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Cytokine profile and oxidative stress parameters in women with initial manifestations of pelvic venous insufficiency

  • Received: 05 March 2022 Revised: 27 June 2022 Accepted: 03 July 2022 Published: 27 July 2022
  • Pelvic venous insufficiency (PVI) in women is widespread and is closely associated with the risk of reproductive disorders (in 15–25% of patients) and a high rate of the disease recurrence after treatment. The factors involved in venous wall damage include atherogenic stimuli and chronic endotoxin aggression due to inflammatory processes. The changes in the initial stages of the disease are usually minor and selective. There is currently an urgent need to identify initial markers of these changes to develop preventive measures for their correction. Therefore, the aim of this study was to determine the cytokine profile parameters' levels, as well as the activity of lipid peroxidation (LPO) and antioxidant defense (AOD) reactions in women with initial manifestations of PVI. Thirty-nine female patients with PVI (mean age 37.4 ± 9.1 years old) were the subjects of the study. The diagnosis was verified by clinical and instrumental examination including ultrasound angioscanning of the pelvic veins and therapeutic and diagnostic laparoscopy, and it was finally confirmed histologically. The control group included 30 nearly healthy women (mean age 33.5 ± 6.3 years old) who underwent surgical sterilization by laparoscopic access. Spectrophotometric, fluorometric and immunoassay methods were used in the study. The cytokine profile in female patients with PVI, as compared to the control group, was characterized by an increased concentration of proinflammatory (interleukin (IL) (IL-6) and IL-8) and anti-inflammatory cytokines (IL-4 and IL-10) and higher ratio values (IL-6/IL-10). The level of primary LPO products, conjugated dienes, was significantly increased and level of final products TBARs values was decreased in comparison to the control. The AOD system main enzyme activity, superoxide dismutase (SOD), was decreased, while the catalase activity increased. In patients with PVI, the glutathione reduced form concentration was lower than in the control group. The results of the study in women with PVI suggest negative changes in the cytokine profile and multidirectional changes in the indicators of the LPO system state in the initial stages of the disease. The control of these changes in patients with PVI should probably be an important component of preventive measures in the initial stages of the disease.

    Citation: Marina A. Darenskaya, Dmitry A. Stupin, Andrey A. Semendyaev, Sergey I. Kolesnikov, Natalya V. Semenova, Lyubov I. Kolesnikova. Cytokine profile and oxidative stress parameters in women with initial manifestations of pelvic venous insufficiency[J]. AIMS Medical Science, 2022, 9(3): 414-423. doi: 10.3934/medsci.2022020

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  • Pelvic venous insufficiency (PVI) in women is widespread and is closely associated with the risk of reproductive disorders (in 15–25% of patients) and a high rate of the disease recurrence after treatment. The factors involved in venous wall damage include atherogenic stimuli and chronic endotoxin aggression due to inflammatory processes. The changes in the initial stages of the disease are usually minor and selective. There is currently an urgent need to identify initial markers of these changes to develop preventive measures for their correction. Therefore, the aim of this study was to determine the cytokine profile parameters' levels, as well as the activity of lipid peroxidation (LPO) and antioxidant defense (AOD) reactions in women with initial manifestations of PVI. Thirty-nine female patients with PVI (mean age 37.4 ± 9.1 years old) were the subjects of the study. The diagnosis was verified by clinical and instrumental examination including ultrasound angioscanning of the pelvic veins and therapeutic and diagnostic laparoscopy, and it was finally confirmed histologically. The control group included 30 nearly healthy women (mean age 33.5 ± 6.3 years old) who underwent surgical sterilization by laparoscopic access. Spectrophotometric, fluorometric and immunoassay methods were used in the study. The cytokine profile in female patients with PVI, as compared to the control group, was characterized by an increased concentration of proinflammatory (interleukin (IL) (IL-6) and IL-8) and anti-inflammatory cytokines (IL-4 and IL-10) and higher ratio values (IL-6/IL-10). The level of primary LPO products, conjugated dienes, was significantly increased and level of final products TBARs values was decreased in comparison to the control. The AOD system main enzyme activity, superoxide dismutase (SOD), was decreased, while the catalase activity increased. In patients with PVI, the glutathione reduced form concentration was lower than in the control group. The results of the study in women with PVI suggest negative changes in the cytokine profile and multidirectional changes in the indicators of the LPO system state in the initial stages of the disease. The control of these changes in patients with PVI should probably be an important component of preventive measures in the initial stages of the disease.


    Abbreviations

    PVI:

    Pelvic venous insufficiency; 

    LPO:

    Lipid peroxidation; 

    AOD:

    Antioxidant defense; 

    TNF:

    Tumor necrosis factor-alpha; 

    IL:

    Interleukin; 

    LH:

    Lipid hydroperoxides; 

    CDs:

    Conjugated dienes; 

    TBARs:

    Thiobarbituric acid reactants; 

    SOD:

    Superoxide dismutase; 

    GSH:

    Reduced glutathione; 

    GR:

    Glutathione reductase; 

    GPO:

    Glutathione peroxidase; 

    G-S-T:

    Glutathione-S-transferase; 

    SEM:

    Standard error of the mean

    Fractional calculus, as the name suggests, extends traditional integer-order calculus to fractional-order differential and integral calculus [1]. In recent decades, applications of fractional differential equations (FDEs) have received widespread attention in various disciplines, including physics [2], engineering [3] and biology [4]. Obtaining analytical solutions to FDEs is often challenging due to the nonlocal and complex nature of fractional derivatives. Therefore, it is crucial to develop efficient numerical methods for solving FDEs.

