Processing math: 87%
Research article

A novel numerical method for solving the Caputo-Fabrizio fractional differential equation

  • Received: 23 September 2022 Revised: 03 January 2023 Accepted: 09 January 2023 Published: 20 February 2023
  • MSC : 26A33, 26C10

  • In this paper, a unique and novel numerical approach—the fractional-order Caputo-Fabrizio derivative in the Caputo sense—is developed for the solution of fractional differential equations with a non-singular kernel. After converting the differential equation into its corresponding fractional integral equation, we used Simpson's 1/3 rule to estimate the fractional integral equation. A thorough study is then conducted to determine the convergence and stability of the suggested method. We undertake numerical experiments to corroborate our theoretical findings.

    Citation: Sadia Arshad, Iram Saleem, Ali Akgül, Jianfei Huang, Yifa Tang, Sayed M Eldin. A novel numerical method for solving the Caputo-Fabrizio fractional differential equation[J]. AIMS Mathematics, 2023, 8(4): 9535-9556. doi: 10.3934/math.2023481

    Related Papers:

    [1] Khadija Gherairi, Zayd Hajjej, Haiyan Li, Hedi Regeiba . Some properties of n-quasi-(m,q)-isometric operators on a Banach space. AIMS Mathematics, 2023, 8(12): 31246-31257. doi: 10.3934/math.20231599
    [2] Hadi Obaid Alshammari . Higher order hyperexpansivity and higher order hypercontractivity. AIMS Mathematics, 2023, 8(11): 27227-27240. doi: 10.3934/math.20231393
    [3] Soon-Mo Jung, Jaiok Roh . Local stability of isometries on 4-dimensional Euclidean spaces. AIMS Mathematics, 2024, 9(7): 18403-18416. doi: 10.3934/math.2024897
    [4] Lijun Ma, Shuxia Liu, Zihong Tian . The binary codes generated from quadrics in projective spaces. AIMS Mathematics, 2024, 9(10): 29333-29345. doi: 10.3934/math.20241421
    [5] Anas Al-Masarwah, Abd Ghafur Ahmad . Subalgebras of type (α, β) based on m-polar fuzzy points in BCK/BCI-algebras. AIMS Mathematics, 2020, 5(2): 1035-1049. doi: 10.3934/math.2020072
    [6] Uğur Gözütok, Hüsnü Anıl Çoban . Detecting isometries and symmetries of implicit algebraic surfaces. AIMS Mathematics, 2024, 9(2): 4294-4308. doi: 10.3934/math.2024212
    [7] Yang Zhang, Shuxia Liu, Liwei Zeng . A symplectic fission scheme for the association scheme of rectangular matrices and its automorphisms. AIMS Mathematics, 2024, 9(11): 32819-32830. doi: 10.3934/math.20241570
    [8] Yuqi Sun, Xiaoyu Wang, Jing Dong, Jiahong Lv . On stability of non-surjective (ε,s)-isometries of uniformly convex Banach spaces. AIMS Mathematics, 2024, 9(8): 22500-22512. doi: 10.3934/math.20241094
    [9] Su-Dan Wang . The q-WZ pairs and divisibility properties of certain polynomials. AIMS Mathematics, 2022, 7(3): 4115-4124. doi: 10.3934/math.2022227
    [10] Peiying Huang, Yiyuan Zhang . H-Toeplitz operators on the Dirichlet type space. AIMS Mathematics, 2024, 9(7): 17847-17870. doi: 10.3934/math.2024868
  • In this paper, a unique and novel numerical approach—the fractional-order Caputo-Fabrizio derivative in the Caputo sense—is developed for the solution of fractional differential equations with a non-singular kernel. After converting the differential equation into its corresponding fractional integral equation, we used Simpson's 1/3 rule to estimate the fractional integral equation. A thorough study is then conducted to determine the convergence and stability of the suggested method. We undertake numerical experiments to corroborate our theoretical findings.



    This paper considers a fractional coupled system on an infinite interval involving the Erdélyi-Kober derivative:

    {Dγ,δ1βu(x)+F(x,u(x),v(x))=0,x(0,+),Dγ,δ2βv(x)+G(x,u(x),v(x))=0,x(0,+),limx0xβ(2+γ)Iδ1+γ,2δ1u(x)=0,limxxβ(1+γ)Iδ1+γ,2δ1u(x)=0,limx0xβ(2+γ)Iδ2+γ,2δ2v(x)=0,limxxβ(1+γ)Iδ2+γ,2δ2v(x)=0, (1.1)

    where δ1,δ2(1,2], γ(2,1), and β>0. Dγ,δ1β, Dγ,δ2β are Erdélyi-Kober fractional derivatives (EKFDs for short), and Iδ1+γ,2δ1,Iδ2+γ,2δ2 are the Erdélyi-Kober fractional integrals. F,G are continuous functions. We discuss the existence of positive solutions for (1.1).

    During the past several decades, fractional equations have been studied widely; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] for instance. From the literature, we can see that there are many fractional derivatives used in differential equations. Among these various definitions, the widely used ones are the Riemann-Liouville and Caputo fractional derivatives, in many works. To generalize the Riemann-Liouville fractional derivative, Erdélyi-Kober defined a new fractional derivative, and we call it the Erdélyi-Kober fractional derivative. Moreover, the Erdélyi-Kober operator is very useful; we can refer to [6,9,14,15,16,17] and the references therein. The Erdélyi-Kober operator is a fractional integration operation which was given by Arthur Erdélyi and Hermann Kober in 1940 [23]. Some of these definitions and results were given in Samko et al. [3], Kiryakova [19], and McBride [20].

    Nowadays, the theory of fractional operators in the Erdélyi-Kober frame has attracted much interest from researchers. The study of fractional systems is also very important, as these systems appear in various applications, especially in biological sciences. Recently, some problems of Erdélyi-Kober type fractional differential equations on infinite intervals received widespread attention from many scholars; see [8,21,22] for example.

    Recently, in [8], the authors investigated the following equation:

    {(Dϑ,σθu)(x)+F(u(x))=0,0x<,limt0xθ(2σ)Iσ+ϑ,2σu(x)=0,limt+xθ(2σ)Iσ+ϑ,2σu(x)=0,

    where σ(1,2), ϑ(1,2), θ>0, and F is a given continuous function, Dϑ,σθ denotes the EKFD, and Iσ+ϑ,2σ denotes the Erdélyi-Kober fractional integral. The authors studied the existence and nonexistence of positive solutions for this problem by utilizing a fixed point result which uses the strongly positive-like operators and eigenvalue criteria.

    In [9], the authors studied a fractional coupled system:

    {cDϱu(τ)=F(τ,u(τ),z(τ),cDς1z(τ),Iξz(τ)),τ[0,T]:=K,2<ϱ3,1<ς1<2,cDςz(τ)=G(τ,u(τ),cDϱ1u(τ),Iζu(τ),z(τ),τ[0,T]:=K,2<ς3,1<ϱ1<2,u(0)=ϕ1(z),u(0)=ε1z(k1),u(T)=γρϑρ(ϖ+v)Γ(ϖ)ϑ0σρv+ρ1z(σ)(ϑρσρ)1ϖdσ:=γJv,ϖρv(ϑ),z(0)=ϕ2(u),z(0)=ε2z(k2),z(T)=δvφv(θ+ω)Γ(θ)φ0σvω+υ1u(σ)(φvσv)1θdσ:=δJω,θvu(φ),

    where cDϱ,cDς1,cDς,cDϱ1 are the Liouville-Caputo fractional derivatives of order 2<ϱ,ς3, 1<ς1,ϱ1<2. Iξ,Iζ are the Riemann-Liouville fractional integrals of order 1<ξ,ζ<2. Jυ,ϖρ,Jω,θv are the Erdélyi-Kober fractional integrals of order ϖ,θ>0, with v,ω>0, ρ, ϑ(,+). F,G:K×(,+)4(,+) and ϕ1,ϕ2:C(K,(,+))(,+) are continuous functions. γ,δ,ε1,ε2 are positive real constants. The existence result was given by the Leray-Schauder alternative, and the uniqueness result was obtained due to Banach's fixed-point theorem. By the same methods, Arioua and Titraoui [18] studied system (1.1). Moreover, In [10], Arioua and Titraoui also investigated a new fractional problem involving the Erdélyi-Kober derivative. Inspired by the above articles, we use different methods to consider the fractional coupled system involving Erdélyi-Kober derivative (1.1). We employ the Guo-Krasnosel'skii fixed point theorem to discuss (1.1) in a special Banach space, and we also use the monotone iterative technique to study this system. Some existence results of positive solutions for system (1.1) are obtained, including the existence results of at least two positive solutions.

    Definition 2.1. (see [2]) Let α(,+). Cnα, nN, denotes a set of all functions f(t),t>0, with f(t)=tpf1(t) with p>α and f1Cn[0,).

    Definition 2.2. (see [1,2]) For a function uCα, the σ-order right-hand Erdélyi-Kober fractional integral is

    (Iγ,σβu)(t)=βtβ(γ+σ)Γ(σ)t0sβ(γ+1)1u(s)(tβsβ)1σds,σ,β>0,γ(,+),

    in which, Γ is the Euler gamma function.

