Opinion paper Special Issues

Bone remodeling platforms: Understanding the need for multicellular lab-on-a-chip systems and predictive agent-based models

  • Received: 31 July 2019 Accepted: 16 October 2019 Published: 19 November 2019
  • The purpose of this paper is to emphasize the need for more complex bone remodeling platforms that allow for investigations of intricate multicellular interactions that regulate this process. We discuss the efforts we have taken to develop lab-on-a-chip systems for bone remodeling and the motivation for pursuing more advanced multicellular models. Further, we discuss mathematical modeling opportunities that will allow experimental results to extend beyond the set laboratory conditions. We advocate for the development of an agent-based model comprised of multiple cellular automata of each bone cell type. In total, this work requires a combination of techniques from bone biology, microfluidics, cell mechanobiology, mechanics, and mathematical modeling. Thus, significant advancements within the field will require a collective contribution from a variety of research laboratories.

    Citation: Sharon L. Truesdell, Marnie M. Saunders. Bone remodeling platforms: Understanding the need for multicellular lab-on-a-chip systems and predictive agent-based models[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1233-1252. doi: 10.3934/mbe.2020063

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  • The purpose of this paper is to emphasize the need for more complex bone remodeling platforms that allow for investigations of intricate multicellular interactions that regulate this process. We discuss the efforts we have taken to develop lab-on-a-chip systems for bone remodeling and the motivation for pursuing more advanced multicellular models. Further, we discuss mathematical modeling opportunities that will allow experimental results to extend beyond the set laboratory conditions. We advocate for the development of an agent-based model comprised of multiple cellular automata of each bone cell type. In total, this work requires a combination of techniques from bone biology, microfluidics, cell mechanobiology, mechanics, and mathematical modeling. Thus, significant advancements within the field will require a collective contribution from a variety of research laboratories.


    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.



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