    In the literature, two main numerical approximation strategies for solving fractional ordinary differential equations (FODEs) have been extensively studied. The first strategy involves a direct approximation of the fractional derivatives in the FODEs, while the second strategy focuses on solving the equivalent Volterra integral equations. For the first strategy, a second-order numerical method for solving FODEs was proposed in [5], where the fractional derivatives are approximated by a weighted sum of function values at specified points. A finite difference method of order (2α) with L1, L2, and L2C formulas was established in [6], where α(0,1) is a scalar representing the fractional order of the derivative. The L1-2 formula, an enhanced version of the L1 formula, was proposed in [7] to achieve a convergence rate of 3α. Other approximation methods with the same convergence rate as obtained in [7] can be found in [8,9]. It is worth noting that the direct discretization method for fractional derivatives described in [8] is specifically designed for linear FODEs. Also, the convergence rates of most of these methods depend on the fractional order α.

    For the second strategy for solving FODEs, the classic (1+α)th-order predicting-correction method was introduced in [10], and a detailed error analysis of this method was carried out in [11]. In [12], a method with the convergence order of min{2,1+2α} was developed. In [13], a one-step numerical integration method with second-order convergence rate, which is independent of α, was proposed. The second method in [8] used piecewise quadratic interpolation polynomials to approximate the Volterra integral equation, achieving a convergence rate of 1+2α. In addition, the perturbed quadratic predictor-corrector (Q-PCP) and decomposed quadratic predictor-corrector (Q-PCD) methods were developed in [14], where the convergence rates of 3α and 3α2 were achieved, respectively. However, the convergence rates of almost all of the aforementioned methods depend on the fractional order α.

    In this paper, we present a third-order numerical scheme for the following general nonlinear FODEs:

    {Ct0Dαtx(t)=f(t,x(t)),t(t0,tf],x(t0)=x0, (1)

    where α=(α1,α2,,αn)(0,1]n is a given vector of fractional orders; x(t)=(x1(t),x2(t),,xn(t))Rn is the state vector at time t; f:R×RnRn is a given thrice continuously differentiable function with respect to all its arguments; t0 is a given initial time; tf is a given terminal time; x0=(x1(t0),x2(t0),,xn(t0))Rn is a given initial state vector; and Ct0Dαtx(t)=(Ct0Dα1tx1(t),Ct0Dα2tx2(t),,Ct0Dαntxn(t)) with Ct0Dαitxi(t) denoting the αith Liouville-Caputo fractional derivative of xi(t).

    We will develop a novel time-stepping numerical method to solve FODEs (1). Specifically, we first transform the FODEs into a set of equivalent Volterra integral equations. Then, we approximate these integral equations by using a third-order Taylor expansion (for the first two mesh points by a second-order Taylor expansion). This approximation leads to implicit nonlinear algebraic equations at each given mesh point. Then, Newton's method is employed to iteratively solve the nonlinear equations. A rigorous convergence analysis of the proposed method is carried out, which shows that the convergence rate from the numerical solution to the exact solution is third order and independent of the fractional order. Finally, the effectiveness and convergence results of the proposed method are illustrated through solving four numerical examples.

    The organization of the rest of the paper is as follows. Section 2 proposes the numerical method for solving nonlinear FODEs. Section 3 provides the convergence and error analysis. Section 4 gives numerical results of four numerical examples. Finally, conclusions are drawn in Section 5.

    Applying the Riemann-Liouville integral to both sides of FODEs (1) yields the following Volterra integral equations [1]:

    xi(t)=x0i+1Γ(αi)tt0(tτ)αi1fi(τ,x(τ))dτ,t[t0,tf], (2)

    for i=1,2,,n, where x0i=xi(t0). For a given positive integer N, define a mesh on [t0,tf] with N+1 mesh points tq,q=0,1,,N, satisfying t0<t1<<tN=tf. For any i=1,2,,n, and q=0,1,,N, Eq (2) can be rewritten as

    xi(tq)=x0i+1Γ(αi)q1j=0tj+1tj(tqτ)αi1fi(τ,x(τ))dτ. (3)

    We now consider the approximation of the integrand fi(τ,x(τ)) on the RHS of Eq (3) in (t0,t1] by the following Taylor expansion:

    fi(τ,x(τ))=ai0+bi0(τt1)+ci0(τt1)2, (4)

    where ai0 and bi0 are coefficients to be determined, and ci0(τt1)2 is the remainder. To determine ai0 and bi0, we omit the remainder and force the following equations to hold:

    {fi(t0,x0)=ai0+bi0(t0t1),fi(t1,x(t1))=ai0.

    Solving these coupled equations gives

    0f1i:=ai0=fi(t1,x(t1)),1f1i:=bi0=fi(t1,x(t1))fi(t0,x0)h1,

    where h1=t1t0. Using these divided differences, Eq (4) can be expressed as

    fi(τ,x(τ))=0f1i+1f1i(τt1)+ci0(τt1)2. (5)

    Substituting the RHS of Eq (5) into the first term of the sum in Eq (3) yields

    1Γ(αi)t1t0(tqτ)αi1[0f1i+1f1i(τt1)+ci0(τt1)2]dτ=Ii0,1h1fi(t0,x0)+(Ii0,0+Ii0,1h1)fi(t1,x(t1))+Ri0, (6)

    where

    Ii0,1=h1(tqt0)αiΓ(αi+1)+(tqt0)αi+1(tqt1)αi+1Γ(αi+2), (7)
    Ii0,0=(tqt0)αi(tqt1)αiΓ(αi+1),Ri0=ci0Γ(αi)t1t0(tqτ)αi1(τt1)2dτ. (8)