    Definition 2.3. (see [2]) Let n1<δn,nN, and for uCα, the σ-order right-hand Erdélyi-Kober fractional derivative is

    (Dγ,σβu)(t)=nj=1(γ+j+tβddt)(Iγ+σ,nσβu)(t),

    where

    nj=1(γ+j+tβddt)(Iγ+σ,nσβu)=(γ+1+tβddt)(γ+n+tβddt)(Iγ+σ,nσβu).

    Lemma 2.1. (see [10]) Let 1<σ2, 2<γ<1, β>0, and hC2α, with 0sβ(γ+m)1h(τ)dτ<, m=1,2. The fractional problem

    {Dγ,σβu(x)+h(x)=0,x>0,limx0xβ(2+γ)Iσ+γ,2δu(x)=0,limxxβ(1+γ)Iσ+γ,2σu(x)=0,

    has a unique solution given by u(x)=0Gσ(x,s)sβ(γ+1)1h(s)ds, where

    Gσ(x,s)={βΓ(σ)[xβ(γ+1)xβ(δ+γ)(xβsβ)σ1],0<sx<,βΓ(σ)xβ(γ+1),0<xs<. (2.1)

    Lemma 2.2. (see [10]) For 1<σ2, 2<γ<1, and β>0, the function Gσ, defined in (2.1), has the following properties:

    (i) Gσ(x,s)1+xβ(1+γ)>0, for x,s>0;

    (ii) Gσ(x,s)1+xβ(1+γ)βΓ(σ), for x,s>0;

    (iii) for 0<τλxτ and s>τλ2, where λ>1,τ>0, we have

    Gσ(x,s)1+xβ(1+γ)β(σ1)τβ(1+γ)Γ(σ)λβ(1γ)(1+τβ(1+γ))=βp(τ)Γ(σ),

    where p(τ)=(σ1)τβ(1+γ)λβ(1+γ)(1+τβ(1+γ)).

    Lemma 2.3. (see [18]) Let 0<σ1,σ21 and F,GC2α with

    0sβ(γ+m)1F(s,u(s),v(s))ds<,m=1,2,
    0sβ(γ+m)1G(s,u(s),v(s))ds<,m=1,2.

    Then, (1.1) has a unique solution given by

    u(x)=0Gσ1(x,s)sβ(γ+1)1F(s,u(s),v(s))ds,
    v(x)=0Gσ2(x,s)sβ(γ+1)1G(s,u(s),v(s))ds,

    where

    Gσ1(x,s)={βΓ(σ1)[xβ(γ+1)xβ(σ1+γ)(xβsβ)σ11],0<sx<,βΓ(σ1)xβ(γ+1),0<xs<, (2.2)
    Gσ2(x,s)={βΓ(σ2)[xβ(γ+1)xβ(σ2+γ)(xβsβ)σ21],0<sx<,βΓ(σ2)xβ(γ+1),0<xs<. (2.3)

    The following result is our main tool.

    Lemma 2.4. (Guo-Krasnosel'skii fixed point theorem; see [37]) P is a cone in a Banach space E, and D1 and D2 are bounded open sets in E with θD1, ¯D1D2. A:P(¯D2D1)P is a completely continuous operator. Consider the following conditions (ⅰ), (ⅱ):

    (i) Aww for wPD1, Aww for wPD2;

    (ii) Aww for wPD1, Aww for wPD2.

    If one of the preceding conditions (ⅰ), (ⅱ) holds, then A has at least one fixed point in P(¯D2D1).

    Next, we present some hypotheses that will play an important role in the subsequent discussion:

    (H1) F,G:(0,+)×(,+)×(,+)(0,+) are continuous and nondecreasing with respect to the second, third variables on (0,+).

    (H2) For (x,u,v)(0,+)×(,+)×(,+),

    F1(x,u,v)=xβ(1+γ)1F(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v),
    F2(x,u,v)=xβ(1+γ)1G(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v),

    such that

    F1(x,u,v)φ1(x)ω1(u)+ψ1(x)ω2(v),
    F2(x,u,v)φ2(x)~ω1(u)+ψ2(x)~ω2(v),

    with ωi,~ωiC((0,+),(0,+)) nondecreasing and φi,ψiL1(0,+), i=1,2.

    (H3) There are positive functions qi,˜qi,i=1,2, with

    qi=0(1+xβ(1+γ))qi(x)dx<,
    ˜qi=0(1+xβ(1+γ))˜qi(x)dx<,

    such that

    xβ(γ+1)1F(x,u,v)F(x,˜u,˜v)∣≤q1(x)u˜u+˜q1(x)v˜v,
    xβ(γ+1)1G(x,u,v)G(x,˜u,˜v)∣≤q2(x)u˜u+˜q2(t)v˜v,

    for any u,v,˜u,˜v(,+) and x(0,+).

    (H4) F,G:(0,+)×(0,+)×(0,+)(0,+) are continuous, such that

    xβ(1+γ)1F(x,u,v)=a1(x)F1(x,u,v),
    xβ(1+γ)1G(x,u,v)=a2(x)G1(x,u,v),

    where a1,a2L1((0,+),(0,+)), F1,G1C((0,+)×(0,+)×(0,+),(0,+)), 0<ττλa1(x)dx<, 0<ττλa2(x)dx<, with τ>0, λ>1. Moreover, xβ(1+γ)1F(x,u,v), xβ(1+γ)1G(x,u,v):[0,+)×(0,+)×(0,+)[0,+) also are continuous.

    Remark 2.1. These conditions ensure the continuity and integrability of nonlinear terms in an infinite interval, which play a very important role in the proof of completely continuity for the relevant integral operators.

    In this section, we use two Banach spaces defined by

    X={uC((0,+),(,+))limx0u(x)1+xβ(1+γ) and limt+u(x)1+xβ(1+γ) exist},

    with the norm

    uX=supx>0u(x)1+xβ(1+γ),

    and

    Y={vC((0,+),(,+))limx0v(x)1+xβ(1+γ) and limx+v(x)1+xβ(1+γ) exist},

    with the norm

    vY=supx>0v(x)1+xβ(1+γ).

    So, (X×Y,(u,v)X×Y) is a Banach space, with the norm (u,v)X×Y=uX+vY.

    Lemma 3.1. If F,G are continuous, then (u,v)X×Y is a solution of system (1.1)(u,v)X×Y is a solution of the following equations:

    {u(x)=0Gσ1(x,s)sβ(γ+1)1F(s,u(s),v(s))ds,v(x)=0Gσ2(x,s)sβ(γ+1)1G(s,u(s),v(s))ds.

    For (u,v)X×Y, we define an operator A:X×YX×Y as follows:

    A(u,v)(x)=(A1(u,v)(x),A2(u,v)(x)),

    where

    A1(u,v)(x)=0Gσ1(x,s)sβ(γ+1)1F(s,u(s),v(s))ds,
    A2(u,v)(x)=0Gσ2(x,s)sβ(γ+1)1G(s,u(s),v(s))ds,

    with Gσi(x,s),i=1,2, given by (2.2) and (2.3).

    Remark 3.1. Let σ1,σ2,β,γ,λ,τR, such that 1<σ1,σ22,β>0,2<γ<1,λ>1,τ>0. If (H2) and (H4) hold, then for (u,v)X×Y with u(x),v(x)>0,

    0sβ(γ+1)1F(s,u(s),v(s))dsητλ2sβ(γ+1)1F(s,u(s),v(s))ds,
    0sβ(γ+1)1G(s,u(s),v(s))dsητλ2sβ(γ+1)1G(s,u(s),v(s))ds,

    where η=max{η1,η2} with η1=1+ιϱ1(λ21),η2=1+ιϱ2(λ21)>1, ϱ1,ϱ2,ι,ι>0.

    Proof. By (H4), for x[τλ2,τ], we know that there exist two constants ϱ1,ϱ2>0, such that

    xβ(γ+1)1F(s,u,v)ϱ1,xβ(γ+1)1G(s,u,v)ϱ2,u,v(0,+).

    So, for (u,v)X×Y with u(x),v(x)>0,

    τλ2sβ(γ+1)1F(s,u(s),v(s))dsττλ2sβ(γ+1)1F(s,u(s),v(s))dsτ(λ21)λ2ϱ1,
    τλ2sβ(γ+1)1G(s,u(s),v(s))dsττλ2sβ(γ+1)1G(s,u(s),v(s))dsτ(λ21)λ2ϱ2,

    and hence,

    λ2τ(λ21)ϱ1τλ2sβ(γ+1)1F(s,u(s),v(s))ds1,
    λ2τ(λ21)ϱ2τλ2sβ(γ+1)1G(s,u(s),v(s))ds1.

    By (H4), we know that there exist two constants ι,ι>0, such that

    xβ(γ+1)1F(x,u(x),v(x))ι,xβ(γ+1)1G(x,u(x),v(x))ι,for  x[0,τλ2].

    Thus,

    τλ20sβ(γ+1)1F(s,u(s),v(s))dsιτλ2,
    τλ20sβ(γ+1)1G(s,u(s),v(s))dsιτλ2.

    Therefore, we can obtain

    0sβ(γ+1)1F(s,u(s),v(s))ds=τλ20sβ(γ+1)1F(s,u(s),v(s))ds+τλ2sβ(γ+1)1F(s,u(s),v(s))dsιτλ2+τλ2sβ(γ+1)1F(s,u(s),v(s))ds(1+ιϱ1(λ21))τλ2sβ(γ+1)1F(s,u(s),v(s))ds=η1τλ2sβ(γ+1)1F(s,u(s),v(s))ds.

