Research article Special Issues

Realizations of kinetic differential equations

  • The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.

    Citation: Gheorghe Craciun, Matthew D. Johnston, Gábor Szederkényi, Elisa Tonello, János Tóth, Polly Y. Yu. Realizations of kinetic differential equations[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 862-892. doi: 10.3934/mbe.2020046

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  • The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.


    We are concerned with Atangana-Baleanu variable order fractional problems:

    {Lu(x)=ABCDα(x)u(x)+a(x)u(x)=f(x,u),x[0,1],B(u)=0, (1.1)

    where 0<α(x)<1, ABCDα(x)(x) denotes the α(x) order Atangana-Baleanu Caputo derivatives, B(u) is the linear boundary condition, which includes initial value condition, periodic condition, final value condition and so on.

    The α(x)(0<α(x)<1) order Atangana-Baleanu Caputo derivatives of a function u(x) is firstly defined by Atangana and Baleanu [1]

    ABCDα(x)u(x)=M(α(x))1α(x)x0Eα(x)(α(x)1α(x)(xt)α(x))u(t)dt, (1.2)

    where Eα(x)(x) is the Mittag-Leffler function.

    Fractional order differential equations (FDEs) have important applications in several fields such as materials, chemistry transmission dynamics, optimal control and engineering [2,3,4,5,6]. In fact, the classical fractional derivatives are defined with weak singular kernels and the solutions of FDEs inherit the weak singularity. The Mittag-Leffler (ML) function was firstly introduced by Magnus Gösta Mittag-Leffler. Recently, it is found that this function has close relation to FDEs arising in real applications.

    Atangana and Baleanu [1] introduced a new fractional derivative by using the ML function, which is nonlocal and nonsingular. The new fractional derivatives is very important and have been applied to several different fields (see e.g. [7,8,9]). Up to now, several numerical algorithms have been developed for solving Atangana-Baleanu FDEs. Akgül et al. [10,11,12] proposed effective difference techniques and kernels based approaches for Atangana-Baleanu FDEs. On the basis of the Sobolev kernel functions, Arqub et al. [13,14,15,16,17] proposed the numerical techniques for Atangana-Baleanu fractional Riccati and Bernoulli equations, Bagley-Torvik and Painlev equations, Volterra and Fredholm integro-differential equations. Yadav et al. [18] introduced the numerical algorithms and application of Atangana-Baleanu FDEs. El-Ajou, Hadid, Al-Smadi et al. [19] developed approximated technique for solutions of population dynamics of Atangana-Baleanu fractional order.

    Reproducing kernel Hilbert space (RKHS) is ideal for function approximation and estimate of fractional derivatives. In recent years, reproducing kernel functions (RKF) theory have been employed to solve linear and nonlinear fractional order problems, singularly perturbed problems, singular integral equations, fuzzy differential equations, and so on (see, e.g. [10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]). However, there exists little discussion on numerical schemes for solving variable order Atangana-Baleanu FDEs.

    In this paper, by using polynomials RKF, we will present a new collocation method for solving variable order Atangana-Baleanu FDEs.

    This work is organized as follows. We summarize fractional derivatives and RKHS theory in Section 2. In Section 3, we develop RKF based collocation technique for Atangana-Baleanu variable order FDEs. Numerical experiments are provided in Section 4. Concluding remarks are included in the last section.

    Definition 2.1. Let H be a Hilbert function space defined on E. The function K:E×ER is known as an RKF of space H if

    (1)K(,t)HforalltE,(2)w(t)=(w(),K(,t)),foralltEandallwH.

    If there exists a RKF in a Hilbert space, then the space is a RKHS.

    Definition 2.2. Symmetric function K:E×ER is known as a positive definite kernel (PDK) if ni,j=1cicjK(xi,xj)0 for any nN, x1,x2,,xnE,c1,c2,,cnR.

    Theorem 2.1. [36] The RKF of an RKHS is positive definite. Besides, every PDK can define a unique RKHS, of which it is the RKF.

    Definition 2.3. Let q>0. The one parameter Mittag-Leffler function of order q is defined by

    Eq(z)=j=0zjΓ(jq+1). (2.1)

    Definition 2.4. Let q1,q2>0. The two-parameter Mittag-Leffler function is defined by

    Eq1,q2(z)=j=0zjΓ(jq1+q2). (2.2)

    For the domains of convergence of the Mittag-Leffler functions, please refer to the following theorem.

    Theorem 2.2. [37] For q1,q2>0, the two-parameter Mittag-Leffler function Eq1,q2(z) is convergent for all zC.

    Definition 2.5. The Sobolev space H1(0,T) is defined as follows

    H1(0,T)={u|uL2(0,T),uL2(0,T)}.

    Definition 2.6. The α(0,1) order Atangana- Baleanu fractional derivative of a function uH1(a,b) is defined

    ABCDαu(x)=M(α)1αx0Eα(α1α(xt)α)u(t)dt, (2.3)

    where M(α) is the normalization term satisfying M(0)=M(1)=1.

    Theorem 2.3. [38] The function k(x,y)=(xy+c)m for c>0,mN is a PDK.

    According to Theorem 2.1, there exists an associated RKHS Qm with k as an RKF.

    To solve (1.1), we will construct the RKF which satisfies the homogenous boundary condition.

    Definition 3.1.

    Qm,0={w(t)w(t)Qm,B(w)=0}.

    Theorem 3.1. The space Qm,0 is an RKHS and its RKF is expressed by

    K(x,y)=k(x,y)Bxk(x,y)Byk(x,y)BxByk(x,y).

    Proof. If Byk(x,y)=0 or Bxk(x,y)=0, then

    K(x,y)=k(x,y).

    If Byk(x,y)0, then

    BxK(x,y)=Bxk(x,y)Bxk(x,y)BxByk(x,y)BxByk(x,y),=0,

    and naturally K(x,y)Qm,0.

    For all u(y)Qm,0, we have u(y)Qm and Byu(y)=0.

    We have

    (u(y),K(x,y))=(u(y),k(x,y))(u(y),Bxk(x,y)Byk(x,y)BxByk(x,y)=u(x)Byk(x,y)BxByk(x,y)(u(y),Bxk(x,y))=u(x)Byk(x,y)BxByk(x,y)Bx(u(y),k(x,y))=u(x)Byk(x,y)BxByk(x,y)Bxu(x)=u(x)0=0.

    Thus, K(x,y) is the RKF of space Qm,0 and the proof is complete.

    Suppose that L:Qm,0H1 is a bounded linear operator. It is easy to proved that its inverse operator L1 is also bounded since both Qm,0 and H1 are Banach spaces.

    Choose N distinct scattered points in [0,1], such as {x1,x2,,xN}. Put ψi(x)=K(x,xi),i=1,2,,N. By using RKF basis, the RKF collocation solution uN(x) for (1.1) can be written as follows

    uN(x)=Ni=1ciψi(x), (3.1)

    where {ci}Ni=1 are undetermined constants.

    Collocating (1.1) at N nodes x1,x2,,xN provides N equations:

    LuN(xk)=Ni=1ciLψi(xk)=f(xk,uN(xk)),k=1,2,,N. (3.2)

    System (3.3) of equations is simplified to the matrix form:

    Ac=f, (3.3)

    where Aik=Lxψk(x)|x=xi,i,k=1,2,,N, f=(f(x1,uN(x1)),f(x2,uN(x2)),,f(xN,uN(xN)).

    Theorem 3.2. If γ>0, then

    ABCDα(x)xγ=M(α(x))1α(x)Γ(γ+1)xγEα(x),γ+1(α(x)1α(x)xα(x)),

    and therefore matrix A can be computed exactly.

    Proof. It is noticed that

    ABCDα(x)xγ=M(α(x))1α(x)x0Eα(x)(α(x)1α(x)(xt)α(x))γtγ1dt=M(α(x))1α(x)x0j=0(α(x)1α(x)(xt)α(x))jΓ(jα(x)+1)γtγ1dt=M(α(x))1α(x)γj=0(α(x)1α(x))jΓ(jα(x)+1)x0(xt)α(x)tγ1dt=M(α(x))1α(x)γj=0(α(x)1α(x))jΓ(jα(x)+1)Γ(jα(x)+1)Γ(γ)Γ(jα(x)+γ+1)xjα(x)+γ=M(α(x))1α(x)Γ(γ+1)xγj=0(α(x)1α(x)xα(x))jΓ(jα(x)+γ+1)=M(α(x))1α(x)Γ(γ+1)xγEα(x),γ+1(α(x)1α(x)xα(x)).

    Since RKF K(x,y) is a polynomials, matrix A in (3.3) can be calculated exactly. The proof is complete.

    If f(x,u) is linear, then (3.3) is a system of linear equations and it is convenient to determine the value of the unknowns {ci}Ni=1. If f(x,u) is nonlinear, then (3.3) is a system of nonlinear equations, we solve it by using the tool "FindRoot" in soft Mathematica 11.0.

    The residual function is defined as

    RN(x)=LuN(x)f(x,uN(x)).

    Theorem 3.3. If a(x) and f(x,u)C4[0,1], then

    RN(x)maxx[x1,xN]RN(x)∣≤ch4,

    where c>0 is a real number, h=max1iNxi+1xi.

    Proof. For the proof, please refer to [22].

    Three experiments are illustrated in this section to show the applicability and effectiveness of the mentioned approach. We take M(α)=1 in the following experiments.

    Problem 4.1

    Solve fractional linear initial value problems (IVPs) as follows:

    {ABCDαu(x)+exu(x)=f(x),x(0,1],u(0)=1,

    where α(x)=0.5x+0.1, f(x)=ex(x2+x3+1)+M(α(x))1α(x)2x2Eα(x),3(α(x)1α(x)xα(x))++M(α(x))1α(x)6x3Eα(x),4(α(x)1α(x)xα(x)). The true solution of this equation is u(x)=x2+x3+1.

    Selecting m=8,N=8, xi=iN,i=1,2,,N, we apply our new method to Problem 4.1. The obtained numerical results are shown in Tables 1. The Mathematica codes for Problem 4.1 is provided as follows:

    tru[x_]=x2+x3+1;p[x_]=Ex;α[x_]=0.5x+0.1;B[x_]=1;a[x_]=1Gamma[2α[x]];K[x_,y_]=(xy+1)8;R[x_,y_]=K[x,y]K[x,0]K[0,y]/K[0,0];w[x_,y_]=p[x]R[x,y];v[x_,d_]=B[α[x]]Gamma[d+1]xdMittagLefflerE[2,d+1,α[x]xα[x]/(1α[x])];fu[x_,y_]=8yv[x,1]+28y2v[x,2]+56y3v[x,3]+70y4v[x,4]+56y5v[x,5]+28y6v[x,6]+8y7v[x,7]+y8v[x,8];m=8;xx=Table[0,{i,1,m}];A=Table[0,{i,1,m},{j,1,m}];For[i=1,im,i++,xx[[i]]=i/m];For[i=1,im,i++,For[j=1,jm,j++,A[[i,j]]=w[xx[[i]],xx[[j]]]+fu[xx[[i]]+xx[[j]]]]];v[x_]=tru[0];f0[x]=p[x]tru[x]+v[x,2]+v[x,3];f[x]=f0[x]p[x]v[x];b=Table[f[xx[[k]]],{i,1,m}];c=LinearSolve[A,b];u[x_]=mi=1c[[i]]R[x,xx[[i]]];u[x_]=u[x]+v[x];
    Table 1.  Errors of numerical results for Problem 4.1.
    Nodes x Exact solution Absolute error Relative error
    0.10 1.011 1.88×1013 1.86×1013
    0.20 1.048 2.57×1013 2.45×1013
    0.30 1.117 9.50×1014 8.50×1014
    0.40 1.224 6.35×1013 5.19×1013
    0.50 1.375 0 0
    0.60 1.576 2.17×1014 1.38×1014
    0.70 1.833 7.65×1013 4.17×1013
    0.80 2.152 8.65×1013 4.02×1013
    0.90 2.539 2.40×1013 9.46×1014
    1.00 3.000 9.09×1013 3.03×1013

     | Show Table
    DownLoad: CSV

    Problem 4.2

    Solve the variable order fractional linear terminal value problems

    {ABCDαu(x)+2u(x)=f(x),x[0,1),u(1)=3,

    where α(x)=sinx, f(x)=2(x4+2)+M(α(x))1α(x)24x4Eα(x),5(α(x)1α(x)xα(x)). The exact solution is u(x)=x4+2.

    Selecting m=8,N=8, xi=i1N,i=1,2,,N, the obtained absolute and relative errors of numerical results using our method are listed in Tables 2.

    Table 2.  Errors of numerical results for Problem 4.2.
    Nodes x Exact solution Absolute error Relative error
    0.00 2.0000 2.75×1010 1.37×1010
    0.10 2.0001 1.02×1010 5.14×1011
    0.20 2.0016 9.96×1011 4.97×1011
    0.30 2.0081 1.08×1010 5.39×1011
    0.40 2.0256 1.12×1010 5.56×1011
    0.50 2.0625 1.10×1010 5.37×1011
    0.60 2.1296 1.05×1010 4.96×1011
    0.70 2.2401 1.08×1010 4.83×1011
    0.80 2.4096 9.36×1011 3.88×1011
    0.90 2.6561 4.38×1011 1.64×1011

     | Show Table
    DownLoad: CSV

    Problem 4.3

    We apply our method to the nonlinear variable order fractional IVPs as follows

    {ABCDαu(x)+sinhxu(x)+sin(u)=f(x),x(0,1],u(0)=1,

    where α(x)=0.5x+0.1, f(x)=sinhx(x+x3+1)+M(α(x))1α(x)xEα(x),2(α(x)1α(x)xα(x))+M(α(x))1α(x)6x3Eα(x),4(α(x)1α(x)xα(x)). Its true solution is u(x)=x+x3+1.

    Choosing m=8,N=8, xi=iN,i=1,2,,N, we plot the absolute and relative errors in Figure 1.

    Figure 1.  Absolute errors (left) and relative errors (right) for Problem 4.3.

    In this work, a new RKF based collocation technique is developed for Atangana-Baleanu variable order fractional problems. The proposed scheme is meshless and therefore it does not require any background meshes. From the numerical results, it is found that the accuracy of obtained approximate solutions is high and can reach to O(1010). Also, for nonlinear fractional problems, our method can yield highly accurate numerical solutions. Hence, our new method is very effective and easy to implement for the considered problems.

    The work was supported by the National Natural Science Foundation of China (No.11801044, No.11326237).

    All authors declare no conflicts of interest in this paper.



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