    For any j=1,2,,q1, and q=2,3,,N, we approximate fi(τ,x(τ)) on (tj,tj+1] by the following third-order Taylor expansion:

    fi(τ,x(τ))=aij+bij(τtj+1)+cij(τtj+1)2+dij(τtj+1)3, (9)

    where aij, bij and cij are coefficients to be determined, and dij(τtj+1)3 is the remainder. Omitting the remainder, we can use the values of fi(τ,x(τ)) at points tj1,tj and tj+1 to determine aij,bij and cij, yielding the following divided differences:

    0fj+1i:=aij=fi(tj+1,x(tj+1)),1fj+1i:=bij=hj+1fi(tj1,x(tj1))hj(hj+hj+1)(hj+hj+1)fi(tj,x(tj))hjhj+1+(hj+2hj+1)fi(tj+1,x(tj+1))hj+1(hj+hj+1),2fj+1i:=cij=fi(tj1,x(tj1))hj(hj+hj+1)fi(tj,x(tj))hjhj+1+fi(tj+1,x(tj+1))hj+1(hj+hj+1),

    where hj+1=tj+1tj; and hj=tjtj1. Using these differences, we rewrite Eq (9) as

    fi(τ,x(τ))=0fj+1i+1fj+1i(τtj+1)+2fj+1i(τtj+1)2+dij(τtj+1)3. (10)

    Substituting the RHS of Eq (10) into the (j+1)th term of the sum in Eq (3) gives

    1Γ(αi)tj+1tj(tqτ)αi1[0fj+1i+1fj+1i(τtj+1)+2fj+1i(τtj+1)2+dij(τtj+1)3]dτ=hj+1Iij,1+Iij,2hj(hj+hj+1)fi(tj1,x(tj1))(hj+hj+1)Iij,1+Iij,2hjhj+1fi(tj,x(tj))+[Iij,0+(hj+2hj+1)Iij,1+Iij,2hj+1(hj+hj+1)]fi(tj+1,x(tj+1))+Rij, (11)

    where

    Iij,2=h2j+1(tqtj)αiΓ(αi+1)2hj+1(tqtj)αi+1Γ(αi+2)2(tqtj+1)αi+2(tqtj)αi+2Γ(αi+3), (12)
    Iij,1=hj+1(tqtj)αiΓ(αi+1)(tqtj+1)αi+1(tqtj)αi+1Γ(αi+2), (13)
    Iij,0=(tqtj)αi(tqtj+1)αiΓ(αi+1),Rij=dijΓ(αi)tj+1tj(tqτ)αi1(τtj+1)3dτ. (14)

    Combining Eqs (3), (6), and (11), we obtain the following equations, which are equivalent to the equations appearing in Eq (3),

    xi(tq)=x0iIi0,1h1fi(t0,x0)+(Ii0,0+Ii0,1h1)fi(t1,x(t1))+q1j=1{hj+1Iij,1+Iij,2hj(hj+hj+1)×fi(tj1,x(tj1))(hj+hj+1)Iij,1+Iij,2hjhj+1fi(tj,x(tj))+[Iij,0+(hj+2hj+1)Iij,1+Iij,2hj+1(hj+hj+1)]fi(tj+1,x(tj+1))}+Rqi, (15)

    for i=1,2,,n, and q=1,2,,N, where Rqi=q1j=0Rij is the cumulative truncation error up to the point tq.

    Omitting the truncation error Rqi in Eq (15), we define the following numerical scheme for Eq (3):

    xqi=x0iIi0,1h1fi(t0,x0)+(Ii0,0+Ii0,1h1)fi(t1,x1)+q1j=1{hj+1Iij,1+Iij,2hj(hj+hj+1)×fi(tj1,xj1)(hj+hj+1)Iij,1+Iij,2hjhj+1fi(tj,xj)+[Iij,0+(hj+2hj+1)Iij,1+Iij,2hj+1(hj+hj+1)]fi(tj+1,xj+1)}, (16)

    for i=1,2,,n, and q=1,2,,N, where xqi represents an approximation of xi(tq) for each feasible i and q.

    We comment that Eq (16) defines an implicit time-stepping scheme for the numerical solution of Eq (1). Since Eq (16) is implicit, we need to use a technique, such as a Newton's method, to solve the nonlinear algebraic equations at each point tq,q=1,2,,N.

    In this section, our focus is on the convergence analysis of the proposed numerical method.

    Note that Eq (16) is a nonlinear system in xq that will be solved by a Newton's iterative method. Thus, we need to show that the Jacobian matrix of this nonlinear system is invertible for any feasible q. For i=1,2,,n, and q=1,2,,N, let

    Fi(xq)=xqix0i+Ii0,1h1fi(t0,x0)(Ii0,0+Ii0,1h1)fi(t1,x1)q1j=1{hj+1Iij,1+Iij,2hj(hj+hj+1)fi(tj1,xj1)(hj+hj+1)Iij,1+Iij,2hjhj+1fi(tj,xj)+[Iij,0+(hj+2hj+1)Iij,1+Iij,2hj+1(hj+hj+1)]fi(tj+1,xj+1)}.

    We then have the following theorem.

    Theorem 3.1. Let F(xq)=(F1(xq),F2(xq),,Fn(xq)) for q{1,2,,N}. Then, the Jacobian matrix of F(xq) is invertible when hq>0 is sufficiently small.

    Proof. From direct computation, we see that the Jacobian matrix of F(xq), denoted as F(xq), can be rewritten as

    F(xq)=In[bqil]n×n,

    for i=1,2,,n,l=1,2,,n, and q=1,2,,N, where In is the n×n identity matrix; and [bqil]n×n is an n×n matrix with

    bqil={(Ii0,0+Ii0,1h1)fixl|(t1,x1),ifq=1,[Iiq1,0+(hq1+2hq)Iiq1,1+Iiq1,2hq(hq1+hq)]fixl|(tq,xq),ifq=2,3,,N. (17)

    Since f is thrice continuously differentiable, fixl is bounded on [t0,tf] for i=1,2,,n, and l=1,2,,n. Let

    M=maxi{1,,n}l{1,,n}{|fixl|}, (18)

    where || is the Euclidean norm.

    Combining Eqs (7), (8), and (17), we obtain

    |b1il||(Ii0,0+Ii0,1h1)|M=hαi1MΓ(αi+2), (19)

    for i=1,2,,n, and l=1,2,,n. Furthermore, combining Eqs (12)–(14) and (17), we obtain

    |bqil||Iiq1,0+(hq1+2hq)Iiq1,1+Iiq1,2hq(hq1+hq)|M<4hαiqMΓ(αi+1), (20)

    for i=1,2,,n, l=1,2,,n, and q=2,3,,N. Note that a key step in the derivation of Eq (20) is given in Appendix.

    For given constants σi(0,1),i=1,2,,n, define

    ˆh=mini{1,,n}{(Γ(αi+1)(1σi)4nM)1αi}. (21)

    Select h1ˆh such that for i=1,2,,n,

    nl=1|b1il|nhαi1MΓ(αi+2)nˆhαiMΓ(αi+2)Γ(αi+1)(1σi)4Γ(αi+2)<1σimax1ιn{1σι}=1min1ιn{σι}. (22)

    Choose hqˆh such that for i=1,2,,n, and q=2,3,,N,

    nl=1|bqil|<4nhαiqMΓ(αi+1)4nˆhαiMΓ(αi+1)=1σimax1ιn{1σι}=1min1ιn{σι}. (23)

    For i=1,2,,n, and q=1,2,,N, combining Eqs (22) and (23), we obtain

    0<min1ιn{σι}<1nl=1|bqil|=1|bqii|nl=1li|bqil||1bqii|nl=1li|bqil|. (24)

    It follows from Eq (24) that F(xq) is strictly diagonally dominant. According to the Levy-Desplanques theorem in [15], F(xq) is non-singular. Therefore, F(xq) is invertible, which completes the proof.

    Theorem 3.2. Let x(tq) be the exact solution of Eq (3), and let xq be the solution of Eq (16) for q=1,2,,N. Then, there exists a positive constant C, independent of α, such that when h1 is chosen to satisfy h1h3/2, it holds that

    x(tq)xqCh3, (25)

    where is the infinity norm; h=max{2,,N}{h} satisfies the condition hˆh (ˆh is as defined in Eq (21)).

    Proof. The proof is carried out in three steps.

    Step 1. The truncation error Rq is of order O(h3)

    Let Rq=(Rq1,Rq2,,Rqn) for q{1,2,,N}. From the definition of Rqi given by Eq (15) and the thrice continuous differentiability of f, we deduce that for i=1,2,,n, and q=1,2,,N,

    |Rqi||Ri0|+q1j=1|Rij||ci0|h21Γ(αi)t1t0(tqτ)αi1dτ+q1j=1|dij|h3j+1Γ(αi)tj+1tj(tqτ)αi1dτdh3Γ(αi)q1j=0tj+1tj(tqτ)αi1dτ=dh3Γ(αi)(tqt0)αiαidh3tαifΓ(αi+1),

    because h1h3/2, where d=maxi{1,,n}j{1,,q1}{|ci0|,|dij|}; and h=max{2,,N}{h}.

    Recall that αi(0,1] for i=1,2,,n. Thus, 2αi1Γ(αi+1)1, as established in Theorem 4.1 of [16]. Also, it is obvious that tαif<max{1,tf}. Therefore, choosing ˆC=2dmax{1,tf}, we obtain

    Rq=O(h3), (26)

    for q=1,2,,N.

    Step 2. The initial step of the mathematical induction process involving Eq (25)

    We first show that Eq (25) holds for the case q=1.

    Subtracting Eq (16) from Eq (15) and applying the Lagrange mean value theorem, we obtain

    xi(t1)x1i=[u1il]1×n(x(t1)x1)+R1i, (27)

    for i=1,2,,n, and l=1,2,,n, where u1il is defined as

    u1il=(Ii0,0+Ii0,1h1)fixl|(t1,ξ1l). (28)

    Here, for l=1,2,,n, ξ1l=xl(t1)+λ1l(x1lxl(t1)) with λ1l(0,1).

    Based on Eq (27), we obtain

    |xi(t1)x1i||[u1il]1×n(x(t1)x1)|+|R1i|n[u1il]1×nx(t1)x1+R1, (29)

    for i=1,2,,n, and l=1,2,,n.

    Similar to the derivation of Eq (19), it follows that [u1il]1×nhαi1MΓ(αi+2), where M is as defined in Eq (18). Thus, Eq (29) can be rewritten as

    |xi(t1)x1i|nhαi1MΓ(αi+2)x(t1)x1+R1max1in{nhαi1MΓ(αi+2)}x(t1)x1+R1,

    for all feasible i, which implies that

    (1max1in{nhαi1MΓ(αi+2)})x(t1)x1R1. (30)

    According to the derivation of Eq (22), if we select h1ˆh (recall that ˆh is as defined in Eq (21)), then 1nhαi1MΓ(αi+2)>min1ιn{σι}>0 is satisfied for all feasible i. Thus, 1max1in{nhαi1MΓ(αi+2)}>min1ιn{σι}>0 holds. From Eqs (26) and (30), we obtain

    x(t1)x1R11max1in{nhαi1MΓ(αi+2)}<R1min1ιn{σι}=O(h3). (31)

    Step 3. Inductive step for Eq (25)

    We now consider the case q2. In Step 2, we have shown that Eq (31), i.e., the case q=1, is valid. To apply the mathematical induction, we assume that

    x(tj)xj=O(h3), (32)

    for j=1,2,,q1. Then, we need to show that x(tq)xq=O(h3).

    Subtracting Eq (16) from Eq (15) and applying the Lagrange mean value theorem, we obtain

    xi(tq)xqi=[u1il]1×n(x(t1)x1)+q1j=1{[vj1il]1×n(x(tj1)xj1)[wjil]1×n(x(tj)xj)}+q2j=1{[zj+1il]1×n(x(tj+1)xj+1)}+[zqil]1×n(x(tq)xq)+Rqi, (33)

    for i=1,2,,n, and l=1,2,,n, where u1il is as defined in Eq (28), while vj1il, wjil, and zj+1il are defined as

    vj1il=hj+1Iij,1+Iij,2hj(hj+hj+1)fixl|(tj1,ξj1l),wjil=(hj+hj+1)Iij,1+Iij,2hjhj+1fixl|(tj,ξjl),zj+1il=[Iij,0+(hj+2hj+1)Iij,1+Iij,2hj+1(hj+hj+1)]fixl|(tj+1,ξj+1l).

    Here, for l=1,2,,n, and k=0,1,,q, ξkl=xl(tk)+λkl(xklxl(tk)) with λkl(0,1).

    Based on Eq (33), we can deduce that

    |xi(tq)xqi||[u1il]1×n(x(t1)x1)+q1j=1{[vj1il]1×n(x(tj1)xj1)[wjil]1×n(x(tj)xj)}+q2j=1{[zj+1il]1×n(x(tj+1)xj+1)}|+n[zqil]1×nx(tq)xq+Rq, (34)

    for i=1,2,,n, and l=1,2,,n.

    Similar to the derivation of Eq (20), we can show that [zqil]1×n<4hαiqMΓ(αi+1). Thus, Eq (34) can be expressed as

    |xi(tq)xqi|<|[u1il]1×n(x(t1)x1)+q1j=1{[vj1il]1×n(x(tj1)xj1)[wjil]1×n(x(tj)xj)}+q2j=1{[zj+1il]1×n(x(tj+1)xj+1)}|+max1in{n4hαiqMΓ(αi+1)}x(tq)xq+Rq,

    for all feasible i and l, which implies that

    (1max1in{n4hαiqMΓ(αi+1)})x(tq)xq<|[u1il]1×n(x(t1)x1)+q1j=1{[vj1il]1×n(x(tj1)xj1)[wjil]1×n(x(tj)xj)}+q2j=1{[zj+1il]1×n(x(tj+1)xj+1)}|+Rq. (35)

    According to the derivation of Eq (23), if we select hqˆh, then 1n4hαiqMΓ(αi+1)min1ιn{σι}>0 is satisfied for all feasible i. Thus, 1max1in{n4hαiqMΓ(αi+1)}min1ιn{σι}>0 holds. From Eqs (26), (32), and (35) and the boundedness of u1il, vj1il, wjil and zj+1il for i=1,2,,n, l=1,2,,n, and j=1,2,,q1, we obtain

    x(tq)xq=O(h3)+Rq1max1in{n4hαiqMΓ(αi+1)}=O(h3).

    This completes the proof.

    Remark 1. The choice of h1h3/2 aligns with the Rannacher time-stepping technique as discussed in [17,18]. This ensures that the overall accuracy of the proposed numerical method is not affected by the first time step.

    Theorem 3.3. Let ˆxq be the numerical solution of Eq (16) obtained by Newton's method for q=1,2,,N. Then, there exists a positive constant ˜C, independent of both h (as defined in Theorem 3.2) and α, such that

    x(tq)ˆxq˜Ch3. (36)

    Proof. From Theorem 2.1 in [19], we note that there exists a well-defined number of iterations such that

    xqˆxq={O(h21),ifq=1,O(h3q),ifq=2,3,,N, (37)

    where h1h3/2 is as defined in Theorem 3.2. Then, Eq (37) can be expressed as

    xqˆxq=O(h3), (38)

    where h is as defined in Theorem 3.2.

    Combining Eqs (25) and (38), we obtain

    x(tq)ˆxqx(tq)xq+xqˆxq=O(h3).

    Therefore, we can infer that Eq (36) holds. The proof is complete.

    Remark 2. The error estimate (36) shows that for q=1,2,,N, the numerical solution ˆxq obtained by Newton's method converges to the exact solution x(tq) with third-order accuracy. Note that this convergence rate is independent of α, which is a significant distinction from the methods reported in [7,8,9,14].

    In this section, we will solve four numerical examples to verify the effectiveness and convergence rate of the numerical method proposed in Section 3. Here, the stopping criterion for Newton's method is set to be 1012, and all computations are conducted in the MATLAB 2022b environment on a PC equipped with a 2.80 GHz Intel Core i7-1165G7 CPU and 16.0 GB RAM.

    For all test examples, the following graded mesh is used:

    tq=(qN)2tf,q=0,1,,N,

    where N is a positive integer. This graded mesh satisfies the condition in Theorem 3.2. The computed maximum error and convergence order are defined as

    Emax:=Eh=maxi{1,,n}q{1,,N}{|xi(tq)ˆxqi|},Order:=log(EhEh)/log(hh),

    respectively, where h is as defined in Theorem 3.2 under the partition number N; and h is the maximum step size under the partition number N.

    Remark 3. Note that other strategies can also be used in our calculations, as long as they comply with Remark 1.

    Consider the following nonlinear FODE [14]:

    C0Dαtx(t)=Γ(9)Γ(9α)t8α3Γ(5+α2)Γ(5α2)t4α2+(t83t4+α2)3x3(t),t(0,1],x(0)=0.

    The exact solution of this nonlinear FODE is x(t)=t83t4+α2. Our proposed numerical method is applied to solve this example. For comparison, we also solve this example by the one-step method in [13]. The computed numerical results and those reported in [14] are listed in Table 1, from which we see that the maximum errors in all methods decrease as the number of mesh points N increases. Notably, the maximum error obtained by our method is significantly smaller than those obtained in [13] for different values of α. Compared with [14], the convergence order computed by our method is comparable to that of Q-PCP for α=0.1 and significantly higher than those of Q-PCD for α=0.8 and α=0.9. This is because the theoretical convergence order of Q-PCP deteriorates as α increases. It is clear from Table 1 that the convergence order obtained by our method approaches to the theoretical value of 3. More importantly, the convergence order is independent of both α and h. Furthermore, we perform 50 tests and take the average time as the computational time for all test examples. From Table 1, we see that the one-step method, utilizing a second-order Taylor expansion and an explicit numerical format, achieves slightly shorter computational time compared to our method at the expense of lower accuracy. Figure 1 illustrates the state trajectories for various α when N=640. From Figure 1, we can see that the numerical solutions obtained by our method closely match the exact solutions, which is consistent with the results in Table 1.

    Table 1.  Maximum error, convergence order, and computational time for Example 1.
    α=0.1 Q-PCP [14] One-step [13] Our method
    N Emax Order Emax Order CPU(s) Emax Order CPU(s)
    10 4.9987E-04 - 8.7421E-02 - 3.8897E-04 7.5535E-04 - 1.2221E-03
    20 7.9647E-05 2.6499 2.3923E-02 1.8696 1.1973E-03 1.8128E-04 2.1391 3.1013E-03
    40 1.1982E-05 2.7328 6.0795E-03 1.9764 4.2962E-03 3.2801E-05 2.5126 8.9735E-03
    80 1.7626E-06 2.7651 1.4810E-03 2.0374 1.6198E-02 5.1679E-06 2.6906 3.1299E-02
    160 2.5521E-07 2.7879 3.5231E-04 2.0717 6.3558E-02 7.5681E-07 2.7842 1.1410E-01
    320 3.6546E-08 2.8039 8.2776E-05 2.0896 2.5346E-01 1.0635E-07 2.8375 4.4582E-01
    640 5.1822E-09 2.8181 1.9326E-05 2.0987 1.0010E+00 1.4574E-08 2.8706 1.7106E+00
    α=0.8 Q-PCD [14] One-step [13] Our method
    N Emax Order Emax Order CPU(s) Emax Order CPU(s)
    10 9.8400E-02 - 2.8733E-02 - 3.9887E-04 1.3935E-02 - 1.2125E-03
    20 9.8625E-03 3.3186 4.8370E-03 2.5705 1.1945E-03 2.4250E-03 2.6209 2.8512E-03
    40 8.7791E-04 3.4898 8.7415E-04 2.4682 4.3209E-03 3.4682E-04 2.8583 9.2150E-03
    80 1.0277E-04 3.0947 1.7301E-04 2.3370 1.6333E-02 4.6295E-05 2.9319 3.0611E-02
    160 1.4646E-05 2.8108 3.7074E-05 2.2224 6.3507E-02 5.9772E-06 2.9668 1.1346E-01
    320 2.2964E-06 2.6731 8.4136E-06 2.1396 2.5051E-01 7.5924E-07 2.9836 4.3508E-01
    640 3.7444E-07 2.6166 1.9831E-06 2.0850 1.0118E+00 9.5666E-08 2.9919 1.7124E+00
    α=0.9 Q-PCD [14] One-step[13] Our method
    N Emax Order Emax Order CPU(s) Emax Order CPU(s)
    10 6.5505E-02 - 2.4339E-02 - 4.0538E-04 1.6829E-02 - 1.2150E-03
    20 7.2229E-03 3.1810 3.8287E-03 2.6683 1.2207E-03 2.8795E-03 2.6462 2.9877E-03
    40 8.5280E-04 3.0823 6.5177E-04 2.5544 4.3127E-03 4.0702E-04 2.8755 9.2930E-03
    80 1.2478E-04 2.7728 1.2335E-04 2.4016 1.6135E-02 5.3973E-05 2.9416 3.1094E-02
    160 2.0420E-05 2.6113 2.5727E-05 2.2614 6.3355E-02 6.9457E-06 2.9715 1.1318E-01
    320 3.4802E-06 2.5527 6.7752E-06 1.9249 2.5034E-01 8.8078E-07 2.9860 4.3424E-01
    640 6.0180E-07 2.5318 1.8000E-06 1.9123 9.9944E-01 1.1088E-07 2.9932 1.7159E+00

     | Show Table
    DownLoad: CSV
    Figure 1.  State trajectories when N=640 for Example 1.

    Consider the following nonlinear FODE [20]:

    C0Dαtx(t)=(x(t)0.01t21)2cos2(4t)+2πJ0(4t)+1+2t1.575π,t(0,10],x(0)=1,

    where J0() is the zero-order Bessel function of the first kind [21].

    This problem is solved again by using our proposed numerical scheme. For α=0.5, the exact solution of this example is x(t)=sin(4t)+0.01t2+1. For comparison, we also solve this example by using the implicit product integration trapezoidal rule (PI 2 Impl.) in [22]. The obtained numerical results are shown in Table 2. From Table 2, we see that the maximum error obtained by our method is much smaller than that obtained in [22] for various values of N. We also observe that the convergence order in [22] increases with the increasing value of N. In contrast, the convergence order computed by our method remains stable and is close to the theoretical value 3. This reconfirms that the convergence order of the proposed numerical method is independent of both α and h. Nevertheless, the computational time of the PI 2 Impl. method is significantly shorter compared to our method, attributed to a lower accuracy of the trapezoidal rule and the efficient treatment of persistent memory of fractional integrals. Figure 2 depicts the state trajectories when N=5120. From Figure 2, it can be seen that the numerical solutions obtained by our method exhibit greater accuracy compared to those obtained using the method reported in [22].

    Table 2.  Maximum error, convergence order, and computational time for Example 2.
    α=0.5 PI 2 Impl. [22] Our method
    N Emax Order CPU(s) Emax Order CPU(s)
    320 9.1818E-01 - 7.8216E-03 3.1282E-01 - 4.9170E-01
    640 7.2655E-01 0.3377 1.3015E-02 2.1987E-02 3.8349 1.9347E+00
    1280 5.0259E-01 0.5317 2.2718E-02 2.6433E-03 3.0580 7.7414E+00
    2560 2.7916E-01 0.8483 3.8415E-02 3.3003E-04 3.0025 3.0429E+01
    5120 1.1294E-01 1.3055 7.8790E-02 4.1351E-05 2.9970 1.2082E+02

     | Show Table
    DownLoad: CSV
    Figure 2.  State trajectories when N=5120 for Example 2.

    Consider the following nonlinear FODE [9]:

    C0Dαtx(t)=Γ(4+α)6t3+t6+2αx2(t),t(0,1],x(0)=0.

    The exact solution of this example is x(t)=t3+α. In contrast to the method proposed in [9] for directly approximating fractional derivatives and the method developed in [23], which employs the three-step Newton polynomial to approximate Volterra integrals, we use our proposed method and the MATLAB code provided in [23] to solve this problem for α=0.3, α=0.6, and α=0.9. The computed numerical results and those reported in [9] are listed in Table 3. From Table 3, we observe that the maximum error obtained by our method is significantly superior to those obtained in [9,23]. Furthermore, we note that our method exhibits a consistent convergence order close to 3 for various fractional orders, indicating its stability and efficiency. We observe that the convergence order obtained in [23] is also close to 3. However, the computational complexity in [23] is higher compared to our method. Specifically, for calculating xq, the values of xq3,xq2, and xq1, q=3,4,,N, are required in [23], while the one-step Euler method and the two-step Adams-Bashforth method are used to obtain the values of x1 and x2, respectively. In contrast, our method only requires the values of xq2 and xq1 when calculating xq for q=2,3,,N. The value of x1 can be calculated by using the second-order Taylor expansion. Thus, it can be seen from Table 3 that the computational time in [23] is higher than that in our method. The state trajectories for different fractional orders with N=2048 are depicted in Figure 3. As it can be seen from Figure 3, our method shows satisfactory accuracy.

    Table 3.  Maximum error, convergence order, and computational time for Example 3.
    α=0.3 Results from [9] Results from [23] Our method
    N Emax Order Emax Order CPU(s) Emax Order CPU(s)
    128 1.0154E-06 - 1.6389E-06 - 9.0903E-02 4.1874E-07 - 6.2773E-02
    256 1.4941E-07 2.7647 1.9683E-07 3.0577 3.6583E-01 5.4184E-08 2.9585 2.3771E-01
    512 2.3628E-08 2.6607 2.3785E-08 3.0488 1.5023E+00 6.9524E-09 2.9665 9.2097E-01
    1024 3.7049E-09 2.6729 2.8896E-09 3.0411 6.1037E+00 8.8676E-10 2.9730 3.6671E+00
    2048 5.6279E-10 2.7187 3.5272E-10 3.0343 2.4766E+01 1.1261E-10 2.9783 1.4613E+01
    α=0.6 Results from [9] Results from [23] Our method
    N Emax Order Emax Order CPU(s) Emax Order CPU(s)
    128 1.0167E-05 - 1.8931E-06 - 9.0620E-02 7.2056E-07 - 6.2624E-02
    256 1.9494E-06 2.3828 2.3429E-07 3.0144 3.6675E-01 9.0807E-08 2.9967 2.3662E-01
    512 3.7176E-07 2.3906 2.9087E-08 3.0098 1.4719E+00 1.1405E-08 2.9974 9.2196E-01
    1024 7.0707E-08 2.3944 3.6190E-09 3.0067 6.1504E+00 1.4297E-09 2.9980 3.6364E+00
    2048 1.3497E-08 2.3891 4.5095E-10 3.0046 2.4819E+01 1.7901E-10 2.9987 1.4423E+01
    α=0.9 Results from [9] Results from [23] Our method
    N Emax Order Emax Order CPU(s) Emax Order CPU(s)
    128 9.1822E-05 - 2.7547E-06 - 9.0707E-02 1.0554E-06 - 6.2696E-02
    256 2.1549E-05 2.0911 3.4432E-07 3.0001 3.6541E-01 1.3237E-07 3.0036 2.3620E-01
    512 5.0416E-06 2.0957 4.3036E-08 3.0001 1.4581E+00 1.6574E-08 3.0018 9.2368E-01
    1024 1.1777E-06 2.0978 5.3791E-09 3.0001 6.2194E+00 2.0735E-09 3.0009 3.6373E+00
    2048 2.7492E-07 2.0989 6.7236E-10 3.0001 2.4452E+01 2.5929E-10 3.0005 1.4573E+01

     | Show Table
    DownLoad: CSV
    Figure 3.  State trajectories when N=2048 for Example 3.

    Consider the following nonlinear FODEs [24]:

    {C0Dα1tx1(t)=12x1(t),C0Dα2tx2(t)=x21(t)+x2(t),t(0,1],x(0)=(1,0). (39)

    The exact solutions of this example are x1(t)=et2 and x2(t)=tet for α1=α2=1.0. We solve this example for (α1,α2)=(0.7,0.9), (α1,α2)=(0.9,0.7), and (α1,α2)=(1.0,1.0) by our proposed method. Since the exact solutions for the cases (α1,α2)=(0.7,0.9) and (α1,α2)=(0.9,0.7) are unknown, we use the numerical solutions for N=2560 as approximations to the exact solutions. The computed maximum error, convergence order, and computational time are presented in Table 4. It is observed from Table 4 that the value of the maximum error decreases as the value of N increases. For different fractional orders, the computed convergence order of our method is roughly 3. To visualize the numerical results, we plot the state trajectories corresponding to different fractional orders in Figure 4.

    Table 4.  Maximum error, convergence order, and computational time for Example 4.
    (α1,α2) (0.7,0.9) (0.9,0.7) (1.0,1.0)
    N Emax Order CPU(s) Emax Order CPU(s) Emax Order CPU(s)
    10 3.6411E-03 - 1.9605E-03 4.9142E-03 - 1.9573E-03 1.4588E-03 - 2.0733E-03
    20 5.0444E-04 2.9626 5.9179E-03 6.8817E-04 2.9465 6.2036E-03 1.9848E-04 2.9898 6.0395E-03
    40 6.6335E-05 2.9816 2.0033E-02 9.0991E-05 2.9736 2.0026E-02 2.5849E-05 2.9959 1.9862E-02
    80 8.5101E-06 2.9897 7.3699E-02 1.1702E-05 2.9861 7.3156E-02 3.2967E-06 2.9983 7.2059E-02
    160 1.0751E-06 2.9983 2.8322E-01 1.4831E-06 2.9936 2.8699E-01 4.1619E-07 2.9993 2.8037E-01
    320 1.3132E-07 3.0402 1.1216E+00 1.8506E-07 3.0094 1.1272E+00 5.2281E-08 2.9997 1.0886E+00
    640 1.5305E-08 3.1045 4.5994E+00 2.1624E-08 3.1008 4.5519E+00 6.5512E-09 2.9998 4.3141E+00

     | Show Table
    DownLoad: CSV
    Figure 4.  State trajectories when N=640 for Example 4.

    This paper has developed the third-order numerical method for solving nonlinear FODEs in the sense of Liouville-Caputo fractional derivatives. The fractional orders of these nonlinear FODEs can differ from each other. At each mesh point of a given mesh, we approximate the equivalent Volterra integral equations by using the third-order Taylor expansion (for the first subinterval, the second-order Taylor expansion is used). This approximation yields the implicit nonlinear algebraic equations that can be iteratively solved by the Newton's method. Furthermore, the convergence analysis and error estimate are performed, showing that the convergence rate of the proposed method is third order, independent of the fractional order. Finally, four non-trivial numerical examples are solved to illustrate the effectiveness and the convergence of the proposed method. In our future research, we will develop effective numerical methods for solving various fractional optimal control problems.

    Xiaopeng Yi: Conceptualization, Investigation, Methodology, Software, Writing-original draft; Chongyang Liu: Conceptualization, Investigation, Methodology, Supervision, Software, Writing-review and editing; Huey Tyng Cheong: Conceptualization, Investigation, Methodology, Supervision, Writing-original draft; Kok Lay Teo: Conceptualization, Investigation, Methodology, Supervision, Writing-review and editing; Song Wang: Conceptualization, Investigation, Methodology, Writing-review and editing. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the Fundamental Research Grant Scheme of Malaysia (grant number FRGS/1/2021/STG06/SYUC/03/1), the National Natural Science Foundation of China (No. 12271307), and the Shandong Province Natural Science Foundation of China (No. ZR2023MA054).

    Prof. Kok Lay Teo is an editorial board member for AIMS Mathematics and was not involved in the editorial review and/or the decision to publish this article.

    The authors declare that they have no competing interests.

    In Eq (20), the following inequality holds:

    |Iiq1,0+(hq1+2hq)Iiq1,1+Iiq1,2hq(hq1+hq)|=|hαiqΓ(αi+1)+hq1hαiq(hq1+hq)Γ(αi+2)hq1hαiq+hαi+1q(hq1+hq)Γ(αi+1)+2hαi+1q(hq1+hq)Γ(αi+3)||hαiqΓ(αi+1)|+|hq1hαiq(hq1+hq)Γ(αi+2)|+|hq1hαiq+hαi+1q(hq1+hq)Γ(αi+1)|+|2hαi+1q(hq1+hq)Γ(αi+3)|<|hαiqΓ(αi+1)|+|hq1hαiq(hq1+hq)Γ(αi+1)|+|hq1hαiq+hαi+1q(hq1+hq)Γ(αi+1)|+|2hαi+1q(hq1+hq)Γ(αi+1)|=hαiqΓ(αi+1)+2hq1hαiq+3hαi+1q(hq1+hq)Γ(αi+1)=3hq1hαiq+4hαi+1q(hq1+hq)Γ(αi+1)<4hαiqΓ(αi+1).

    Acknowledgments



    None.

    Conflict of interest



    All authors declare no conflicts of interest in this paper.

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