    Similarly,

    0sβ(γ+1)1G(s,u(s),v(s))ds(1+ιϱ2(λ21))τλ2sβ(γ+1)1G(s,u(s),v(s))ds=η2τλ2sβ(γ+1)1G(s,u(s),v(s))ds.

    Take η=max{η1,η2}, and thus

    0sβ(γ+1)1F(s,u(s),v(s))dsητλ2sβ(γ+1)1F(s,u(s),v(s))ds,
    0sβ(γ+1)1G(s,u(s),v(s))dsητλ2sβ(γ+1)1G(s,u(s),v(s))ds,

    hold.

    Define two cones

    K1={uXu(x)>0,x>0;minx[τλ,τ]u(x)1+xβ(1+γ)p(τ)ηuX},
    K2={vYv(x)>0,x>0;minx[τλ,τ]v(x)1+xβ(1+γ)p(τ)ηvY}.

    Obviously, K1×K2={(u,v)X×Yu(x)>0,v(x)>0,x>0;  minx[τλ,τ]u(x)1+xβ(1+γ)p(τ)ηuX,minx[τλ,τ]v(x)1+xβ(1+γ)p(τ)ηvY} is also a cone. For convenience, we first list the following definitions:

    F0=lim(u,v)(0+,0+)supx>0F1(t,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    f=lim(u,v)(+,+)infx>0F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    f0=lim(u,v)(0+,0+)infx>0F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    F=lim(u,v)(+,+)supx>0F1(t,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    G0=lim(u,v)(0+,0+)supx>0G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    g=lim(u,v)(+,+)infx>0G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    g0=lim(u,v)(0+,0+)infx>0G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v,
    G=lim(u,v)(+,+)supx>0G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)u+v.

    Lemma 4.1. If assumptions (H1) and (H2) hold, then A:K1×K2K1×K2 is completely continuous.

    Proof. First, we show A:K1×K2K1×K2. By (H1) and (H2), for (u,v)K1×K2,

    A1(u,v)X=supt>0|A1(u,v)(x)|1+xβ(1+γ)=supx>00Gσ1(x,s)1+xβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))dsβΓ(σ1)0sβ(γ+1)1F(s,u(s),v(s))ds=βΓ(σ1)0sβ(γ+1)1F(s,(1+sβ(1+γ))u(s)1+sβ(1+γ),(1+sβ(1+γ))v(s)1+sβ(1+γ))ds=βΓ(σ1)0F1(s,u(s)1+sβ(1+γ),v(s)1+sβ(1+γ))βΓ(σ1)[ω1(uX)0φ1(s)ds+ω2(vY)0ψ1(s)ds]<+.

    Similarly,

    A2(u,v)YβΓ(σ1)[~ω1(uX)0φ2(s)ds+~ω2(vY)0ψ2(s)ds]<+.

    By (H1) and Lemma 2.2, for (u,v)K1×K2, we have A1(u,v)(x)>0,A2(u,v)(x)>0,x>0. From Lemma 2.2 and Remark 3.1, for x[τλ,τ],τ>0, and λ>1,

    |A1(u,v)(x)|1+xβ(1+γ)=0Gσ1(x,s)1+xβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))ds=τλ20Gσ1(x,s)1+xβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))ds+0τλ2Gσ1(x,s)1+xβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))ds0τλ2Gσ1(t,s)1+tβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))dsβp(τ)Γ(σ1)0τλ2sβ(γ+1)1F(s,u(s),v(s))dsβp(τ)ηΓ(σ1)0sβ(γ+1)1F(s,u(s),v(s))dsp(τ)ηA1(u,v)X.

    So, A1(u,v)(x)1+xβ(1+γ)p(τ)ηA1(u,v)X. Similarly, A2(u,v)(x)1+xβ(1+γ)p(τ)ηA2(u,v)Y. Therefore,

    minx[τλ,τ]A1(u,v)(x)1+xβ(1+γ)p(τ)ηA1(u,v)X,
    minx[τλ,τ]A2(u,v)(x)1+xβ(1+γ)p(τ)ηA2(u,v)Y.

    That is, A:K1×K2K1×K2 is true.

    Second, it will give a simply prove that A is continuous. Let D={(u,v)|(u,v)K1×K2,(u,v)X×YK,K>0}, a bounded subset in K1×K2. Let (un,vn)D be a sequence that converges to (u,v) in K1×K2. Then (un,vn)X×YK. From Lemma 2.2,

    A1(un,vn)A1(u,v)X=supx>0A1(un,vn)(x)A1(u,v)(x)1+xβ(1+γ)βΓ(σ1)0sβ(γ+1)1F(s,un(s),vn(s))ds0sβ(γ+1)1F(s,u(s),v(s))dsβΓ(σ1)0sβ(γ+1)1(F(s,un(s),vn(s))F(s,u(s),v(s)))ds.

    By (H2),

    sβ(γ+1)1F(s,un(s),vn(s))=sβ(γ+1)1F(s,(1+sβ(1+γ))un(s)1+sβ(1+γ),(1+sβ(1+γ))vn(s)1+sβ(1+γ))=F1(s,un(s)1+sβ(1+γ),vn(s)1+sβ(1+γ))φ1(s)ω1(unX)+ψ1(s)ω2(vnY)L1(0,).

    By the continuity of sβ(γ+1)1F(s,u(s),v(s)) and the Lebesgue dominated convergence theorem,

    0sβ(γ+1)1F(s,un(s),vn(s))ds0sβ(γ+1)1F(s,u(s),v(s))ds,n.

    Therefore, A1(un,vn)A1(u,v)X0,n. Similarly, A2(un,vn)A2(u,v)Y0,n.

    So, A(un,vn)A(u,v)X×Y0,n. That is, A is continuous in D. In the end, we know that A(D) is relatively compact on (0,) and is equi-convergent at by [18]. Therefore, A:K1×K2K1×K2 is completely continuous.

    Theorem 4.1. Assume that (H2) and (H4) hold. If F0=0,G0=0,f=,g=, then the system (1.1) has at least one positive solution.

    Proof. We divide the proof into several steps.

    Step 1. A:K1×K2K1×K2 is completely continuous. This result easily follows from Lemma 4.1.

    Step 2. We show that there exist R1>0 and D1={(u,v)X×Y,(u,v)X×Y<R1} such that A(u,v)X×Y(u,v)X×Y, (u,v)(K1×K2)D1.

    Because F0=0,G0=0, we choose R1>0, such that

    F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ϵ1(u+v),
    G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ϵ2(u+v),

    for 0<u+vR1,x>0, where ϵ1,ϵ2>0 satisfy

    ϵ112Γ(σ1)β0a1(s)ds,ϵ212Γ(σ2)β0a2(s)ds.

    So, for (u,v)K1×K2 and (u,v)X×Y=R1, by Lemma 2.2,

    A1(u,v)(x)1+xβ(1+γ)=0Gσ1(x,s)1+xβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))dsβΓ(σ1)0sβ(γ+1)1F(s,u(s),v(s))ds,
    A2(u,v)(x)1+xβ(1+γ)=0Gσ2(x,s)1+xβ(1+γ)sβ(γ+1)1G(s,u(s),v(s))dsβΓ(σ2)0sβ(γ+1)1G(s,u(s),v(s))ds.

    By (H4),

    A1(u,v)(x)1+xβ(1+γ)βΓ(σ1)0a1(s)F1(s,u(s),v(s))ds=βΓ(σ1)0a1(s)F1(s,(1+sβ(1+γ))u(s)1+sβ(1+γ),(1+sβ(1+γ))v(s)1+sβ(1+γ))dsβΓ(σ1)0a1(s)ϵ1u(s)+v(s)1+sβ(1+γ)dsβΓ(σ1)ϵ1(u,v)X×Y0a1(s)ds12(u,v)X×Y.

    Similarly,

    A2(u,v)(x)1+xβ(1+γ)βΓ(σ2)ϵ2(u,v)X×Y0a2(s)ds12(u,v)X×Y.

    Therefore,

    A(u,v)X×Y(u,v)X×Y, for (u,v)K1×K2, and (u,v)X×Y=R1.

    Let D1={(u,v)X×Y,(u,v)X×Y<R1}. Then,

    A(u,v)X×Y(u,v)X×Y, for (u,v)(K1×K2)D1.

    Step 3. We show that there exist R2>0 and D2={(u,v)X×Y,(u,v)X×Y<R2} such that

    A(u,v)X×Y(u,v)X×Y, for (u,v)(K1×K2)D2.

    Because f=,g=, there exists R>0, such that

    F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)m1(u+v),
    G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)m2(u+v),

    for u+vR,x>0, where m1,m2>0 satisfy

    m112η1ηΓ(σ1)βp2(τ)τλτa1(s)ds,m212η2ηΓ(σ2)βp2(τ)τλτa2(s)ds,η=max{η1,η2}.

    Let R2max{R1,ηRp(τ)}, and D2={(u,v)X×Y,(u,v)X×Y<R2}. Then, D1D2.

    Thus, for  (u,v)K1×K2, (u,v)X×Y=R2, we have

    u(x)1+xβ(1+γ)minx[τλ,τ]u(x)1+xβ(1+γ)p(τ)η1uX,
    v(x)1+xβ(1+γ)minx[τλ,τ]v(x)1+xβ(1+γ)p(τ)η2vY.

    So,

    u(x)+v(x)1+xβ(1+γ)p(τ)η1uX+p(τ)η2vYp(τ)η(uX+vY)=p(τ)η(u,v)X×Y=p(τ)ηR2R.

    By (H4), for x[τλ,τ], we can obtain

    A1(u,v)(x)1+xβ(1+γ)βp(τ)η1Γ(σ1)0sβ(γ+1)1F(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)0a1(s)F1(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)0a1(s)F1(s,(1+sβ(1+γ))u(s)1+sβ(1+γ),(1+sβ(1+γ))v(s)1+sβ(1+γ))dsβp(τ)η1Γ(σ1)m10a1(s)u(s)+v(s)1+sβ(1+γ)dsβp(τ)η1Γ(σ1)m10a1(s)dsp(τ)η1uX+βp(τ)η1Γ(σ1)m10a1(s)dsp(τ)η2vYβp(τ)η1Γ(σ1)m1τλτa1(s)dsp(τ)η1uX+βp(τ)η1Γ(σ1)m1τλτa1(s)dsp(τ)η2vY=βp2(τ)η1Γ(σ1)m1τλτa1(s)ds(1η1uX+1η2vY)βp2(τ)η1Γ(σ1)m1τλτa1(s)ds1η(u,v)X×Y12(u,v)X×Y.

    Similarly, A2(u,v)(x)1+xβ(1+γ)12(u,v)X×Y. Therefore,

    A(u,v)X×Y(u,v)X×Y, for (u,v)(K1×K2)D2.

    Finally, by Lemma 2.4, A has a fixed point in (K1×K1)(¯D2D1). So, (1.1) has at least one positive solution.

    Theorem 4.2. Assume that (H2) and (H4) hold. If f0=,g0=,F=0,G=0, then (1.1) has at least one positive solution.

    Proof. We divide the proof into several steps.

    Step 1. A:K1×K2K1×K2 is completely continuous. This result easily follows from Lemma 4.1.

    Step 2. We show that there exist r1>0 and D1={(u,v)X×Y,(u,v)X×Y<r1} such that

    A(u,v)X×Y(u,v)X×Y, for (u,v)(K1×K2)D1.

    Because f0=,g0=, there exists r1>0 such that

    F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)M1(u+v),
    G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)M2(u+v),

    for 0<u+vr1,x>0, where M1,M2>0, satisfy

    M112η1ηΓ(σ1)βp2(τ)τλτa1(s)ds,M212η2ηΓ(σ2)βp2(τ)τλτa2(s)ds,η=max{η1,η2}.

    Let D1={(u,v)X×Y,(u,v)X×Y<r1}. So, for (u,v)K1×K2 with (u,v)X×Y=r1, and x[τλ,τ], then by (H4),

    A1(u,v)(x)1+xβ(1+γ)βp(τ)η1Γ(σ1)0sβ(γ+1)1F(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)0a1(s)F1(s,u(s),v(s))ds=βp(τ)η1Γ(σ1)0a1(s)F1(s,(1+sβ(1+γ))u(s)1+sβ(1+γ),(1+sβ(1+γ))v(s)1+sβ(1+γ))dsβp(τ)η1Γ(σ1)M10a1(s)u(s)+v(s)1+sβ(1+γ)dsβp(τ)η1Γ(σ1)M10a1(s)dsp(τ)η1uX+βp(τ)η1Γ(σ1)M10a1(s)dsp(τ)η2vYβp(τ)η1Γ(σ1)M1τλτa1(s)dsp(τ)η1uX+βp(τ)η1Γ(σ1)M1τλτa1(s)dsp(τ)η2vY=βp2(τ)η1Γ(σ1)M1τλτa1(s)ds(1η1uX+1η2vY)βp2(τ)η1Γ(σ1)M1τλτa1(s)ds1η(u,v)X×Y12(u,v)X×Y.

    Similarly, A2(u,v)(x)1+xβ(1+γ)12(u,v)X×Y. Thus,

    A(u,v)X×Y(u,v)X×Y, for (u,v)(K1×K2)D1.

    Step 3. We show that there exist r2>0 and D2={(u,v)X×Y,(u,v)X×Y<r2} such that

    A(u,v)X×Y(u,v)X×Y for (u,v)(K1×K2)D2.

    Because F=0,G=0, there exists r>0, such that

    F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ϵ1(u+v),
    G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ϵ2(u+v),

    for u+v>r,x>0, where ϵ1,ϵ2>0 satisfy

    ϵ112Γ(σ1)β0a1(s)ds,ϵ212Γ(σ2)β0a2(s)ds.

    Let D2={(u,v)X×Y,(u,v)X×Y<r2}, where r2>max{r1,r}. Then D1D1. We define two functions U1,U2 as follows:

    U1:(,+)(,+),U1(a)=sup0<u+vasupx>0F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v),
    U2:(,+)(,+),U2(a)=sup0<u+vasupx>0G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v).

    For (u,v)K1×K2 and (u,v)X×Y=r2,

    U1(r2)=sup0<u+vr2supx>0F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ϵ1sup0<u+vr2(u+v)=ϵ1r2=ϵ1(u,v)X×Y,
    U2(r2)=sup0<u+vr2supx>0G1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ϵ2sup0<u+vr2(u+v)=ϵ2r2=ϵ2(u,v)X×Y.

    By Lemma 2.2 and (H4),

    A1(u,v)(x)1+xβ(1+γ)βΓ(σ1)0sβ(γ+1)1F(s,u(s),v(s))ds=βΓ(σ1)0a1(s)F1(s,u(s),v(s))ds=βΓ(σ1)0a1(s)F1(s,(1+sβ(1+γ))u(s)1+sβ(1+γ),(1+sβ(1+γ))v(s)1+sβ(1+γ))dsβΓ(σ1)0a1(s)sup0<u+vr2supx>0F1(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)ds=βΓ(σ1)0a1(s)U1(r2)dsβΓ(σ1)0a1(s)dsϵ1(u,v)X×Y12(u,v)X×Y.

    Similarly, A2(u,v)(x)1+xβ(1+γ)12(u,v)X×Y. Therefore, A(u,v)X×Y(u,v)X×Y, for (u,v)(K1×K2)D2. Finally, by Lemma 2.4, A has a fixed point in (K1×K1)(¯D2D1). So, the system (1.1) has at least one positive solution.

    In the section, we obtain the multiplicity of positive solution of (1.1) by using the monotone iterative technique.

    Theorem 5.1. If (H1) and (H2) hold, then (1.1) has two positive solutions (u,v) and (w,z) satisfying 0(u,v)X×YΥ and 0(w,z)X×YΥ, where Υ is a positive preset constant. Moreover, limn(un,vn)=(u,v) and limn(wn,zn)=(w,z), where (un,vn) and (wn,zn) are given by

    (un(x),vn(x))=(A1(un1,vn1)(x),A2(un1,vn1)(x)),n=1,2,, (5.1)

    with

    (u0(x),v0(x))=(Υ1[1+xβ(γ+1)],Υ2[1+xβ(γ+1)]),Υ1,Υ2>0,Υ1+Υ2Υ,

    and

    (wn(x),zn(x))=(A1(wn1,zn1)(x),A2(wn1,zn1)(x)),n=1,2,, (5.2)

    with (w0(x),z0(x))=(0,0). In addition,

    (w0(x),z0(x))(w1(x),z1(x))(wn(x),zn(x))(w,z)(u,v)(un(x),vn(x))(u1(x),v1(x))(u0(x),v0(x)). (5.3)

    Proof. First, from Lemma 4.1, A(K1×K2)K1×K2 for (u,v)K1×K2. Let

    Υ1=βΓ(σ1)[ω1(Υ)0φ1(s)ds+ω2(Υ)0ψ1(s)ds]<,
    Υ2=βΓ(σ2)[~ω1(Υ)0φ2(s)ds+~ω2(Υ)0ψ2(s)ds]<,

    and ΥΥ1+Υ2 with DΥ={(u,v)K1×K2:(u,v)X×YΥ}. For any (u,v)DΥ, from (H2) and Lemma 2.2,

    A1(u,v)X=supx>0|A1(u,v)(x)|1+xβ(1+γ)=supx>00Gσ1(x,s)1+tβ(1+γ)sβ(γ+1)1F(s,u(s),v(s))dsβΓ(σ1)0sβ(γ+1)1F(s,u(s),v(s))dsβΓ(σ1)[ω1(u(s)1+sβ(1+γ))0φ1(s)ds+ω2(v(s)1+sβ(1+γ))0ψ1(s)ds]βΓ(σ1)[ω1(uX)0φ1(s)ds+ω2(vY)0ψ1(s)ds]βΓ(σ1)[ω1(Υ)0φ1(s)ds+ω2(Υ)0ψ1(s)ds]=Υ1.

    Similarly, A2(u,v)YΥ2 for (u,v)DΥ. Thus,

    A(u,v)X×Y=A1(u,v)X+A2(u,v)YΥ1+Υ2Υ.

    That is, A(DΥ)DΥ. We construct two sequences as follows:

    (un,vn)=A(un1,vn1),(wn,zn)=A(wn1,zn1),  n=1,2,3,.

    Obviously, (u0(x),v0(x)),(w0(x),z0(x))DΥ. Because A(DΥ)DΥ, (un,vn),(wn,zn)DΥ,n=1,2,. We need to show that there exist (u,v) and (w,z) satisfying limn(un,vn)=(u,v) and limn(wn,zn)=(w,z) which are two monotone sequences for approximating positive solutions of the system (1.1).

    For x(0,+),(un,vn)DΥ, from Lemma 2.2 and (5.1),

    u1(x)=A1(u0,v0)(x)=0Gσ1(x,s)sβ(γ+1)1F(s,u0(s),v0(s))dsβΓ(σ1)0(1+tβ(1+γ))sβ(γ+1)1F(s,u0(s),v0(s))dsβΓ(σ1)(1+xβ(1+γ))[ω1(u0(s)1+sβ(1+γ))0φ1(s)ds+ω2(v0(s)1+sβ(1+γ))0ψ1(s)ds]βΓ(σ1)(1+xβ(1+γ))[ω1(u0X)0φ1(s)ds+ω2(v0Y)0ψ1(s)ds]βΓ(σ1)(1+xβ(1+γ))[ω1(Υ)0φ1(s)ds+ω2(Υ)0ψ1(s)ds]=(1+xβ(1+γ))Υ1=u0(x)

    and

    v1(x)=A2(u0,v0)(x)=0Gσ2(x,s)sβ(γ+1)1G(s,u0(s),v0(s))dsβΓ(σ2)0(1+xβ(1+γ))sβ(γ+1)1G(s,u0(s),v0(s))dsβΓ(σ2)(1+tβ(1+γ))[~ω1(u0(s)1+sβ(1+γ))0φ2(s)ds+~ω2(v0(s)1+sβ(1+γ))0ψ2(s)ds]βΓ(σ2)(1+tβ(1+γ))[~ω1(u0X)0φ2(s)ds+~ω2(v0Y)0ψ2(s)ds]βΓ(σ2)(1+xβ(1+γ))[~ω1(Υ)0φ2(s)ds+~ω2(Υ)0ψ2(s)ds]=(1+xβ(1+γ))Υ2=v0(x),

    that is,

    (u1(x),v1(x))=(A1(u0,v0)(x),A2(u0,v0)(x))((1+xβ(1+γ))Υ1,(1+xβ(1+γ))Υ2)=(u0(x),v0(x)).

    So, by the condition (H1),

    (u2(x),v2(x))=(A1(u1,v1)(x),A2(u1,v1)(x))(A1(u0,v0)(x),A2(u0,v0)(x))=(u1(x),v1(x)).

    For x(0,+), the sequences {(un,vn)}n=0 satisfy (un+1(x),vn+1(x))(un(x),vn(x)). By the iterative sequences (un+1,vn+1)=A(un,vn) and the complete continuity of the operator A, (un,vn)(u,v), and A(u,v)=(u,v).

    Similarly, for the sequences {(wn,zn)}n=0, we have

    (w1(x),z1(x))=(A1(w0,z0)(x),A2(w0,z0)(x))=(0Gσ1(x,s)sβ(γ+1)1F(s,w0(s),z0(s))ds,0Gσ2(x,s)sβ(γ+1)1G(s,w0(s),z0(s))ds)(0,0)=(w0(x),z0(x)).

    Then, by the condition (H1),

    (w2(x),z2(x))=(A1(w1,z1)(x),A2(w1,z1)(x))(A1(w0,z0)(x),A2(w0,z0)(x))=(w1(x),z1(x)).

    Analogously, for x(0,+), we have (wn+1(x),zn+1(x))(wn(x),zn(x)). By the iterative sequences (wn+1,zn+1)=A(wn,zn) and the complete continuity of the operator A, (wn,zn)(w,z), and A(w,z)=(w,z).

    Finally, we prove that (u,v) and (w,z) are the minimal and maximal positive solutions of (1.1). Assume that (ς(x),μ(x)) is any positive solution of (1.1). Then, A(ς(x),μ(x))=(ς(x),μ(x)), and

    (w0(x),z0(x))=(0,0)(ς(x),μ(x))((1+xβ(1+γ))Υ1,(1+xβ(1+γ))Υ2)=(u0(x),v0(x)).

    Therefore,

    (w1(x),z1(x))=(A1(w0,z0)(x),A2(w0,z0)(x))(ς(x),μ(x))(A1(u0,v0)(x),A2(u0,v0)(x))=(u1(x),v1(x)).

    That is, (w1(x),z1(x))(ς(x),μ(x))(un(x),vn(x)). So, (5.3) holds. By (H1), (0,0) is not a solution of (1.1). From (5.1), (w,z) and (u,v) are two extreme positive solutions of (1.1), which can be constructed via limitS of two monotone iterative sequences in (5.1) and (5.2).

    Example 6.1. We consider the following system:

    {D32,531u(x)+x32(u1+x12)2ex+x32(v1+x12)2ex=0,t(0,+),D32,321v(x)+x52e2x2(u1+x12)2ln(1+(u1+x12)2)+x52e2x2(u1+x12)2ln(1+(u1+x12)2),x(0,+),limx0x12I16,13u(x)=0,limxx12I16,13u(x)=0,limx0x12I0,12v(x)=0,limxx12I0,12v(x)=0, (6.1)

    where σ1=53,σ2=32,γ=32,β=1,

    F(x,u,v)=x32ex[(u1+x12)2+(v1+x12)2],
    G(x,u,v)=x52e2x2[(u1+x12)2ln(1+(u1+x12)2)+(u1+x12)2ln(1+(u1+x12)2)].

    First, for F1(x,u,v)=xβ(1+γ)1F(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)=ex(u2,v2), we choose ω1(u)=u2C((0,+),(0,+)), ω2(v)=v2C((0,+),(0,+)), and φ1(x)=ψ1(x)=exL1(0,+). Then,

    F1(x,u,v)∣≤φ1(x)ω1(u)+ψ1(t)ω2(v),  (0,+)×(,+)×(,+).

    Similarly, for F2(x,u,v)=xβ(1+γ)1G(x,(1+xβ(1+γ))u,(1+xβ(1+γ))v)=xe2x2[u2ln(u2+1)+v2ln(v2+1)], we choose ~ω1(u)=u2ln(u2+1)C((0,+),(0,+)), ~ω2(v)=v2ln(v2+1)C((0,+),(0,+)), and φ2(x)=ψ2(x)=xe2x2L1(0,+). Then,

    F2(x,u,v)∣≤φ2(x)~ω1(u)+ψ2(x)~ω2(v),  (0,+)×(,+)×(,+).

    So, the condition (H2) holds. Obviously, F,G:(0,+)×(0,+)×(0,+)(0,+) are continuous.

    x32F(x,u,v)=ex[(u1+x12)2+(v1+x12)2]=a1(x)F1(x,u,v),
    x32G(x,u,v)=xe2x2[(u1+x12)2ln(1+(u1+x12)2)+(u1+x12)2ln(1+(u1+x12)2)]=a2(x)G1(x,u,v),

    where a1(x)=ex,a2(x)=xe2x2, F1(x,u,v)=(u1+x12)2+(v1+x12)2, G1(t,u,v)=(u1+x12)2ln(1+(u1+x12)2)+(u1+x12)2ln(1+(u1+x12)2). So, x32f(x,u,v),x32G(x,u,v):[0,+)×(0,+)×(0,+)[0,+) are continuous. Hence, the condition (H4) holds. Finally,

    F0=lim(u,v)(0+,0+)u2+v2u+v=0,G0=lim(u,v)(0+,0+)u2ln(u2+1)+v2ln(v2+1)u+v=0,
    f=lim(u,v)(+,+)u2+v2u+v=,g=lim(u,v)(+,+)u2ln(u2+1)+v2ln(v2+1)u+v=.

    Therefore, from Theorem 4.1, (6.1) has at least one positive solution (u(x), v(x)) . Further,

    \begin{cases} u(x) = \frac 3{2\Gamma(\frac 23)}[x^{\frac 12}\int_0^\infty s^{-\frac 32}F(s, u(s), v(s))ds-x^{-\frac 83}\int_x^\infty (x-s)^{\frac 23}s^{-\frac 32}F(s, u(s), v(s))ds], \\ v(x) = \frac 2{\sqrt{\pi}}[x^{\frac 12}\int_0^\infty s^{-\frac 32}G(s, u(s), v(s))ds-\int_x^\infty (x-s)^{\frac 12}s^{-\frac 32}G(s, u(s), v(s))ds]. \end{cases}

    Example 6.2. We consider the following system:

    \begin{align} \begin{cases} D_{1}^{-\frac{3}{2}, \frac{3}{2}}u(x)+x^{\frac{5}{2}}e^{-2x^{2}+1}[\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}]+ x^{\frac{5}{2}}e^{-2x^{2}+1} [\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\pi] = 0, x\in (0, +\infty), \\ D_{1}^{-\frac{3}{2}, \frac{7}{6}}v(x)+x^{\frac{3}{2}}e^{-x}[\arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi]+ x^{\frac{3}{2}}e^{-x} [\arctan(\ln((\frac{v}{1+x^{\frac{1}{2}}})^{2}+1))+1], x\in (0, +\infty), \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{0 \frac{1}{2}}u(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{0, \frac{1}{2}}u(x) = 0, \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{-\frac{1}{3}, \frac{5}{6}}v(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{-\frac{1}{3}, \frac{5}{6}}v(x) = 0, \end{cases} \end{align} (6.2)

    where \sigma_{1} = \frac{3}{2}, \sigma_{2} = \frac{7}{6}, \gamma = -\frac{3}{2}, \beta = 1 ,

    F(x, u, v) = x^{\frac{5}{2}}e^{-2x^{2}+1}[\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}]+x^{\frac{5}{2}}e^{-2x^{2}+1} [\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}],
    G(x, u, v) = x^{\frac{3}{2}}e^{-x}[\arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi]+x^{\frac{3}{2}}e^{-x} [\arctan(\ln((\frac{v}{1+x^{\frac{1}{2}}})^{2}+1))+1].

    First, for

    F_{1}(x, u, v) = x^{\beta(1+\gamma)-1}F(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = xe^{-2x^{2}+1}[\arctan u^{2}+\frac{1}{\pi}+\arctan v^{2}+\pi],

    we choose \omega_{1}(u) = \arctan u^{2}+\frac{1}{\pi}\in C((0, +\infty), (0, +\infty)), \omega_{2}(v) = \arctan v^{2}+\pi\in C((0, +\infty), (0, +\infty)) , and \varphi_{1}(x) = \psi_{1}(x) = xe^{-2x^{2}+1}\in L^{1}(0, +\infty) . Then,

    \mid F_{1}(x, u, v)\mid\leq\varphi_{1}(x)\omega_{1}(\mid u\mid)+\psi_{1}(x)\omega_{2}(\mid v\mid), \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty).

    Similarly, for

    F_{2}(x, u, v) = x^{\beta(1+\gamma)-1}g(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = e^{-x}[\arctan(\ln(u^{2}+1))+ \frac{3}{2}\pi+\arctan(\ln(v^{2}+1))+1],

    we choose \widetilde{\omega_{1}}(u) = \arctan(\ln(u^{2}+1))+\frac{3}{2}\pi\in C((0, +\infty), (0, +\infty)) , \widetilde{\omega_{2}}(v) = \arctan(\ln(v^{2}+1))+1\in C((0, +\infty), (0, +\infty)) , and \varphi_{2}(x) = \psi_{2}(x) = e^{-x}\in L^{1}(0, +\infty) . Then,

    \mid F_{2}(x, u, v)\mid\leq\varphi_{2}(x)\widetilde{\omega_{1}}(\mid u\mid)+\psi_{2}(x)\widetilde{\omega_{2}}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty).

    That is, (H_{2}) holds. Second, F, G:(0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow (0, +\infty) are continuous. And

    x^{-\frac{3}{2}}F(x, u, v) = xe^{-2x^{2}+1}[\arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}+\arctan(\frac{v} {1+x^{\frac{1}{2}}})^{2}+\pi] = a_{1}(x)F_{1}(x, u, v),
    x^{-\frac{3}{2}}G(x, u, v) = e^{-x}[\arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi+\arctan(\ln((\frac{v} {1+x^{\frac{1}{2}}})^{2}+1))+1] = a_{2}(x)G_{1}(x, u, v),

    where a_{1}(x) = xe^{-2x^{2}+1}, a_{2}(x) = e^{-x} , F_{1}(x, u, v) = \arctan(\frac{u}{1+x^{\frac{1}{2}}})^{2}+\frac{1}{\pi}+\arctan(\frac{v}{1+x^{\frac{1}{2}}})^{2}+\pi , G_{1}(x, u, v) = \arctan(\ln((\frac{u}{1+x^{\frac{1}{2}}})^{2}+1))+\frac{3}{2}\pi+\arctan(\ln((\frac{v}{1+x^{\frac{1}{2}}})^{2}+1))+1 . So, x^{-\frac{3}{2}}F(x, u, v), x^{-\frac{3}{2}}G(x, u, v):[0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow [0, +\infty) are continuous. That is, (H_{4}) holds. In addition,

    f_{0} = \lim\limits_{(u, v)\rightarrow (0^{+}, 0^{+})} \frac{\arctan u^{2}+\frac{1}{\pi}+\arctan v^{2}+\pi}{u+v} = \infty,
    g_{0}^{\ast} = \lim\limits_{(u, v)\rightarrow (0^{+}, 0^{+})} \frac{\arctan(\ln(u^{2}+1))+\frac{3}{2}\pi+\arctan(\ln(v^{2}+1))+1}{u+v} = \infty,
    F_{\infty} = \lim\limits_{(u, v)\rightarrow (+\infty, +\infty)} \frac{\arctan u^{2}+\frac{1}{\pi}+\arctan v^{2}+\pi}{u+v} = 0,
    G_{\infty}^{\ast} = \lim\limits_{(u, v)\rightarrow (+\infty, +\infty)} \frac{\arctan(\ln(u^{2}+1))+\frac{3}{2}\pi+\arctan(\ln(v^{2}+1))+1}{u+v} = 0.

    Therefore, from Theorem 4.2, (6.2) has at least one positive solution (u(x), v(x)) . Further,

    \begin{cases} u(x) = \frac 2{\sqrt{\pi}}[x^{\frac 12}\int_0^\infty s^{-\frac 32}F(s, u(s), v(s))ds-\int_x^\infty (x-s)^{\frac 12}s^{-\frac 32}F(s, u(s), v(s))ds], \\ v(x) = \frac 6{\Gamma(\frac 16)}[x^{\frac 12}\int_0^\infty s^{-\frac 32}G(s, u(s), v(s))ds-x^{\frac 13}\int_x^\infty (x-s)^{\frac 16}s^{-\frac 32}G(s, u(s), v(s))ds]. \end{cases}

    Example 6.3. We consider the following system:

    \begin{align} \begin{cases} D_{1}^{-\frac{3}{2}, \frac{5}{3}}u(x)+x^{\frac{3}{2}}\frac{e^{-x}}{3}\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+ x^{\frac{5}{2}}\ln(\mid\frac{v}{1+x^{\frac{1}{2}}}\mid+1)\frac{e^{-2x^{2}+1}}{10} = 0, x\in (0, +\infty), \\ D_{1}^{-\frac{3}{2}, \frac{3}{2}}v(x)+x^{\frac{5}{2}}e^{-2x^{2}+1}\arctan(\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+ \frac{1}{\sqrt{\pi}})+x^{\frac{5}{2}}\frac{e^{-2x^{2}+1}}{5} \mid\frac{v}{1+x^{\frac{1}{2}}}\mid = 0, x\in (0, +\infty), \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{\frac{1}{6}, \frac{1}{3}}u(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{\frac{1}{6}, \frac{1}{3}}u(x) = 0, \\ \lim\nolimits_{x\rightarrow 0} x^{\frac{1}{2}}I^{0, \frac{1}{2}}v(x) = 0, \lim\nolimits_{x\rightarrow \infty} x^{-\frac{1}{2}}I^{0, \frac{1}{2}}v(x) = 0, \end{cases} \end{align} (6.3)

    where \sigma_{1} = \frac{5}{3}, \sigma_{2} = \frac{3}{2}, \gamma = -\frac{3}{2}, \beta = 1 ,

    F(x, u, v) = x^{\frac{3}{2}}\frac{e^{-x}}{3}\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+x^{\frac{5}{2}}\ln(\mid\frac{v} {1+x^{\frac{1}{2}}}\mid+1)\frac{e^{-2x^{2}+1}}{10},
    G(x, u, v) = x^{\frac{5}{2}}e^{-2x^{2}+1}\arctan(\mid\frac{u}{1+x^{\frac{1}{2}}}\mid+\frac{1}{\sqrt{\pi}})+ x^{\frac{5}{2}}\frac{e^{-2x^{2}+1}}{5} \mid\frac{v}{1+x^{\frac{1}{2}}}\mid.

    Obviously, F, G:(0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty)\rightarrow (0, +\infty) are continuous and nondecreasing with respect to the second and the third variables on (0, +\infty) . That is, (H_{1}) holds. Next,

    F_{1}(x, u, v) = x^{\beta(1+\gamma)-1}F(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = \frac{e^{-x}}{3}\mid u\mid+x\frac{e^{-2x^{2}+1}}{10}\ln(\mid v\mid+1).

    We choose \omega_{1}(u) = \mid u\mid\in C((0, +\infty), (0, +\infty)) , \omega_{2}(v) = \ln(\mid v\mid+1)\in C((0, +\infty), (0, +\infty)) , and \varphi_{1}(x) = \frac{e^{-x}}{3}, \psi_{1}(x) = \frac{xe^{-2x^{2}+1}}{10}\in L^{1}(0, +\infty) . Then,

    \mid F_{1}(x, u, v)\mid\leq\varphi_{1}(x)\omega_{1}(\mid u\mid)+\psi_{1}(x)\omega_{2}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty).

    Similarly, for

    F_{2}(x, u, v) = x^{\beta(1+\gamma)-1}G(x, (1+x^{-\beta(1+\gamma)})u, (1+x^{-\beta(1+\gamma)})v) = xe^{-2x^{2}+1}\arctan(\mid u\mid+\frac{1}{\sqrt{\pi}})+x\frac{e^{-2x^{2}+1}}{5}\mid v\mid,

    we choose \widetilde{\omega_{1}}(u) = \arctan(\mid u\mid+\frac{1}{\sqrt{\pi}})\in C((0, +\infty), (0, +\infty)) , \widetilde{\omega_{2}}(v) = \mid v\mid\in C((0, +\infty), (0, +\infty)) , and \varphi_{2}(x) = xe^{-2x^{2}+1}, \psi_{2}(x) = x\frac{e^{-2x^{2}+1}}{5}\in L^{1}(0, +\infty) . Then,

    \mid F_{2}(x, u, v)\mid\leq\varphi_{2}(x)\widetilde{\omega_{1}}(\mid u\mid)+\psi_{2}(x)\widetilde{\omega_{2}}(\mid v\mid), \ \ (0, +\infty)\times(-\infty, +\infty)\times(-\infty, +\infty).

    That is, (H_{2}) holds. Therefore, from Theorem 5.1, (6.3) has two positive solutions (u^{\ast}, v^{\ast}) and (w^{\ast}, z^{\ast}) with (0, 0)\leq (u^{\ast}(x), v^{\ast}(x)), (w^{\ast}(x), z^{\ast}(x))\leq ((1+x^{\frac 12})\Upsilon_{1}, (1+x^{\frac 12})\Upsilon_{2}) , where \Upsilon_{1}+\Upsilon_{2}\leq \Upsilon , and \Upsilon satisfies

    \frac{95.58}{191.86}\Upsilon-0.69\arctan (\Upsilon+0.56)\geq \frac 1{36}.

    This paper studies the Erdélyi-Kober fractional coupled system (1.1), where the variable is in an infinite interval. We give some proper conditions and set a special Banach space. We obtain the existence of at least one positive solution for (1.1) by using the Guo-Krasnosel'skii fixed point theorem, and we get the existence of at least two positive solutions for (1.1) by using the monotone iterative technique. Our methods and results are different from ones in [18]. Moreover, we give three examples to show the plausibility of our main results. For future work, we intend to use other fixed point theorems to solve some Erdélyi-Kober fractional differential equations.

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

    This paper is supported by the Fundamental Research Program of Shanxi Province (202303021221068).

    The authors declare that they have no competing interests.



    [1] M. Abdulhameed, D. Vieru, R. Roslanc, Magnetohydrodynamic electroosmotic flow of Maxwell fluids with Caputo-Fabrizio derivatives through circular tubes, Comput. Math. Appl., 74 (2017), 2503–2519. http://doi.org/10.1016/j.camwa.2017.07.040 doi: 10.1016/j.camwa.2017.07.040
    [2] T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11–27. http://doi.org/10.1016/S0034-4877(17)30059-9 doi: 10.1016/S0034-4877(17)30059-9
    [3] H. Abboubakar, P. Kumar, N. A. Rangaig, S. Kumar, A malaria model with Caputo-Fabrizio and Atangana-Baleanu derivatives, Int. J. Model. Simul. Sci. Comput., 12 (2021), 2150013. http://doi.org/10.1142/S1793962321500136 doi: 10.1142/S1793962321500136
    [4] J. F. G. Aguilar, H. Y. Martinez, C. C. Ramon, I. C. Ordunia, R. F. E. Jimenez, V. H. O. Peregrino, Modeling of a mass-spring-damper system by fractional derivatives with and without a singular Kernel, Entropy, 17 (2015), 6289–6303. http://doi.org/10.3390/e17096289 doi: 10.3390/e17096289
    [5] B. S. T. Alkahtani, A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 539–546. http://doi.org/10.1016/j.chaos.2016.03.012 doi: 10.1016/j.chaos.2016.03.012
    [6] I. Area, J. J. Nieto, Fractional-order logistic differential equation with Mittag–Leffler-type kernel, Fractal Fract., 5 (2021), 273. http://doi.org/10.3390/fractalfract5040273 doi: 10.3390/fractalfract5040273
    [7] S. Arshad, D. Baleanu, J. Huang, Y. Tang, M. M. Al Qurashi, Dynamical analysis of fractional order model of immunogenic tumors, Adv. Mech. Eng., 8 (2016), 1–13. https://doi.org/10.1177/1687814016656704 doi: 10.1177/1687814016656704
    [8] A. Atangana, A. Secer, A note on fractional order derivatives and table of fractional derivatives of some special function, Abstr. Appl. Anal., 2013 (2013), 279681. http://doi.org/10.1155/2013/279681 doi: 10.1155/2013/279681
    [9] D. Avci, M. Yavuz, N. Ozdemir, Fundamental solutions to the Cauchy and Dirichlet problems for a heat conduction equation equipped with the Caputo-Fabrizio differentiation, In: Heat conduction: methods, applications and research, Nova Science Publishers, 2019, 95–107.
    [10] D. Baleanu, S. Arshad, A. Jajarmi, W. Shokat, F. A. Ghassabzade, M. Wali, Dynamical behaviours and stability analysis of a generalized fractional model with a real case study, J. Adv. Res., in press. http://doi.org/10.1016/j.jare.2022.08.010
    [11] M. Bologna, P. Grigolini, B. J. West, Physics of fractal operators, New York: Springer, 2003. http://doi.org/10.1007/978-0-387-21746-8
    [12] M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1–11. http://doi.org/10.18576/pfda/020101 doi: 10.18576/pfda/020101
    [13] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. http://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
    [14] S. Das, I. Pan, Kriging based surrogate modeling for fractional order control of microgrids, IEEE Trans. Smart Grid, 6 (2015), 36–44. http://doi.org/10.1109/TSG.2014.2336771 doi: 10.1109/TSG.2014.2336771
    [15] H. Dehestani, Y. Ordokhani, An efficient approach based on Legendre–Gauss–Lobatto quadrature and discrete shifted Hahn polynomials for solving Caputo–Fabrizio fractional Volterra partial integro-differential equations, J. Comput. Appl. Math., 403 (2022), 113851. http://doi.org/10.1016/j.cam.2021.113851 doi: 10.1016/j.cam.2021.113851
    [16] N. Djeddi, S. Hasan, M. Al-Smadi, S. Momani, Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative, Alex. Eng. J., 59 (2020), 5111–5122. http://doi.org/10.1016/j.aej.2020.09.041 doi: 10.1016/j.aej.2020.09.041
    [17] J. Dison, S. Mekee, Weakly singular discrete Gronwall inequalities, Z. Angew. Math. Mech., 66 (1986), 535–544. https://doi.org/10.1002/zamm.19860661107 doi: 10.1002/zamm.19860661107
    [18] F. Evirgen, M. Yavuz, An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative, ITM Web Conf., 22 (2018), 01009. http://doi.org/10.1051/itmconf/20182201009 doi: 10.1051/itmconf/20182201009
    [19] M. Farman, H. Besbes, K. S. Nisar, M. Omri, Analysis and dynamical transmission of Covid-19 model by using Caputo-Fabrizio derivative, Alex. Eng. J., 66 (2023), 597–606. http://doi.org/10.1016/j.aej.2022.12.026 doi: 10.1016/j.aej.2022.12.026
    [20] M. A. Firoozjaee, H. Jafari, A. Lia, D. Baleanu, Numerical approach of Fokker-Planck equation with Caputo-Fabrizio fractional derivative using Ritz approximation, J. Comput. Appl. Math., 339 (2018), 367–373. http://doi.org/10.1016/j.cam.2017.05.022 doi: 10.1016/j.cam.2017.05.022
    [21] J. F. Gómez-Aguilar, M. G. López-López, V. M. Alvarado-Martínez, J. Reyes-Reyes, M. Adam-Medina, Modeling diffusive transport with a fractional derivative without singular kernel, Physic A, 447 (2016), 467–481. http://doi.org/10.1016/j.physa.2015.12.066 doi: 10.1016/j.physa.2015.12.066
    [22] A. Horani, R. Khalil, M. Sababheh, A. Yousef, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. http://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
    [23] N. Harrouche, S. Momani, S. Hasan, M. Al-Smadi, Computational algorithm for solving drug pharmacokinetic model under uncertainty with non-singular kernel type Caputo-Fabrizio fractional derivative, Alex. Eng. J., 60 (2021), 4347–4362. http://doi.org/10.1016/j.aej.2021.03.016 doi: 10.1016/j.aej.2021.03.016
    [24] A. Jajarmi, D. Baleanu, A new fractional analysis on the interaction of HIV with CD_{4}^{+} T-cells, Chaos Soliton. Fract., 113 (2018), 221–229. http://doi.org/10.1016/j.chaos.2018.06.009 doi: 10.1016/j.chaos.2018.06.009
    [25] A. Jajarmi, S. Arshad, D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A, 535 (2019), 122524. https://doi.org/10.1016/j.physa.2019.122524 doi: 10.1016/j.physa.2019.122524
    [26] T. Jin, X. Yang, H. Xia, H. Ding, Reliability index and option pricing formulas of the first-hitting time model based on the uncertain fractional order differential equation with Caputo type, Fractals, 29 (2021), 2150012. https://doi.org/10.1142/S0218348X21500122 doi: 10.1142/S0218348X21500122
    [27] T. Jin, X. Yang, Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market, Math. Comput. Simulat., 190 (2021), 203–221. http://doi.org/10.1016/j.matcom.2021.05.018 doi: 10.1016/j.matcom.2021.05.018
    [28] G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. Math. Lett., 18 (2005), 817–826. http://doi.org/10.1016/j.aml.2004.09.012 doi: 10.1016/j.aml.2004.09.012
    [29] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 51 (2006), 1367–1376. http://doi.org/10.1016/j.camwa.2006.02.001 doi: 10.1016/j.camwa.2006.02.001
    [30] A. Keten, M. Yavuz, D. Baleanu, Nonlocal Cauchy problem via a fractional operator involving power kernel in Banach spaces, Fractal Fract., 3 (2019), 27. http://doi.org/10.3390/fractalfract3020027 doi: 10.3390/fractalfract3020027
    [31] A. Khan, T. Akram, A. Khan, S. Ahmad, K. Nonlaopon, Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with non-singular kernels, AIMS Mathematics, 8 (2023), 1251–1268. http://doi.org/10.3934/math.2023063 doi: 10.3934/math.2023063
    [32] K. Khan, A. Ali, M. De la Sen, M. Irfan, Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels, AIMS Mathematics, 7 (2022), 1580–1602. http://doi.org/10.3934/math.2022092 doi: 10.3934/math.2022092
    [33] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, London and New York: Elsevier, 2006.
    [34] J. Klafter, S. C. Lim, R. Metzler, Fractional dynamics: recent advances, Singapore: World Scientific, 2011. http://doi.org/10.1142/8087
    [35] C. Li, J. Lu, J. Wang, Observer-based robust stabilisation of a class of non-linear fractional-order uncertain systems: an linear matrix inequalitie approach, IET Control Theory Appl., 6 (2012), 2757–2764. http://doi.org/10.1049/iet-cta.2012.0312 doi: 10.1049/iet-cta.2012.0312
    [36] J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015) 87–92. http://doi.org/10.12785/pfda/010202 doi: 10.12785/pfda/010202
    [37] Q. Liu, T. Jin, M. Zhu, C. Tian, F. Li, D. Jiang, Uncertain currency option pricing based on the fractional differential equation in the Caputo sense, Fractal Fract., 6 (2022), 407. http://doi.org/10.3390/fractalfract6080407 doi: 10.3390/fractalfract6080407
    [38] C. Ludwin, Blood alcohol content, Undergraduate Journal of Mathematical Modeling: One + Two, 3 (2011), 1. http://doi.org/10.5038/2326-3652.3.2.1 doi: 10.5038/2326-3652.3.2.1
    [39] R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2006), 1–104. http://doi.org/10.1615/critrevbiomedeng.v32.i1.10 doi: 10.1615/critrevbiomedeng.v32.i1.10
    [40] I. A. Mirza, D. Vierub, Fundamental solutions to advection-diffusion equation with time-fractional Caputo-Fabrizio derivative, Comput. Math. Appl., 73 (2017), 1–10. http://doi.org/10.1016/j.camwa.2016.09.026 doi: 10.1016/j.camwa.2016.09.026
    [41] V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khan, P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Physica A, 523 (2019), 48–65. http://doi.org/10.1016/j.physa.2019.02.018 doi: 10.1016/j.physa.2019.02.018
    [42] S. Momani, N. Djeddi, M. Al-Smadi, S. Al-Omari, Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method, Appl. Numer. Math., 170 (2021), 418–434. http://doi.org/10.1016/j.apnum.2021.08.005 doi: 10.1016/j.apnum.2021.08.005
    [43] J. J. Nieto, Solution of a fractional logistic ordinary differential equation, Appl. Math. Lett., 123 (2022), 107568. http://doi.org/10.1016/j.aml.2021.107568 doi: 10.1016/j.aml.2021.107568
    [44] S. Noeiaghdam, S. Micula, J. J. Nieto, A novel technique to control the accuracy of a nonlinear fractional order model of covid-19: Application of the CESTAC method and the CADNA library, Mathematics, 9 (2021), 1321. http://doi.org/10.3390/math9121321 doi: 10.3390/math9121321
    [45] H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagt, Digital computation of the fractional Fourier transform, IEEE Transactionson Signal Processing, 44 (1996), 2141–2150. http://doi.org/10.1109/78.536672 doi: 10.1109/78.536672
    [46] A. J. J. Obaid, Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional-order: Allen Cahn model, Chaos Soliton. Fract., 89 (2016), 552–559. http://doi.org/10.1016/j.chaos.2016.03.026 doi: 10.1016/j.chaos.2016.03.026
    [47] K. B. Oldham, J. Spanier, The fractional calculus, New York: Academic Press, 1974.
    [48] K. M. Owolabi, A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Soliton. Fract., 105 (2017), 111–119. http://doi.org/10.1016/j.chaos.2017.10.020 doi: 10.1016/j.chaos.2017.10.020
    [49] P. Pandey, J. F. Gómez-Aguilar, M. K. A. Kaabar, Z. Sirid, A. A. Mousa, Mathematical modeling of COVID-19 pandemic in India using Caputo-Fabrizio fractional derivative, Comput. Biol. Med., 145 (2022), 105518. http://doi.org/10.1016/j.compbiomed.2022.105518 doi: 10.1016/j.compbiomed.2022.105518
    [50] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386.
    [51] J. RongLoh, A. Isah, C. Phang, Y. T. Toh, On the new properties of Caputo-Fabrizio operator and its application in deriving shifted Legendre operational matrix, Appl. Numer. Math., 132 (2018), 138–153. http://doi.org/10.1016/j.apnum.2018.05.016 doi: 10.1016/j.apnum.2018.05.016
    [52] Q. Rubbab, M. Nazeer, F. Ahmad, Y. Chu, M. I. Khan, S. Kadry, Numerical simulation of advection–diffusion equation with caputo-fabrizio time fractional derivative in cylindrical domains: Applications of pseudo-spectral collocation method, Alex. Eng. J., 60 (2021), 1731–1738. http://doi.org/10.1016/j.aej.2020.11.022 doi: 10.1016/j.aej.2020.11.022
    [53] S. G. Samko, A. A. Kilbas, O. I. Maritchev, Integrals and derivatives of the fractional order and some of their applications, (Russian), Minsk, Belarus: Nauka i Tekhnika, 1987.
    [54] L. Shi, S. Tayebi, O. A. Arqub, M. S. Osman, P. Agarwal, W. Mahamoud, et al., The novel cubic B-spline method for fractional Painleve and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense, Alex. Eng. J., 65 (2023), 413–426. http://doi.org/10.1016/j.aej.2022.09.039 doi: 10.1016/j.aej.2022.09.039
    [55] W. R. Schneider, W. Wyess, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), 134–144. http://doi.org/10.1063/1.528578 doi: 10.1063/1.528578
    [56] V. E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, Berlin, Heidelberg: Springer, 2010. http://doi.org/10.1007/978-3-642-14003-7
    [57] X. Yang, H. M. Srivastava, J. A. M. Tenreiro, A new fractional derivative without singular kernel: application to the modelling of the steady heat flow, Thermal Sci., 20 (2016), 753–756. http://doi.org/10.2298/TSCI151224222Y doi: 10.2298/TSCI151224222Y
    [58] M. Yavuz, N. Özdemir, European vanilla option pricing model of fractional order without singular kernel, Fractal Fract., 2 (2018), 3. http://doi.org/10.3390/fractalfract2010003 doi: 10.3390/fractalfract2010003
    [59] M. Yavuz, N. Özdemir, Comparing the new fractional derivative operators involving exponential and Mittag Leffler kernel, Discrete Contin. Dyn. Syst. S, 13 (2020), 995–1006. http://doi.org/10.3934/dcdss.2020058 doi: 10.3934/dcdss.2020058
    [60] T. A. Yıldız, S. Arshad, D. Baleanu, New observations on optimal cancer treatments for a fractional tumor growth model with and without singular kernel, Chaos Soliton. Fract., 117 (2018), 226–239. http://doi.org/10.1016/j.chaos.2018.10.029 doi: 10.1016/j.chaos.2018.10.029
    [61] M. Yavuz, E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Physica A, 525 (2019), 373–393. http://doi.org/10.1016/j.physa.2019.03.069 doi: 10.1016/j.physa.2019.03.069
    [62] M. Yavuz, N. Ozdemir, Analysis of an epidemic spreading model with exponential decay law, Mathematical Sciences & Applications E-Notes, 8 (2020), 142–154. http://doi.org/10.36753/mathenot.691638 doi: 10.36753/mathenot.691638
    [63] H. Yépez-Martínez, J. F. Gómez-Aguilar, A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM), J. Comput. Appl. Math., 346 (2019), 247–260. http://doi.org/10.1016/j.cam.2018.07.023 doi: 10.1016/j.cam.2018.07.023
    [64] F. Youbi, S. Momani, S. Hasan, M. Al-Smadi, Effective numerical technique for nonlinear Caputo-Fabrizio systems of fractional Volterra integro-differential equations in Hilbert space, Alex. Eng. J., 61 (2022), 1778–1786. http://doi.org/10.1016/j.aej.2021.06.086 doi: 10.1016/j.aej.2021.06.086
    [65] T. Zhang, Y. Li, Exponential Euler scheme of multi-delay Caputo–Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. http://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
    [66] D. Zhao, M. Luo, Representations of acting processes and memory effects: general fractional derivative and its application to theory of heat conduction with finite wave speeds, Appl. Math. Comput., 346 (2019), 531–544. http://doi.org/10.1016/j.amc.2018.10.037 doi: 10.1016/j.amc.2018.10.037
    [67] A. Zappone, E. Jorswieck, Energy efficiency in wireless networks via fractional programming theory found, Trends Commun. Inf. Theory, 11 (2014), 185–396. http://doi.org/10.1561/0100000088 doi: 10.1561/0100000088
  • Reader Comments
  • © 2023 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2797) PDF downloads(341) Cited by(10)

Figures and Tables

Figures(9)  /  Tables(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog