Research article Topical Sections

Effect of heterologous protein expression on Escherichia coli biofilm formation and biocide susceptibility

  • Received: 09 September 2016 Accepted: 10 November 2016 Published: 17 November 2016
  • Escherichia coli is recognized as an excellent model for biofilm studies and one of the favourite hosts for recombinant protein expression. This work assesses the influence of heterologous protein production on biofilm formation and susceptibility to chemical treatment. Biofilm formation by two E. coli strains was compared using a flow cell system. One strain contained the commercial pET28A plasmid and the other a plasmid derivative with the same backbone but containing the enhanced green fluorescent protein (eGFP) gene. The susceptibility of biofilms to the biocide benzyldimethyldodecylammonium chloride (BDMDAC) was also assessed. It was found that the eGFP-expressing strain formed thicker biofilms with a higher cell density than the non-producing strain. Biofilms of both strains were neither completely inactivated nor removed by biocide treatment. Similar inactivation efficiencies were obtained, although biofilm cohesion was higher for the non-producing strain.

    Citation: Luciana C. Gomes, Filipe J. Mergulhão. Effect of heterologous protein expression on Escherichia coli biofilm formation and biocide susceptibility[J]. AIMS Microbiology, 2016, 2(4): 434-446. doi: 10.3934/microbiol.2016.4.434

    Related Papers:

    [1] Nadiyah Hussain Alharthi, Abdon Atangana, Badr S. Alkahtani . Numerical analysis of some partial differential equations with fractal-fractional derivative. AIMS Mathematics, 2023, 8(1): 2240-2256. doi: 10.3934/math.2023116
    [2] Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park . A study on the fractal-fractional tobacco smoking model. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767
    [3] Abdon Atangana, Seda İğret Araz . Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations. AIMS Mathematics, 2024, 9(3): 5763-5793. doi: 10.3934/math.2024280
    [4] Manal Alqhtani, Khaled M. Saad . Numerical solutions of space-fractional diffusion equations via the exponential decay kernel. AIMS Mathematics, 2022, 7(4): 6535-6549. doi: 10.3934/math.2022364
    [5] Abdon Atangana, Ali Akgül . Analysis of a derivative with two variable orders. AIMS Mathematics, 2022, 7(5): 7274-7293. doi: 10.3934/math.2022406
    [6] Emile Franc Doungmo Goufo, Abdon Atangana . On three dimensional fractal dynamics with fractional inputs and applications. AIMS Mathematics, 2022, 7(2): 1982-2000. doi: 10.3934/math.2022114
    [7] Rahat Zarin, Amir Khan, Pushpendra Kumar, Usa Wannasingha Humphries . Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Mathematics, 2022, 7(10): 18897-18924. doi: 10.3934/math.20221041
    [8] Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046
    [9] Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
    [10] Muhammad Aslam, Muhammad Farman, Hijaz Ahmad, Tuan Nguyen Gia, Aqeel Ahmad, Sameh Askar . Fractal fractional derivative on chemistry kinetics hires problem. AIMS Mathematics, 2022, 7(1): 1155-1184. doi: 10.3934/math.2022068
  • Escherichia coli is recognized as an excellent model for biofilm studies and one of the favourite hosts for recombinant protein expression. This work assesses the influence of heterologous protein production on biofilm formation and susceptibility to chemical treatment. Biofilm formation by two E. coli strains was compared using a flow cell system. One strain contained the commercial pET28A plasmid and the other a plasmid derivative with the same backbone but containing the enhanced green fluorescent protein (eGFP) gene. The susceptibility of biofilms to the biocide benzyldimethyldodecylammonium chloride (BDMDAC) was also assessed. It was found that the eGFP-expressing strain formed thicker biofilms with a higher cell density than the non-producing strain. Biofilms of both strains were neither completely inactivated nor removed by biocide treatment. Similar inactivation efficiencies were obtained, although biofilm cohesion was higher for the non-producing strain.


    Fractional calculus is a generalization of classical calculus and many researchers have paid attention to this science as they encounter many of these issues in the real world. Most of these issues do not have analytical exact solution. Which made many researchers interest and search in numerical and approximate methods to obtain solutions using these methods. There are many of these methods, such as the homotopy analysis [1,2,3,4], He's variational iteration method [5,6], Adomians decomposition method [7,8,9], Fourier spectral methods [10], finite difference schemes [11], collocation methods [12,13,14]. To find out more about the fractal calculus, refer to the following references [15,16]. More recently, a new concept was introduced for the fractional operator, as this operator has two orders, the first representing the fractional order, and the second representing the fractal dimension. In our work we aim to applied the idea of fractal-fractional derivative of orders β,k to a reaction-diffusion equation with q-th nonlinear. To this end [17], we replace the derivative with respect to t by the fractal-fractional derivatives power (FFP) law, the fractal-fractional exponential(FFE) law and the fractal-fractional Mittag-Leffler (FFM) law kernels which corresponds to the [18], Caputo-Fabrizio (CF) [19] and the Atangana-Baleanu (AB) [20] fractional derivatives, respectively. This topic has attracted many researchers and has been applied to research related to the real world, such as [21,22,23,24,25,26]. Some recent developments in the area of numerical techniques can be found in [27,28,29,30,31].

