Processing math: 79%
Research article Special Issues

Entropy Analysis for boundary layer Micropolar fluid flow

  • Received: 28 September 2019 Accepted: 09 February 2020 Published: 21 February 2020
  • MSC : 74A15, 76D10, 76S05

  • This paper reports entropy generation analysis of radiative micropolar fluid flow in porous medium. The mathematical model depicting convective boundary layer flow due to a vertically moving infinite plate bounding the porous medium on one side is solved numerically. An implicit finite difference method together with Gauss elimination method is used. The numerically computed velocity and temperature fields are employed to analyze entropy. The plots for entropy generation number for various sets of parameters are drawn. It is found that entropy generation number Ns decreases with increasing values of heat sink parameter Q and radiation parameter N whereas it increases with increasing values of Grashoff number Gr, Brinkman number Br. The Bejan number shows pronounced variations for the parameters entering into the problem.

    Citation: Paresh Vyas, Rajesh Kumar Kasana, Sahanawaz Khan. Entropy Analysis for boundary layer Micropolar fluid flow[J]. AIMS Mathematics, 2020, 5(3): 2009-2026. doi: 10.3934/math.2020133

    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
  • This paper reports entropy generation analysis of radiative micropolar fluid flow in porous medium. The mathematical model depicting convective boundary layer flow due to a vertically moving infinite plate bounding the porous medium on one side is solved numerically. An implicit finite difference method together with Gauss elimination method is used. The numerically computed velocity and temperature fields are employed to analyze entropy. The plots for entropy generation number for various sets of parameters are drawn. It is found that entropy generation number Ns decreases with increasing values of heat sink parameter Q and radiation parameter N whereas it increases with increasing values of Grashoff number Gr, Brinkman number Br. The Bejan number shows pronounced variations for the parameters entering into the problem.


    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 , (w_{n}, z_{n})\rightarrow (w^{\ast}, z^{\ast}) , and A(w^{\ast}, z^{\ast}) = (w^{\ast}, z^{\ast}) .

    Finally, we prove that (u^{\ast}, v^{\ast}) and (w^{\ast}, z^{\ast}) are the minimal and maximal positive solutions of (1.1). Assume that (\varsigma(x), \mu(x)) is any positive solution of (1.1). Then, A(\varsigma(x), \mu(x)) = (\varsigma(x), \mu(x)) , and

    (w_{0}(x), z_{0}(x)) = (0, 0)\leq(\varsigma(x), \mu(x))\leq((1+x^{-\beta(1+\gamma)})\Upsilon_{1}, (1+x^{-\beta(1+\gamma)})\Upsilon_{2}) = (u_{0}(x), v_{0}(x)).

    Therefore,

    (w_{1}(x), z_{1}(x)) = (A_{1}(w_{0}, z_{0})(x), A_{2}(w_{0}, z_{0})(x))\leq(\varsigma(x), \mu(x))\leq(A_{1}(u_{0}, v_{0})(x), A_{2} (u_{0}, v_{0})(x)) = (u_{1}(x), v_{1}(x)).

    That is, (w_{1}(x), z_{1}(x))\leq(\varsigma(x), \mu(x))\leq(u_{n}(x), v_{n}(x)) . So, (5.3) holds. By (H_{1}) , (0, 0) is not a solution of (1.1). From (5.1), (w^{\ast}, z^{\ast}) and (u^{\ast}, v^{\ast}) 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:

    \begin{align} \begin{cases} D_{1}^{-\frac{3}{2}, \frac{5}{3}}u(x)+x^{\frac{3}{2}}(\frac{u}{1+x^{\frac{1}{2}}})^{2}e^{-x}+x^{\frac{3}{2}} (\frac{v}{1+x^{\frac{1}{2}}})^{2}e^{-x} = 0, t\in (0, +\infty), \\ D_{1}^{-\frac{3}{2}, \frac{3}{2}}v(x)+x^{\frac{5}{2}}e^{-2x^{2}}(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u} {1+x^{\frac{1}{2}}})^{2}) +x^{\frac{5}{2}}e^{-2x^{2}}(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}), 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} (6.1)

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

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

    \mid F_{1}(x, u, v)\mid\leq\varphi_{1}(x)\omega_{1}(\mid u\mid)+\psi_{1}(t)\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}}[u^{2}\ln(u^{2}+1)+ v^{2}\ln(v^{2}+1)] , we choose \widetilde{\omega_{1}}(u) = u^{2}\ln(u^{2}+1)\in C((0, +\infty), (0, +\infty)) , \widetilde{\omega_{2}}(v) = v^{2}\ln(v^{2}+1)\in C((0, +\infty), (0, +\infty)) , and \varphi_{2}(x) = \psi_{2}(x) = xe^{-2x^{2}}\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).

    So, the condition (H_{2}) holds. Obviously, F, G:(0, +\infty)\times(0, +\infty)\times(0, +\infty)\rightarrow (0, +\infty) are continuous.

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

    where a_{1}(x) = e^{-x}, a_{2}(x) = xe^{-2x^{2}} , F_{1}(x, u, v) = (\frac{u}{1+x^{\frac{1}{2}}})^{2}+(\frac{v}{1+x^{\frac{1}{2}}})^{2} , G_{1}(t, u, v) = (\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}) +(\frac{u}{1+x^{\frac{1}{2}}})^{2}\ln(1+(\frac{u}{1+x^{\frac{1}{2}}})^{2}) . 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. Hence, the condition (H_{4}) holds. Finally,

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

    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] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
    [2] A. C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl., 38 (1966), 480-496.
    [3] M. M. Khonsari, On the self-excited whirl orbits of a journal in a sleeve, bearing lubricated with micropolar fluids, Acta Mech., 81 (1990), 235-244.
    [4] M. M. Khonsari, D. Brewe, On the performance of finite journal bearing lubricated with micropolar fluids, STLE Tribol. Transm., 32 (1989), 155-160. doi: 10.1080/10402008908981874
    [5] B. Hadimoto, T. Tokioka, Two-dimensional shear flows of linear micropolar fluids, Int. J. Eng. Sci., 7 (1969), 515-522. doi: 10.1016/0020-7225(69)90036-6
    [6] F. Lockwood, M. Benchaita, S. Friberg, Study of polyotropic liquid crystals in viscometric flow and elasto hydrodynamic contact, ASLE Tribol. Transm., 30 (1987), 539-548.
    [7] J. D. Lee, A. C. Eringen, Boundary effects of orientation of nematic liquid crystals, J. Chem. Phys., 55 (1971), 509-512.
    [8] V. Kolpashchikov, N. P. Migun, P. P. Prokhorenko, Experimental determinations of material micropolar coefficients, Int. J. Eng. Sci., 21 (1983), 405-411. doi: 10.1016/0020-7225(83)90123-4
    [9] T. Ariman, M. A. Turk, N. D. Sylvester, Micro continuum fluid mechanics: A review, Int. J. Eng. Sci., 11 (1973), 905-930. doi: 10.1016/0020-7225(73)90038-4
    [10] T. Ariman, M. A. Turk, N. D. Sylvester, Application of micro continuum fluid mechanics, Int. J. Eng. Sci., 12 (1974), 273-293. doi: 10.1016/0020-7225(74)90059-7
    [11] G. Ahmedi, Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate, Int. J. Eng. Sci., 14 (1976), 639-646. doi: 10.1016/0020-7225(76)90006-9
    [12] S. K. Jena, M. N. Mathur, Free convection in the laminar boundary layer flow of thermomicropolar fluid past a non-isothermal vertical flat plate with suction/injection, Acta Mech., 42 (1982), 227-238. doi: 10.1007/BF01177194
    [13] E. M. Abo-Eldahab, M. A. El-Aziz, Flow and heat transfer in a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium with uniform free stream, Appl. Math. Comput., 162 (2005), 881-899.
    [14] M. M. Rashidi, N. Kavyani, S. Abelman, Investigation of entropy generation in MHD and slip flowover a rotating porous disk with variable properties, Int. J. Heat Mass Tran., 70 (2014), 892-917. doi: 10.1016/j.ijheatmasstransfer.2013.11.058
    [15] D. Gupta, L. Kumar, O. A. Bég, et al. Finite element simulation of mixed convection flow of micropolar fluid over a shrinking sheet with thermal radiation, P. I. Mech. Eng. E, 228 (2014), 61-72.
    [16] N. Ali, A. Zaman, O. A. Bég, Numerical simulation of unsteady micropolar hemodynamics in a tapered catheterized artery with a combination of stenosis and aneurysm, Med. Biol. Eng. Comput., 54 (2015), 1423-1436.
    [17] M. J. Uddin, M. N. Kabir, Y. M. Alginahi, Lie group analysis and numerical solution of magnetohydrodynamic free convective slip flow of micropolar fluid over a moving plate with heat transfer, Comput. Math. Appl., 70 (2015), 846-856. doi: 10.1016/j.camwa.2015.06.002
    [18] I. Dražić, N. Črnjarić-Žic, L. Simčić, A shear flow problem for compressible viscous micropolar fluid: Derivation of the model and numerical solution, Math. Comput. Simulat., 162 (2019), 249-267. doi: 10.1016/j.matcom.2019.01.013
    [19] A. A. Farooq, D. Tripathib, T. Elnaqeeb, On the propulsion of micropolar fluid inside a channel due to ciliary induced metachronal wave, Appl. Math. Comput., 347 (2019), 225-235.
    [20] H. H. Sherief, M. S. Faltas, S. El-Sapa, Interaction between two rigid spheres moving in a micropolar fluid with slip surfaces, J. Mol. Liq., 290 (2019), 111165.
    [21] M. S. Uddin, K. Bhattacharyya, S. Shafie, Micropolar fluid flow and heat transfer over an exponentially permeable shrinking sheet, Popul. Power Res., 5 (2016), 310-317.
    [22] R. D. Cess, The Interaction of thermal radiation with free convection heat transfer, Int. J. Heat Mass Tran., 9 (1966), 1269-1277. doi: 10.1016/0017-9310(66)90119-0
    [23] A. A. Hayday, D. A. Bowlus, R. A. McGraw, Free convection from a vertical flat plate with step discontinuities in surface temperature, ASME J. Heat Tran., 89 (1967), 244-250. doi: 10.1115/1.3614371
    [24] T. T. Kao, Laminar free convective heat transfer response along a vertical plat plate with step jump in surface temperature, Heat Mass Transfer, 2 (1975), 419-428.
    [25] P. Cheng, W. J. Minkowycz, Flow about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res., 82 (1977), 2040-2044. doi: 10.1029/JB082i014p02040
    [26] A. Raptis, N. Kafousias, Heat transfer in flow through a porous medium bounded by an infinite vertical plate under the action of a magnetic field, Energy Res., 6 (1982), 241-245. doi: 10.1002/er.4440060305
    [27] M. V. A. Bianchi, R. Viskanta, Momentum and heat transfer on a continuous flat surface moving in a parallel counterflow free stream, Warme-und Stoffubertrangung, 29 (1993), 89-94. doi: 10.1007/BF01560077
    [28] H. S. Thakhar, R. S. R. Gorla, V. M. Soundalgekar, Radiation effect on MHD free convection flow of a radiating gas past a semi-infinite vertical plate, Int. J. Numer. Method Heat, 6 (1996), 77-83.
    [29] A. J. Chamkha, Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption, Int. J. Eng. Sci., 24 (2004), 217-230.
    [30] M. M. Abdelkhalek, Heat and Mass transfer in MHD free convection from a moving permeable vertical surface by a perturbation technique, Commun. Nonlinear Sci., 14 (2009), 2091-2102. doi: 10.1016/j.cnsns.2008.06.001
    [31] O. D. Makinde, Similarity solution for natural convection from a moving vertical plate with internal heat generation and a convective boundary condition, Therm. Sci., 15 (2011), 5137-5143.
    [32] D. Srinivasacharya, O. Surender, Non-similar solution for natural convective boundary layer flow of ananofluid past a vertical plate embedded in a doubly stratified porous medium, Int. J. Heat Mass Tran., 71 (2014), 431-438. doi: 10.1016/j.ijheatmasstransfer.2013.12.001
    [33] A. Khalid, I. Khan, S. Shafie, Heat transfer in ferrofluid with cylindrical shape nanoparticles past a vertical plate with ramped wall temperature embedded in a porous medium, J. Mol. Liq., 221 (2016), 1175-1183. doi: 10.1016/j.molliq.2016.06.105
    [34] S. R. Mishra, I. Khanb, Q. M. Al-mdallalc, et al. Free convective micropolar fluid flow and heat transfer over a shrinking sheet with heat source, Case Stud. Therm. Eng., 11 (2018), 113-119. doi: 10.1016/j.csite.2018.01.005
    [35] H. Chen, J. Ma, H. Liu, Least square spectral collocation method for nonlinear heat transfer in moving porous plate with convective and radiative boundary conditions, Int. J. Therm. Sci., 132 (2018), 335-343. doi: 10.1016/j.ijthermalsci.2018.06.020
    [36] A. Bejan, A study of entropy generation in fundamental convective heat transfer, J. Heat Trans., 101 (1979), 718-725. doi: 10.1115/1.3451063
    [37] A. Bejan, Second law analysis in heat transfer, Energy, 5 (1980), 721-732.
    [38] P. Vyas, S. Soni, Entropy analysis for MHD casson fluid flow in a channel subjected to weakly temperature dependent convection coefficient and hyderodynamic slip, J. Rajasthan Acad. Phys. Sci., 15 (2016), 1-18.
    [39] P. Vyas, N. Srivastava, Entropy analysis of generalized MHD Couette flow inside a composite duct with asymmetric convective cooling, Arab. J. Sci. Eng., 40 (2015), 603-614. doi: 10.1007/s13369-014-1562-0
    [40] N. Srivastava, P. Vyas, S. Soni, Entropy generation analysis for oscillatory flow in a vertical channel filled with Porous Medium, IEEE International Conference on Recent Advances and Innovations in Engineering (ICRAIE-2016), December 23-25, Jaipur, India.
    [41] M. F. Modest, Heat Transfer, 2 Eds., Academic press, 2003.
    [42] D. Srinivasacharya, K. H. Bindu, Entropy generation due to micropolar fluid flow between concentric cylinders with slip and convective boundary conditions, Ain Shams Eng. J., 9 (2018), 245-255. doi: 10.1016/j.asej.2015.10.016
    [43] D. Srinivasacharya, K. H. Bindu, Entropy generation of micropolar fluid flow in an inclined porous pipe with convective boundary conditions, Sadhna, 42 (2017), 729-740. doi: 10.1007/s12046-017-0639-3
    [44] S. K. Asha, C. K. Deepa, Entropy generation for peristaltic blood flow of a magneto-micropolar fluid with thermal radiation in a tapered asymmetric channel, Results Eng., 3 (2019), 100024.
    [45] A. Z. Sahin, Second law analysis of laminar viscous flow through a duct subjected to constant wall temperature, J. Heat Trans., 120 (1998), 76-83. doi: 10.1115/1.2830068
    [46] J. V. R. Murthy, J. Srinivas, Second law analysis for Poiseuille flow of immiscible micropolar fluids in a channel, Int. J. Heat Mass Tran., 65 (2013), 254-264. doi: 10.1016/j.ijheatmasstransfer.2013.05.048
    [47] C. K. Chen, Y. T. Yang, K. H. Chang, The effect of thermal radiation on entropy generation due to micro-polar fluid flow along a wavy surface, Entropy, 13 (2011),1595-1610. doi: 10.3390/e13091595
    [48] A. Shahsavar, P. T. Sardari, D. Toghraie, Free convection heat transfer and entropy generation analysis of water-Fe3O4/CNT hybridnanofluid in a concentric annulus, Int. J. Numer. Meth. Heat Fluid Flow, 424 (2018), 0961-5539.
    [49] E. Manay, E. F. Akyürek, B. Sahin, Entropy generation of nanofluid flow in a microchannel heat sink, Results Phys., 9 (2018), 615-624. doi: 10.1016/j.rinp.2018.03.013
    [50] P. Barnoon, D. Toghrai, R. B. Dehkordi, et al. MHD mixed convection and entropy generation in a lid-driven cavity with rotating cylinders filled by a nanofluid using two phase mixture model, J. Magn. Magn. Mater., 483 (2019), 224-248. doi: 10.1016/j.jmmm.2019.03.108
    [51] P. Barnoon, D. Toghraie, F. Eslami, et al. Entropy generation analysis of different nanofluid flows in the space between two concentric horizontal pipes in the presence of magnetic field: Single-phase and two-phase approaches, Comput. Math. Appl., 73 (2019), 662-692.
    [52] A. A. A. A. Abdullah, O. A. Akbari, A. Heydari, et al. The numerical modeling of water/FMWCNT nanofluid flow and heat transfer in a backward-facing contracting channel, Physica B, 537 (2018), 176-183. doi: 10.1016/j.physb.2018.02.022
    [53] S. M. Seyyedi, A. S. Hashemi-Tilehnoee, M. Waqas, et al. Entropy generation and economic analyses in a nanofluid filled L-shaped enclosure subjected to an oriented magnetic field, Appl. Therm. Eng., 168 (2019), 114789.
    [54] M. Maskaniyan, M. Nazari, S. Rashidi, et al. Natural convection and entropy generation analysis inside a channel with a porous plate mounted as a cooling system, Therm. Sci. Eng. Prog., 6 (2018),186-193. doi: 10.1016/j.tsep.2018.04.003
    [55] P. Gholamalipour, M. Siavashi, M. H. Doranehgard, Eccentricity effects of heat source inside a porous annulus on the natural convection heat transfer and entropy generation of Cu-water nanofluid, Int. Commun. Heat Mass Tran., 109 (2019), 104367.
    [56] P. Vyas, S. Khan, Entropy analysis for MHD dissipative Casson fluid flow in porous medium due to stretching cylinder, Acta Tech., 61 (2016), 299-315.
    [57] P. Vyas, N. Srivastava, Entropy analysis for magnetohyrodynamic fluid flow in porous medium due to a non-isothermal stretching sheet, J. Rajasthan Acad. Phys. Sci.,14 (2015), 323-336.
    [58] Y. J. Kim, Heat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porous medium, Transport in Porous Med., 56 (2004), 17-37. doi: 10.1023/B:TIPM.0000018420.72016.9d
    [59] A. A. Raptis, V. M. Soundalgekar, MHD flow past a steadily moving infinite vertical porous plate with constant heat flux, Nucl. Eng. Des., 72 (1982), 373-379. doi: 10.1016/0029-5493(82)90050-4
    [60] P. Vyas, A. Rai, K. S. Shekhawat, Dissipative heat and mass transfer in porous medium due to continuously moving plate, Appl. Math. Sci., 6 (2012), 4319-4330.
    [61] A. A. Raptis, Flow of a micropolar fluid past a continuously moving plate by the presence of rotation, Int. J. Heat Mass Tran., 41 (1998), 2865-2866. doi: 10.1016/S0017-9310(98)00006-4
    [62] F. Atlan, M. E. A. El-Mikkawy, A new symbolic algorithm for solving general opposite-bordered tridiagonal linear systems, Am. J. Comput. Math., 5 (2015), 258-266. doi: 10.4236/ajcm.2015.53023
    [63] J. Jia, S. Li, New algorithms for numerically solving a class of bordered tridiagonal systems of linear equations, Comput. Math. Appl., 78 (2019), 144-151.< doi: 10.1016/j.camwa.2019.02.028
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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(4706) PDF downloads(555) Cited by(14)

Figures and Tables

Figures(8)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog