carcasses were identified at each step of the processing line, with prevalence varying between 10.0% and 25.0%, whereas this prevalence varied between 0% and 25.0% for the environmental surfaces sampled. Based on the results obtained, enterotoxigenic C. perfringens strains could potentially represent a risk for the consumer.

    Citation: Marie-Lou Gaucher, Alexandre Thibodeau, Philippe Fravalo, Marie Archambault, Julie Arsenault, Sylvain Fournaise, Ann Letellier, Sylvain Quessy. Broiler chicken carcasses and their associated abattoirs as a source of enterotoxigenic Clostridium perfringens: Prevalence and critical steps for contamination[J]. AIMS Microbiology, 2018, 4(3): 439-454. doi: 10.3934/microbiol.2018.3.439

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  • Clostridium perfringens ranks among the three most frequent bacterial pathogens causing human foodborne diseases in Canada, and poultry meat products are identified as a source of infection for humans. The objective of the current study was to estimate the proportion of broiler chicken flocks, carcasses and various environmental samples from critical locations of the slaughter plant positive for the presence of C. perfringens enterotoxin encoding gene (cpe). From the 16 visits conducted, 25% of the 79 flocks sampled, 10% of the 379 carcasses sampled and 5% of the 217 environmental samples collected were found positive for cpe. The proportion of cpe-positive carcasses was statistically different between surveyed plants, with 17.0% for one abattoir and 2.2% for the other. For the most contaminated plant, cpe-positive carcasses were identified at each step of the processing line, with prevalence varying between 10.0% and 25.0%, whereas this prevalence varied between 0% and 25.0% for the environmental surfaces sampled. Based on the results obtained, enterotoxigenic C. perfringens strains could potentially represent a risk for the consumer.


    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]

    $ CF0αtu(t)=11αt0u(s)exp[α1α(ts)]ds,0<α<1, $ (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 $.

    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) $

    $ CF0αtu(tk+12)=11αkj=1tj+12tj12u(s)exp[α1α(tk+12s)]ds+11αt12t0u(s)exp[α1α(tk+12s)]ds=11αkj=1tj+12tj12[u(tj)+u(tj)(stj)+12u(ξj)(stj)2]exp[α1α(tk+12s)]ds+11αt12t0[u(t12)+u(t12)(st12)+12u(ξ0)(st12)2]exp[α1α(tk+12s)]ds=1αkj=1uj+1uj12Δt(Mk+12j+12Mk+12j12)+1αu1u0Δt(Mk+1212Mk+120)+Rk+121+Rk+122CF0Dαtu(tk+12)+Rk+12, $ (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

    $ Mkj=exp[α1α(tktj)],     Rk+12=Rk+121+Rk+122, $ (2.2)
    $ Rk+121=11αkj=1tj+12tj12[u(tj)(stj)+12u(ξj)(stj)2]exp[α1α(tk+12s)]ds+11αt12t0[u(t12)(st12)+12u(ξ0)(st12)2]exp[α1α(tk+12s)]ds,Rk+122=11αO(Δ2t)tk+12t0exp[α1α(tk+12s)]ds. $ (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 $.

    $ CF0Dαtu(tk+12)=1αkj=1uj+1uj12Δt(Mk+12j+12Mk+12j12)+1αu1u0Δt(Mk+1212Mk+120). $ (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

    $ |Rk+12|CΔ2t, $ (2.5)

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

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

    $ |Rk+12||Rk+121|+|Rk+122||11αkj=1u(tj)tj+12tj12(stj)exp[α1α(tk+12s)]ds|+|11αu(t12)t12t0(st12)exp[α1α(tk+12s)]ds|+|12(1α)kj=1tj+12tj12u(ξj)(stj)2exp[α1α(tk+12s)]ds|+|12(1α)t12t0u(ξ0)(st12)2exp[α1α(tk+12s)]ds|+|11αtk+12t0O(Δ2t)exp[α1α(tk+12s)]ds|=I1+I2+I3+I4+I5. $ (2.6)

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

    $ I1maxt[0,T]|u(t)|Δ2t8(1α)kj=1{exp[α1α(tk+12tj+12)]exp[α1α(tk+12tj12)]}+maxt[0,T]|u(t)|αΔ2t8(1α)2tk+12t12exp[α1α(tk+12s)]dsmaxt[0,T]|u(t)|Δ2t4(1α)(1Mk+1212)CΔ2t. $ (2.7)

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

    $ I2=|Δt2(1α)u(t12)(tεt12)exp[α1α(tk+12tε)]|maxt[0,T]|u(t)|Δ2t4(1α)CΔ2t, $ (2.8)

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

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

    $ I3maxt[0,T]|u(t)|Δ2t8(1α)tk+12t12exp[α1α(tk+12s)]dsCΔ2t. $ (2.9)

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

    $ I4maxt[0,T]|u(t)|Δ2t8(1α)t12t0exp[α1α(tk+12s)]dsCΔ2t. $ (2.10)

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

    $ I511α|O(Δ2t)|(1Mk+120)CΔ2t. $ (2.11)

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

    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

    $ CF0αtu(tk+12)=Ch(tk+12)+Cl(tk+12)=11αtk12t0u(s)exp[α1α(tk+12s)]ds+11αtk+12tk12u(s)exp[α1α(tk+12s)]ds. $ (3.1)

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

    $ Cl(tk+12)=uk+1uk12αΔt(1Mk+12k12)+Rk+12l, $ (3.2)

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

    $ Rk+12l=11αtk+12tk12[u(tk)(stk)+12u(ξk)(stk)2]exp[α1α(tk+12s)]ds+11α[u(tk)uk+1uk12Δt]tk+12tk12exp[α1α(tk+12s)]ds,k1. $ (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

    $ Ch(tk+12)=11αtk32t0u(s)exp[α1α(tk+12s)]ds+11αtk12tk32u(s)exp[α1α(tk+12s)]dsC(1)h(tk+12)+C(2)h(tk+12), $ (3.4)

    and when $ k = 1 $,

    $ Ch(t32)=11αt12t0u(s)exp[α1α(t32s)]dsC(2)h(t32). $ (3.5)

    Careful calculations show that

    $ C(1)h(tk+12)=exp(αΔtα1)Ch(tk12),k2. $ (3.6)

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

    $ C(2)h(tk+12)={ukuk22αΔt(Mk+12k12Mk+12k32)+Rk+12h,if k2u1u0αΔt(M3212M320)+R32h,if k=1, $ (3.7)

    where, for $ k\geq 2 $,

    $ Rk+12h=11αtk12tk32[u(tk1)(stk1)+12u(ξk1)(stk1)2]exp[α1α(tk+12s)]ds+11α[u(tk1)ukuk22Δt]tk12tk32exp[α1α(tk+12s)]ds, $ (3.8)

    and, for $ k = 1 $,

    $ R32h=11αt12t0[u(t12)(st12)+12u(ξ0)(st12)2]exp[α1α(tk+12s)]ds+11α[u(t12)u1u0Δt]t12t0exp[α1α(t32s)]ds. $ (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

    $ |Rk+12l|CΔ3t,|Rk+12h|CΔ3t, $ (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

    $ |Rk+12l|11αmaxt[0,T]|u(t)||tk+12tk12(stk)exp[α1α(tk+12s)]ds|+12(1α)maxt[0,T]|u(t)|tk+12tk12(stk)2exp[α1α(tk+12s)]ds+CΔ2t1αtk+12tk12exp[α1α(tk+12s)]ds.L1+L2+L3. $ (3.11)

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

    $ L1CΔ2t(Mk+12k+12Mk+12k12)+Ctk+12tk12(stk)2exp[α1α(tk+12s)]dsCΔ2t[1(1αΔt1α|O(Δ2t)|)]+CΔ3tCΔ3t. $ (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

    $ CF0Fαtu(tk+12)=uk+1uk12αΔt(1Mk+12k12)+Fh(tk+12),k1, $ (3.13)

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

    $ Fh(tk+12)={exp(αΔtα1)Fh(tk12)+ukuk22αΔt(Mk+12k12Mk+12k32),if k2u1u0αΔt(M3212M320),if k=1. $ (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

    $ |CF0αtu(tk+12)CF0Fαtu(tk+12)|CΔ2t, $ (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

    $ |CF0αtu(tk+12)CF0Fαtu(tk+12)||Ch(tk+12)Fh(tk+12)|+|Rk+12l|,k2. $ (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

    $ Ch(tk+12)Fh(tk+12)=exp(αΔtα1)[Ch(tk12)Fh(tk12)]+Rk+12h. $ (3.17)

    We introduce some notations to simplify the presentation. Let

    $ Tk+1=Ch(tk+12)Fh(tk+12),L=exp(αΔtα1). $ (3.18)

    Then, the recursive formula (3.17) reads that

    $ Tk+1=Lk1T2+Rk+1h, $ (3.19)

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

    $ Rk+1h=Lk2R2+12h+Lk3R3+12h++Rk+12h. $ (3.20)

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

    $ |T2|=|R32h|CΔ3t, $ (3.21)

    and

    $ |Rk+1h|CΔ3t(Lk2+Lk3++1)=CΔ3t1Lk11L. $ (3.22)

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

    $ 1L=1exp(αΔtα1)αΔt1ααΔt2(1α). $ (3.23)

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

    $ |Tk+1|CΔ2t. $ (3.24)

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

    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}}} $:

    $ u(tk+12)=12(uk+uk+1)+O(Δ2t),dudt|t=tk+12=uk+1ukΔt+O(Δ2t). $ (4.1)

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

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

    $ {CF0αtu(t)+u(t)=g1(t), tˉI,u(0)=φ0. $ (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:

    $ g1(t)=2tα+t22(1α)α2[1exp(α1αt)]. $ (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 $

    $ (12+1M120αΔt)U1=(12+1M120αΔt)U0+g1(t12), $ (4.4)

    Case $ k\geq 1 $

    $ (12+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk112UkMk+1212Mk+120αΔt(U1U0)12αΔtk1j=1(Uj+1Uj1)(Mk+12j+12Mk+12j12)+g1(tk+12). $ (4.5)

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

    $ (12+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk112Uk+g1(tk+12)Fh(tk+12), $ (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
    $ 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 Figure 1, 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 Figure 2 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} $.
    Figure 2.  CPU time for Example $ 1 $ with $ \alpha = 0.1 $.

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

    $ {du(t)dt+CF0αtu(t)=g2(t), tˉI,u(0)=ψ0, $ (4.7)

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

    $ g2(t)=22α[exp(2t)exp(α1αt)]+2exp(2t). $ (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 $

    $ (1Δt+1M120αΔt)U1=(1Δt+1M120αΔt)U0+g2(t12), $ (4.9)

    Case $ k\geq 1 $

    $ (1Δt+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk1+1ΔtUkMk+1212Mk+120αΔt(U1U0)12αΔtk1j=1(Uj+1Uj1)(Mk+12j+12Mk+12j12)+g2(tk+12). $ (4.10)

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

    $ (1Δt+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk1+UkΔt+g1(tk+12)Fh(tk+12). $ (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
    $ 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 Table 2, 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 Figure 3 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 $.

    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.

    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).

    The authors declare no conflict of interest.

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