    Merkin and Needham [32] considered the reaction-diffusion travelling waves that can develop in a coupled system involving simple isothermal autocatalysis kinetics. They assumed that reactions took place in two separate and parallel regions, with, in I, the reaction being given by quadratic autocatalysis

    F+G2G(ratek1fg), (1.1)

    together with a linear decay step

    GH(ratek2g) (1.2)

    where f and g are the concentrations of reactant F and autocatalyst H, the ki(i=1,2) are the rate constants and H is some inert product of reaction. The reaction in region II was the quadratic autocatalytic step (1.1) only. The two regions were assumed to be coupled via a linear diffusive interchange of the autocatalytic species G. We shall consider a similar system as I, but with cubic autocatalysis

    F+2G3G(ratek3fg2) (1.3)

    together with a linear decay step

    GH(ratek4g). (1.4)

    For q-th autocatalytic, we have

    F+qG(q+1)G(ratek3fgq),1q2, (1.5)

    together with a linear decay step

    GH(rate k4g). (1.6)

    This yields to the following system

    η1t=2η1ξ2+ν(η2η1)η1ζq1, (1.7)
    ζ1t=2ζ1ξ2κζ1+η1ζq1, (1.8)
    η2t=2η2ξ2+ν(η1η2)η2ζq2, (1.9)
    ζ2t=2ζ2ξ2+η2ζq2 (1.10)

    where ν represents the couple between (I) and (II) and κ represents the strength of the auto-catalyst decay. For more details see [32]. Omitting the diffusion terms in the system (1.7)-(1.10), one has the following ordinary differential equations

    η1t=ν(η2η1)η1ζq1, (1.11)
    ζ1t=κζ1+η1ζq1, (1.12)
    η2t=ν(η1η2)η2ζq2, (1.13)
    ζ2t=η2ζq2. (1.14)

    Now we provide some basic definitions that be needed in this work. As for the theorems and proofs related to the three fractal-fractional operators, they are found in details in [17]. Thus we suffice in this work by constructing the algorithms and making the numerical simulations of the set of Eqs (1.7)-(1.10) with the three fractal-fractional operators.

    Definition 1. If η(t) is continuous and fractal differentiable on (a,b) of order k, then the fractal-fractional derivative of η(t) of order β in Riemann Liouville sense with the power law is given by [17]:

    {FFP}0Dβ,ktη(t)=1Γ(1β)ddtkt0(tτ)βη(τ)dτ,(0<β,k1), (1.15)

    and the fractal-fractional integral of η(t) is given by

    FFP0Iβ,ktη(t)=kΓ(β)t0τk1(tτ)β1η(τ)dτ. (1.16)

    Definition 2. If η(t) is continuous in the (a,b) and fractal differentiable on (a,b) with order k, then the fractal-fractional derivative of η(t) of order β in Riemann Liouville sense with the exponential decay kernel is given by [17]:

    FFE0Dβ,ktη(t)=M(β)1βddtkt0eβ1β(tτ)η(τ)dτ,(0<β,k1), (1.17)

    and the fractal-fractional integral of η(t) is given by

    FFE0Iβ,ktη(t)=(1β)ktk1M(β)η(t)+βkM(β)t0τk1η(τ)dτ (1.18)

    where M(β) is the normalization function such that M(0)=M(1)=1.

    Definition 3. If η(t) is continuous in the (a,b) and fractal differentiable on (a,b) with order k, then the fractal-fractional derivative of η(t) of order β in Riemann Liouville sense with the Mittag-Leffler type kernel is given by [17]:

    FFE0Dβ,ktη(t)=A(β)1βddtkt0Eβ(β1β(tτ))η(τ)dτ,(0<β,k1), (1.19)

    and the fractal-fractional integral of η(t) is given by

    FFE0Iβ,ktη(t)=(1β)ktk1A(β)η(t)+βkA(β)Γ(β)t0τk1(tτ)β1η(τ)dτ, (1.20)
    dη(t)dtk=limτtη(τ)η(t)τktk (1.21)

    where where A(β)=1β+βΓ(β) is a normalization function such that A(0)=A(1)=1.

    Our contribution to this paper is to construct the successive approximations and evaluate the numerical solutions of the FFRDE. These successive approximations allow us to study the behavior of numerical solutions based on power, exponential, and the Mittag-Leffler kernels. Also we can study the behavior of approximate solutions in the case of nonlinearity of the FFRDE in general. To our best knowledge, this is the first study of the FFRDE using fractal-fractional with these kernels. The importance of these results lies in the fact that they highlight the possibility of using these results for the benefit of chemical and physical researchers, by trying to link the numerical results of these mathematical models with the laboratory results. These results also contribute to the reliance on numerical results in the case of many models related to the real world, which often cannot find an analytical solution. The structure of this paper is summarized as follows: In sections, two, three and four, the FFRDE is presented with the three kernels that proposed in this work and construct the successive approximations. In section Five, numerical solutions for the FFRDE are discussed with a study of their behavior. Section Six the conclusion is presented.

    The new model is obtained by replacing the ordinary derivative with the the fractal-fractional derivative the power law kernel as [17]

    FFP0Dβtη1(t)=ν(η2(t)η1(t))η1(t)ζq1(t), (2.1)
    FFP0Dβtζ1(t)=κζ1(t)+η1(t)ζq1(t), (2.2)
    FFP0Dβtη2(t)=ν(η1(t)η2(t))η2(t)ζq2(t), (2.3)
    FFP0Dβtζ2(t)=η2(t)ζq2(t). (2.4)

    By following the procedure in [17], we can obtain the following successive approximations:

    η1(t)η1(0)=kΓ(β)t0τk1(tτ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (2.5)
    ζ1(t)ζ2(0)=kΓ(β)t0τk1(tτ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (2.6)
    η2(t)η3(0)=kΓ(β)t0τk1(tτ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (2.7)
    ζ2(t)ζ2(0)=kΓ(β)t0τk1(tτ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ (2.8)

    where

    φ1(η1,ζ1,η2,ζ2,τ)=(ν(η2(τ)η1(τ))η1(τ)ζq1(τ)), (2.9)
    φ2(η1,ζ1,η2,ζ2,τ)=(κζ1(τ)+η1(τ)ζq1(τ)), (2.10)
    φ3(η1,ζ1,η2,ζ2,τ)=(ν(η1(τ)η2(τ))η2(τ)ζq2(τ)), (2.11)
    φ4(η1,ζ1,η2,ζ2,τ)=η2(τ)ζq2(τ). (2.12)

    Equation (2.5)-(2.8) can be reformulated as

    η1(t)η1(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (2.13)
    ζ1(t)ζ1(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ2(η1(τ),ζ1(τ),η2(τ),ζ2(τ),τ)dτ, (2.14)
    η2(t)η2(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (2.15)
    ζ2(t)ζ2(0)=kΓ(β)nm=0tm+1tmτk1(tn+1τ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ. (2.16)

    Using the two-step Lagrange polynomial interpolation, we obtain

    η1(t)η1(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q1,m(τ)dτ, (2.17)
    ζ1(t)ζ1(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q2,m(τ)dτ, (2.18)
    η2(t)η2(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q3,m(τ)dτ, (2.19)
    ζ2(t)ζ2(0)=kΓ(β)nm=0tm+1tm(tn+1τ)β1Q4,m(τ)dτ, (2.20)

    where,

    Q1,m(τ)=τtm1tmtm1tk1mφ1(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ1(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1), (2.21)
    Q2,m(τ)=τtm1tmtm1tk1mφ2(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ2(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1), (2.22)
    Q3,m(τ)=τtm1tmtm1tk1mφ3(η1(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ3(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1), (2.23)
    Q4,m(τ)=τtm1tmtm1tk1mφ4(η4(τm),ζ1(τm),η2(τm),ζ2(τm),τm)τtmtmtm1×tk1m1φ4(η1(τm1),ζ1(τm1),η2(τm1),ζ2(τm1),τm1). (2.24)

    These integrals are evaluated directly and the numerical solutions of (2.1)-(2.4) involving the FFP derivative are given by

    η1(tn+1)=η1(0)+khβΓ(β+2)nm=0tk1mφ1(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ1(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.25)
    ζ1(tn+1)=ζ1(0)+khβΓ(β+2)nm=0tk1mφ2(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ2(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.26)
    η2(tn+1)=η2(0)+khβΓ(β+2)nm=0tk1mφ3(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ4(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.27)
    ζ2(tn+1)=ζ2(0)+khβΓ(β+2)nm=0tk1mφ4(η1(tm),ζ1(tm),η2(tm),ζ2(tm),tm)Ξ1(n,m)tk1m1φ4(η1(τm1),ζ1(tm1),η2(tm1),ζ2(tm1),tm1)Ξ2(n,m)), (2.28)
    Ξ1(n,m)=((n+1m)β(nm+2+β)(nm)β×(nm+2+2β)), (2.29)
    Ξ2(n,m)=((n+1m)β+1(nm)β(nm+1+β)). (2.30)

    Considering the FFE derivative, we have from [17]

    FFE0Dβtη1(t)=ν(η2(t)η1(t))η1(t)ζq1(t), (3.1)
    FFE0Dβtζ1(t)=κζ1(t)+η1(t)ζq1(t), (3.2)
    FFE0Dβtη2(t)=ν(η1(t)η2(t))η2(t)ζq2(t), (3.3)
    FFE0Dβtζ2(t)=η2(t)ζq2(t). (3.4)

    For the successive approximations of the system (3.1)-(3.4), we follow the same procedures as in [17], we obtain

    η1(t)η1(0)=ktk1(1β)M(β)φ1(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ1(η1,ζ1,η2,ζ2,τ)dτ, (3.5)
    ζ1(t)ζ1(0)=ktk1(1β)M(β)φ2(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ2(η1,ζ1,η2,ζ2,τ)dτ, (3.6)
    η2(t)η2(0)=ktk1(1β)M(β)φ3(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ3(η1,ζ1,η2,ζ2,τ)dτ, (3.7)
    ζ2(t)ζ2(0)=ktk1(1β)M(β)φ4(η1,ζ1,η2,ζ2,t)+βM(β)t0kτk1φ4(η1,ζ1,η2,ζ2,τ)dτ. (3.8)

    Using t=tn+1 the following is established

    η1(tn+1)η1(0)=ktk1(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ1(η1,ζ1,η2,ζ2,τ)dτ, (3.9)
    ζ1(tn+1)ζ1(0)=ktk1(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ2(η1,ζ1,η2,ζ2,τ)dτ, (3.10)
    η2(tn+1)η2(0)=ktk1(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ3(η1,ζ1,η2,ζ2,τ)dτ, (3.11)
    ζ2(tn+1)ζ2(0)=ktk1(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn)+βM(β)tn+10kτk1φ4(η1,ζ1,η2,ζ2,τ)dτ. (3.12)

    Further, we have the following:

    η1(tn+1)η1(tn)=ktk1n(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ1(η1,ζ1,η2,ζ2,τ)dτ, (3.13)
    ζ1(tn+1)ζ1(tn)=ktk1n(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ2(η1,ζ1,η2,ζ2,τ)dτ, (3.14)
    η2(tn+1)η2(tn)=ktk1n(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ3(η1,ζ1,η2,ζ2,τ)dτ, (3.15)
    ζ2(tn+1)ζ2(tn)=ktk1n(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn1)+βM(β)tn+1tnkτk1φ4(η1,ζ1,η2,ζ2,τ)dτ. (3.16)

    It follows from the Lagrange polynomial interpolation and integrating the following expressions:

    η1(tn+1)η1(tn)=ktk1n(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ1(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ1(η1,ζ1,η2,ζ2,tn)tk1n1φ1(η1,ζ1,η2,ζ2,tn1), (3.17)
    ζ1(tn+1)ζ1(tn)=ktk1n(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ2(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ2(η1,ζ1,η2,ζ2,tn)tk1n1φ2(η1,ζ1,η2,ζ2,tn1), (3.18)
    η2(tn+1)η2(tn)=ktk1n(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ3(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ3(η1,ζ1,η2,ζ2,tn)tk1n1φ3(η1,ζ1,η2,ζ2,tn1), (3.19)
    ζ2(tn+1)ζ2(tn)=ktk1n(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn)ktk1n1(1β)M(β)φ4(η1,ζ1,η2,ζ2,tn1)+khβ2M(β)×(3tk1nφ4(η1,ζ1,η2,ζ2,tn)tk1n1φ4(η1,ζ1,η2,ζ2,tn1). (3.20)

    Finally, it is appropriate to write the successive approximations of the system (3.1)-(3.4) as follows:

    η1(tn+1)η1(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ1(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ1(η1,ζ1,η2,ζ2,tn1), (3.21)
    ζ1(tn+1)ζ1(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ2(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ2(η1,ζ1,η2,ζ2,tn1), (3.22)
    η2(tn+1)η2(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ3(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ3(η1,ζ1,η2,ζ2,tn1), (3.23)
    ζ2(tn+1)ζ2(tn)=ktk1n((1β)M(β)+3hβ2M(β))φ4(η1,ζ1,η2,ζ2,tn)ktk1n1((1β)M(β)+hβ2M(β))φ4(η1,ζ1,η2,ζ2,tn1). (3.24)

    Considering the FFM derivative, we have [18]

    FFM0Dβtη1(t)=ν(η2(t)η1(t))η1(t)ζq1(t), (4.1)
    FFM0Dβtζ1(t)=κζ1(t)+η1(t)ζq1(t), (4.2)
    FFM0Dβtη2(t)=ν(η1(t)η2(t))η2(t)ζq2(t), (4.3)
    FFM0Dβtζ2(t)=η2(t)ζq2(t). (4.4)

    Also, for this system (4.1)-(4.4), we follow the same treatment that was done in [17] to obtain the successive approximate solutions as follows:

    η1(t)η1(0)=ktk1(1β)A(β)φ1(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (4.5)
    ζ1(t)ζ1(0)=ktk1(1β)A(β)φ2(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (4.6)
    η2(t)η2(0)=ktk1(1β)A(β)φ3(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (4.7)
    ζ2(t)ζ2(0)=ktk1(1β)A(β)φ4(η1,ζ1,η2,ζ2,t)+βA(β)Γ(β)t0kτk1(tτ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ. (4.8)

    At tn+1 we obtain the following

    η1(tn+1)η1(0)=ktk1n(1β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (4.9)
    ζ1(tn+1)ζ1(0)=ktk1n(1β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (4.10)
    η2(tn+1)η2(0)=ktk1n(1β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (4.11)
    ζ2(tn+1)ζ2(0)=ktk1n(1β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)tn+10kτk1(tn+1τ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ, (4.12)

    The integrals involving in (4.9)-(4.12) can be approximated as:

    η1(tn+1)η1(0)=ktk1n(1β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ1(η1,ζ1,η2,ζ2,τ)dτ, (4.13)
    ζ1(tn+1)ζ1(0)=ktk1n(1β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ2(η1,ζ1,η2,ζ2,τ)dτ, (4.14)
    η2(tn+1)η2(0)=ktk1n(1β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ3(η1,ζ1,η2,ζ2,τ)dτ, (4.15)
    ζ2(tn+1)ζ2(0)=ktk1n(1β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+βA(β)Γ(β)nm=0tm+1tmkτk1(tn+1τ)β1φ4(η1,ζ1,η2,ζ2,τ)dτ. (4.16)

    The following numerical schemes after approximating the expressions τk1φi(η1,ζ1,η2,ζ2,τ),i=1,2,3,4 in the interval [tm,tm+1] in (4.13)-(4.16) are given by

    η1(tn+1)η1(0)=ktk1n(1β)A(β)φ1(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)nm=0[tk1mφ1(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ1(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)], (4.17)
    ζ1(tn+1)ζ1(0)=ktk1n(1β)A(β)φ2(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)nm=0[tk1mφ2(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ2(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)], (4.18)
    η2(tn+1)η2(0)=ktk1n(1β)A(β)φ3(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(β+2)nm=0[tk1mφ3(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ3(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)], (4.19)
    ζ2(tn+1)ζ2(0)=ktk1n(1β)A(β)φ4(η1(tn),ζ1(tn),η2(tn),ζ2(tn),tn)+khβA(β)Γ(α+2)nm=0[tk1mφ4(η1(tm),ζ1(tm),η2(tm),ζ2(tm),(tm))Ξ1(n,m)tk1m1φ4(η1(tm1),ζ1(tm1),η2(tm1),ζ2(tm1),(tm1))Ξ2(n,m)]. (4.20)

    In this section, we study in detail the effect of the non-linear term in general, as well as the effect of the fractal-fractional order on the numerical solutions that we obtained by using successive approximations in the above sections. First we begin by satisfying the effective of the numerical solutions of the proposed system when β=1 and k=1.

    We compare only for the power kernel with a known numerical method which is the finite differences method. This is because all numerical solutions based on the three fractal-fractional operators that presented in this paper are very close each other when β=1 and k=1. Figure 1 illustrates the comparison between numerical solutions (2.25)-(2.28) and numerical solutions computed by using the finite differences method with k and β. The parameters that used are γ=0.4,κ=0.004,h=0.02. From this figure we note that an excellent agreement. And the accurate is increasing as we take small h. From, Figure 1(a) and 1(c), we can see, that the profiles for η1 and η2 are very similar, but the profiles of ζ1 and ζ2 are more distinct with ζ2>ζ2. For Figure 1(b), the profiles of ζ1 and ζ2 are very close than in Figure 1(a) and 1(c), also for ζ1 and ζ2. Figures 2 and 3 show that the behavior of the approximate solutions based on FFP, FFE and FFM, when the degree of the non-linear term is cubic and for different values of k and β. For the parameters γ and κ, we fixed them in all computations. The remain parameters are the same as in Figure 1. Similarly, in Figures 4 and 5, the approximate solutions are plotted in the case of a non-linear with quadratic degree and for different values of k and β. Finally in Figures 6 and 7, the approximate solutions are shown in the case of non-linear with fractional order and for different values for k and β. For the Figures 2 and 3 which the nonlinear is cubic, all the profiles are distinct. Similarly with Figures 6 and 7 when the nonlinear is quadratic. From Figures 4 and 5, we can see in the case of fraction non-linear, the profiles of η1 and η2 are very close to each other than the profiles of ζ1 and ζ2.

    Figure 1.  Comparison between the numerical solutions (2.25)-(2.28) and numerical based on finite difference methods for β=1,k=1,γ=0.4,κ=0.001,h=0.01.(a)q=2;(b)q=1;(c)q=1.8; (Green solid color: Numerical solutions (2.25)-(2.28); Red dashed color: FDM).
    Figure 2.  Graph of the numerical solutions with q=2 for β=0.8,k=1,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 3.  Graph of the numerical solutions with q=2 for β=0.7,k=0.8,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 4.  Graph of the numerical solutions with q=1 for β=0.8,k=1,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 5.  Graph of the numerical solutions with q=1 for β=0.7,k=0.8,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 6.  Graph of the numerical solutions with q=1.8 for β=0.8,k=1,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).
    Figure 7.  Graph of the numerical solutions with q=1.8 for β=0.7,k=0.8,γ=0.4,κ=0.001,h=0.01 (a) FFP; (b) FFE; (c) FFM; (Red color: η1; Blue color: ζ1; Green color: η2; Cyan color: ζ2).

    In this paper, numerical solutions of the of the fractal-fractional reaction diffusion equations with general nonlinear have been studied. We introduced the FFRDE in three instances of fractional derivatives based on power, exponential, and Mittag-Leffler kernels. After that, we used the fundamental fractional calculus with the help of Lagrange polynomial functions. We obtained the iterative and approximate formulas in the three cases. We studied the effect of the non-linear term order, in the case of cubic, quadratic, and fractional for different values of the fractal-fractional derivative order. The accuracy of the numerical solutions in the classic case of the FFRDE was tested in the case of power kernel, where all the numerical solutions in the classic case of integer order coincide to each other, and the comparison result has excellent agreement. In all calculations was used the Mathematica Program Package.

    The authors would like to express their Gratitudes to the ministry of education and the deanship of scientific research-Najran University-Kingdom of Saudi Arabia for their financial and Technical support under code number (NU/ESCI/17/025).

    The authors declare that there is no conflict of interests regarding the publication of this paper.

    [1] Mergulhão FJM, Monteiro GA, Cabral JMS, et al. (2004) Design of bacterial vector systems for the production of recombinant proteins in Escherichia coli. J Microbiol Biotechnol 14: 1–14.
    [2] Sanchez-Garcia L, Martín L, Mangues R, et al. (2016) Recombinant pharmaceuticals from microbial cells: a 2015 update. Microb Cell Fact 15: 1–7. doi: 10.1186/s12934-015-0402-6
    [3] Overton TW (2014) Recombinant protein production in bacterial hosts. Drug Discov Today 19: 590–601. doi: 10.1016/j.drudis.2013.11.008
    [4] Baneyx F (1999) Recombinant protein expression in Escherichia coli. Curr Opin Biotechnol 10: 411–421. doi: 10.1016/S0958-1669(99)00003-8
    [5] Pines O, Inouye M (1999) Expression and secretion of proteins in E. coli. Mol Biotechnol 12: 25–34. doi: 10.1385/MB:12:1:25
    [6] Ong CL, Beatson SA, McEwan AG, et al. (2009) Conjugative plasmid transfer and adhesion dynamics in an Escherichia coli biofilm. Appl Environ Microbiol 75: 6783–6791. doi: 10.1128/AEM.00974-09
    [7] Ghigo JM (2001) Natural conjugative plasmids induce bacterial biofilm development. Nature 412: 442–445. doi: 10.1038/35086581
    [8] Reisner A, Höller BM, Molin S, et al. (2006) Synergistic effects in mixed Escherichia coli biofilms: conjugative plasmid transfer drives biofilm expansion. J Bacteriol 188: 3582–3588. doi: 10.1128/JB.188.10.3582-3588.2006
    [9] Reisner A, Haagensen JA, Schembri MA, et al. (2003) Development and maturation of Escherichia coli K-12 biofilms. Mol Microbiol 48: 933–946. doi: 10.1046/j.1365-2958.2003.03490.x
    [10] May T, Okabe S (2008) Escherichia coli harboring a natural IncF conjugative F plasmid develops complex mature biofilms by stimulating synthesis of colanic acid and curli. J Bacteriol 190: 7479–7490. doi: 10.1128/JB.00823-08
    [11] Yang X, Ma Q, Wood TK (2008) The R1 conjugative plasmid increases Escherichia coli biofilm formation through an envelope stress response. Appl Environ Microbiol 74: 2690–2699. doi: 10.1128/AEM.02809-07
    [12] Król JE, Nguyen HD, Rogers LM, et al. (2011) Increased transfer of a multidrug resistance plasmid in Escherichia coli biofilms at the air-liquid interface. Appl Environ Microbiol 77: 5079–5088. doi: 10.1128/AEM.00090-11
    [13] Norman A, Hansen LH, She Q, et al. (2008) Nucleotide sequence of pOLA52: a conjugative IncX1 plasmid from Escherichia coli which enables biofilm formation and multidrug efflux. Plasmid 60: 59–74. doi: 10.1016/j.plasmid.2008.03.003
    [14] Burmølle M, Bahl MI, Jensen LB, et al. (2008) Type 3 fimbriae, encoded by the conjugative plasmid pOLA52, enhance biofilm formation and transfer frequencies in Enterobacteriaceae strains. Microbiology 154: 187–195. doi: 10.1099/mic.0.2007/010454-0
    [15] May T, Ito A, Okabe S (2009) Induction of multidrug resistance mechanism in Escherichia coli biofilms by interplay between tetracycline and ampicillin resistance genes. Antimicrob Agents Chemother 53: 4628–4639. doi: 10.1128/AAC.00454-09
    [16] Castonguay MH, van der Schaaf S, Koester W, et al. (2006) Biofilm formation by Escherichia coli is stimulated by synergistic interactions and co-adhesion mechanisms with adherence-proficient bacteria. Res Microbiol 157: 471–478. doi: 10.1016/j.resmic.2005.10.003
    [17] Gallant CV, Daniels C, Leung JM, et al. (2005) Common β-lactamases inhibit bacterial biofilm formation. Mol Microbiol 58: 1012–1024. doi: 10.1111/j.1365-2958.2005.04892.x
    [18] Lim JY, Yoon J, Hovde CJ (2010) A brief overview of Escherichia coli O157:H7 and its plasmid O157. J Microbiol Biotechnol 20: 5–14.
    [19] Burland V, Shao Y, Perna NT, et al. (1998) The complete DNA sequence and analysis of the large virulence plasmid of Escherichia coli O157:H7. Nucleic Acids Res 26: 4196–4204. doi: 10.1093/nar/26.18.4196
    [20] Lim JY, La HJ, Sheng H, et al. (2010) Influence of plasmid pO157 on Escherichia coli O157:H7 Sakai biofilm formation. Appl Environ Microbiol 76: 963–966. doi: 10.1128/AEM.01068-09
    [21] Huang CT, Peretti SW, Bryers JD (1993) Plasmid retention and gene expression in suspended and biofilm cultures of recombinant Escherichia coli DH5α (pMJR1750). Biotechnol Bioeng 41: 211–220. doi: 10.1002/bit.260410207
    [22] Huang CT, Peretti SW, Bryers JD (1994) Effects of inducer levels on a recombinant bacterial biofilm formation and gene expression. Biotechnol Lett 16: 903–908. doi: 10.1007/BF00128622
    [23] Bryers JD, Huang CT (1995) Recombinant plasmid retention and expression in bacterial biofilm cultures. Wat Sci Tech 31: 105–115.
    [24] O’Connell HA, Niu C, Gilbert ES (2007) Enhanced high copy number plasmid maintenance and heterologous protein production in an Escherichia coli biofilm. Biotechnol Bioeng 97: 439–446. doi: 10.1002/bit.21240
    [25] Teodósio JS, Simões M, Mergulhão FJ (2012) The influence of nonconjugative Escherichia coli plasmids on biofilm formation and resistance. J Appl Microbiol 113: 373–382. doi: 10.1111/j.1365-2672.2012.05332.x
    [26] Mergulhão FJ, Taipa MA, Cabral JM, et al. (2004) Evaluation of bottlenecks in proinsulin secretion by Escherichia coli. J Biotechnol 109: 31–43. doi: 10.1016/j.jbiotec.2003.10.024
    [27] Gomes LC, Carvalho D, Briandet R, et al. (2016) Temporal variation of recombinant protein expression in Escherichia coli biofilms analysed at single-cell level. Process Biochem 51: 1155–1161. doi: 10.1016/j.procbio.2016.05.016
    [28] Ferreira C, Pereira AM, Pereira MC, et al. (2011) Physiological changes induced by the quaternary ammonium compound benzyldimethyldodecylammonium chloride on Pseudomonas fluorescens. J Antimicrob Chemother 66: 1036–1043. doi: 10.1093/jac/dkr028
    [29] Ferreira C, Pereira AM, Melo LF, et al. (2010) Advances in industrial biofilm control with micro-nanotechnology, In: Méndez-Vilas A, editor. Current Research, Technology and Education Topics in Applied Microbiology and Microbial Biotechnology. Badajoz: Formatex, 845–854.
    [30] Yanischperron C, Vieira J, Messing J (1985) Improved M13 phage cloning vectors and host strains: nucleotide sequences of the M13mpl8 and pUC19 vectors. Gene 33: 103–119. doi: 10.1016/0378-1119(85)90120-9
    [31] Sambrook J, Russell DW (2001) Molecular Cloning: a Laboratory Manual. New York: Cold Spring Harbor Laboratory Press.
    [32] Teodósio JS, Simões M, Melo LF, et al. (2011) Flow cell hydrodynamics and their effects on E. coli biofilm formation under different nutrient conditions and turbulent flow. Biofouling 27: 1–11.
    [33] Gomes LC, Silva LN, Simões M, et al. (2015) Escherichia coli adhesion, biofilm development and antibiotic susceptibility on biomedical materials. J Biomed Mater Res A 103: 1414–1423. doi: 10.1002/jbm.a.35277
    [34] Mergulhão FJ, Monteiro GA (2007) Analysis of factors affecting the periplasmic production of recombinant proteins in Escherichia coli. J Microbiol Biotechnol 17: 1236–1241.
    [35] Bentley WE, Mirjalili N, Andersen DC, et al. (1990) Plasmid-encoded protein: The principal factor in the “metabolic burden” associated with recombinant bacteria. Biotechnol Bioeng 35: 668–681. doi: 10.1002/bit.260350704
    [36] Sørensen HP, Mortensen KK (2005) Advanced genetic strategies for recombinant protein expression in Escherichia coli. J Biotechnol 115: 113–128. doi: 10.1016/j.jbiotec.2004.08.004
    [37] Cunningham DS, Koepsel RR, Ataai MM, et al. (2009) Factors affecting plasmid production in Escherichia coli from a resource allocation standpoint. Microb Cell Fact 8: 1475–2859.
    [38] Hoffmann F, Weber J, Rinas U (2002) Metabolic adaptation of Escherichia coli during temperature-induced recombinant protein production: 1. Readjustment of metabolic enzyme synthesis. Biotechnol Bioeng 80: 313–319.
    [39] Hoffmann F, Rinas U (2004) Stress induced by recombinant protein production in Escherichia coli, In: Enfors S-O, editor. Physiological Stress Responses in Bioprocesses, Berlin: Springer-Verlag Berlin Heidelberg, 73–92.
    [40] Landini P (2009) Cross-talk mechanisms in biofilm formation and responses to environmental and physiological stress in Escherichia coli. Res Microbiol 160: 259–266. doi: 10.1016/j.resmic.2009.03.001
    [41] Kurland CG, Dong H (1996) Bacterial growth inhibition by overproduction of protein. Mol Microbiol 21: 1–4. doi: 10.1046/j.1365-2958.1996.5901313.x
    [42] Glick BR (1995) Metabolic load and heterologous gene expression. Biotechnol Adv 13: 247–261. doi: 10.1016/0734-9750(95)00004-A
    [43] Xia XX, Qian ZG, Ki CS, et al. (2010) Native-sized recombinant spider silk protein produced in metabolically engineered Escherichia coli results in a strong fiber. Proc Natl Acad Sci USA 107: 14059–14063. doi: 10.1073/pnas.1003366107
    [44] Yang YX, Qian ZG, Zhong JJ, et al. (2016) Hyper-production of large proteins of spider dragline silk MaSp2 by Escherichia coli via synthetic biology approach. Process Biochem 51: 484–490. doi: 10.1016/j.procbio.2016.01.006
    [45] Dong H, Nilsson L, Kurland CG (1995) Gratuitous overexpression of genes in Escherichia coli leads to growth inhibition and ribosome destruction. J Bacteriol 177: 1497–1504. doi: 10.1128/jb.177.6.1497-1504.1995
    [46] Georgiou G, Shuler ML, Wilson DB (1988) Release of periplasmic enzymes and other physiological effects of β-lactamase overproduction in Escherichia coli. Biotechnol Bioeng 32: 741–748. doi: 10.1002/bit.260320603
    [47] Williams I, Venables WA, Lloyd D, et al. (1997) The effects of adherence to silicone surfaces on antibiotic susceptibility in Staphylococcus aureus. Microbiology 143: 2407–2413. doi: 10.1099/00221287-143-7-2407
    [48] Simões M, Pereira MO, Vieira MJ (2005) Effect of mechanical stress on biofilms challenged by different chemicals. Water Res 39: 5142–5152. doi: 10.1016/j.watres.2005.09.028
    [49] Araújo PA, Mergulhão FJM, Melo LF, et al. (2014) The ability of an antimicrobial agent to penetrate a biofilm is not correlated with its killing or removal efficiency. Biofouling 30: 675–683. doi: 10.1080/08927014.2014.904294
    [50] Cloete TE, Jacobs L, Brözel VS (1998) The chemical control of biofouling in industrial water systems. Biodegradation 9: 23–37. doi: 10.1023/A:1008216209206
    [51] Ahimou F, Semmens MJ, Haugstad G, et al. (2007) Effect of protein, polysaccharide, and oxygen concentration profiles on biofilm cohesiveness. Appl Environ Microbiol 73: 2905–2910. doi: 10.1128/AEM.02420-06
  • This article has been cited by:

    1. Jia-Bao Liu, Saad Ihsan Butt, Jamshed Nasir, Adnan Aslam, Asfand Fahad, Jarunee Soontharanon, Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator, 2022, 7, 2473-6988, 2123, 10.3934/math.2022121
    2. Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park, A study on the fractal-fractional tobacco smoking model, 2022, 7, 2473-6988, 13887, 10.3934/math.2022767
    3. Hasib Khan, Muhammad Ibrahim, Abdel-Haleem Abdel-Aty, M. Motawi Khashan, Farhat Ali Khan, Aziz Khan, A fractional order Covid-19 epidemic model with Mittag-Leffler kernel, 2021, 148, 09600779, 111030, 10.1016/j.chaos.2021.111030
    4. Krunal B. Kachhia, Chaos in fractional order financial model with fractal–fractional derivatives, 2023, 7, 26668181, 100502, 10.1016/j.padiff.2023.100502
    5. Hari M. Srivastava, Khaled Mohammed Saad, Walid M. Hamanah, Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations, 2022, 10, 2227-7390, 1089, 10.3390/math10071089
    6. Hasnaa H Alzahrani, Marco Lucchesi, Kassem Mustapha, Olivier P Le Maître, Omar M Knio, Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation, 2021, 5, 2399-6528, 085014, 10.1088/2399-6528/ac1507
    7. Jagdev Singh, Arpita Gupta, Dumitru Baleanu, On the analysis of an analytical approach for fractional Caudrey-Dodd-Gibbon equations, 2022, 61, 11100168, 5073, 10.1016/j.aej.2021.09.053
    8. Xiaojun Zhou, Yue Dai, A spectral collocation method for the coupled system of nonlinear fractional differential equations, 2022, 7, 2473-6988, 5670, 10.3934/math.2022314
    9. Nauman Ahmed, Ali Raza, Ali Akgül, Zafar Iqbal, Muhammad Rafiq, Muhammad Ozair Ahmad, Fahd Jarad, New applications related to hepatitis C model, 2022, 7, 2473-6988, 11362, 10.3934/math.2022634
    10. Esra Karatas Akgül, Wasim Jamshed, Kottakkaran Sooppy Nisar, S.K. Elagan, Nawal A. Alshehri, On solutions of gross domestic product model with different kernels, 2022, 61, 11100168, 1289, 10.1016/j.aej.2021.06.067
    11. Rubayyi T. Alqahtani, Shabir Ahmad, Ali Akgül, On Numerical Analysis of Bio-Ethanol Production Model with the Effect of Recycling and Death Rates under Fractal Fractional Operators with Three Different Kernels, 2022, 10, 2227-7390, 1102, 10.3390/math10071102
    12. A. DLAMINI, EMILE F. DOUNGMO GOUFO, M. KHUMALO, CHAOTIC BEHAVIOR OF MODIFIED STRETCH–TWIST–FOLD FLOW UNDER FRACTAL-FRACTIONAL DERIVATIVES, 2022, 30, 0218-348X, 10.1142/S0218348X22402071
    13. Anwar Zeb, Abdon Atangana, Zareen A. Khan, Salih Djillali, A robust study of a piecewise fractional order COVID-19 mathematical model, 2022, 61, 11100168, 5649, 10.1016/j.aej.2021.11.039
    14. Shabir Ahmad, Aman Ullah, Ali Akgül, Manuel De la Sen, A study of fractional order Ambartsumian equation involving exponential decay kernel, 2021, 6, 2473-6988, 9981, 10.3934/math.2021580
    15. Raheel Kamal, Gul Rahmat, Kamal Shah, Ricardo Escobar, On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations, 2021, 2021, 1563-5147, 1, 10.1155/2021/4640467
    16. Kaihong Zhao, Shuang Ma, Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses, 2022, 7, 2473-6988, 3169, 10.3934/math.2022175
    17. Kamsing Nonlaopon, Muhammad Naeem, Ahmed M. Zidan, Rasool Shah, Ahmed Alsanad, Abdu Gumaei, Muhammad Imran Asjad, Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators, 2021, 2021, 1099-0526, 1, 10.1155/2021/7979365
    18. Saima Rashid, Rehana Ashraf, Ebenezer Bonyah, Azhar Hussain, On Analytical Solution of Time-Fractional Biological Population Model by means of Generalized Integral Transform with Their Uniqueness and Convergence Analysis, 2022, 2022, 2314-8888, 1, 10.1155/2022/7021288
    19. Khadija Tul Kubra, Rooh Ali, Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan, 2023, 2363-6203, 10.1007/s40808-023-01747-w
    20. ZAREEN A. KHAN, KAMAL SHAH, BAHAAELDIN ABDALLA, THABET ABDELJAWAD, A NUMERICAL STUDY OF COMPLEX DYNAMICS OF A CHEMOSTAT MODEL UNDER FRACTAL-FRACTIONAL DERIVATIVE, 2023, 31, 0218-348X, 10.1142/S0218348X23401813
    21. Kamal Shah, Thabet Abdeljawad, On complex fractal-fractional order mathematical modeling of CO 2 emanations from energy sector, 2024, 99, 0031-8949, 015226, 10.1088/1402-4896/ad1286
    22. Samy A. Abdelhafeez, Anas A. M. Arafa, Yousef H. Zahran, Ibrahim S. I. Osman, Moutaz Ramadan, Adapting Laplace residual power series approach to the Caudrey Dodd Gibbon equation, 2024, 14, 2045-2322, 10.1038/s41598-024-57780-x
    23. Krunal B. Kachhia, Prit P. Parmar, A novel fractional mask for image denoising based on fractal–fractional integral, 2024, 11, 26668181, 100833, 10.1016/j.padiff.2024.100833
    24. Harpreet Kaur, Amanpreet Kaur, Palwinder Singh, Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel, 2024, 13, 2192-8029, 10.1515/nleng-2022-0380
    25. Muhammad Farman, Changjin Xu, Perwasha Abbas, Aceng Sambas, Faisal Sultan, Kottakkaran Sooppy Nisar, Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach, 2025, 142, 10075704, 108525, 10.1016/j.cnsns.2024.108525
    26. Ashish Rayal, System of fractal-fractional differential equations and Bernstein wavelets: a comprehensive study of environmental, epidemiological, and financial applications, 2025, 100, 0031-8949, 025236, 10.1088/1402-4896/ada592
    27. Mashael M. AlBaidani, Abdul Hamid Ganie, Adnan Khan, Fahad Aljuaydi, An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator, 2025, 9, 2504-3110, 199, 10.3390/fractalfract9040199
  • Reader Comments
  • © 2016 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(6631) PDF downloads(1080) Cited by(1)

Figures and Tables

Figures(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog