Research article

The modified Kies-Fréchet distribution: Properties, inference and application

  • The present article introduces a new distribution called the modified Kies-Fréchet (MKF) distribution that extends the Fréchet distribution and provides two new sub-models called modified Kies inverse-exponential and modified Kies inverse-Rayleigh distributions. The MKF model can provide left-skewed, symmetrical, right-skewed, J-shaped, and reversed-J shaped densities. The MKF density was expressed as a linear combination of Fréchet densities. We derive some basic mathematical properties of the MKF model. The MKF parameters are estimated using some classical estimation methods called, the maximum likelihood, Anderson-Darling, least-squares, Cramér-von Mises, and weighted least squares. The performances of these estimators were explored by a detailed simulation study. Finally, the flexibility of the MKF model is checked using a real data set, showing that it can provide close fit as compared with other competing models.

    Citation: Mashail M. Al Sobhi. The modified Kies-Fréchet distribution: Properties, inference and application[J]. AIMS Mathematics, 2021, 6(5): 4691-4714. doi: 10.3934/math.2021276

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  • The present article introduces a new distribution called the modified Kies-Fréchet (MKF) distribution that extends the Fréchet distribution and provides two new sub-models called modified Kies inverse-exponential and modified Kies inverse-Rayleigh distributions. The MKF model can provide left-skewed, symmetrical, right-skewed, J-shaped, and reversed-J shaped densities. The MKF density was expressed as a linear combination of Fréchet densities. We derive some basic mathematical properties of the MKF model. The MKF parameters are estimated using some classical estimation methods called, the maximum likelihood, Anderson-Darling, least-squares, Cramér-von Mises, and weighted least squares. The performances of these estimators were explored by a detailed simulation study. Finally, the flexibility of the MKF model is checked using a real data set, showing that it can provide close fit as compared with other competing models.



    Reaction-diffusion systems have attracted a considerable amount of attention in recent years. They arise naturally in various biological, chemical and physical models to describe the spatio-temporal concentration change of one or more species which involve both local reaction and diffusion simultaneously. The reaction-diffusion system consists of a set of partial differential equations (PDEs) to represent the behaviour of each species individually. Recently, the interest in the study of reaction-diffusion systems has been extended to fractional reaction-diffusion systems [1,2,3,4,5] which from one side exhibit self-organization phenomena and from the other side introduce a new parameter to these systems, which is a fractional derivative index. It gives a greater degree of freedom for diversity of self-organization phenomena and it can exhibit new types of solutions that are not possible to find in classical reaction-diffusion equations with integer derivatives [6,7,8]. From numerical point of view, several methods have been developed, including, for example, finite element schemes [9], finite volume [] and finite difference methods [11,12] and spectral ones [13,14].

    In this paper, we consider a system of two coupled nonlinear time-fractional reaction-diffusion equations (TF-RDEs)

    {αtu(t,x)k1xxu(t,x)+f(t,x,u,v)=0,αtv(t,x)k2xxv(t,x)+g(t,x,u,v)=0,0<α<1, (1.1)

    where αt is the Riemann-Liouville fractional derivative operator [15,16,17,18,19,20]

    αtw(t,x)=1Γ(1α)tt0w(s,x)(ts)αds (1.2)

    u(t,x) and v(x,t) being the field variables with t and x independent variables, k1>0 and k2>0 the diffusion coefficients, and f=f(t,x,u,v) and g=g(t,x,u,v) the nonlinear interaction terms. This kind of fractional problem has been numerically studied in a lot of papers, see, for example [21,22,23,24].

    The main aim of this paper is to solve the system of reaction-diffusion equations (1.1) by applying the procedure recently proposed that combines the Lie symmetry analysis with the numerical methods to get exact and numerical solutions. This procedure was applied to a wide class of FDES: time-fractional advection–diffusion–reaction equations [25,26], space-fractional advection–diffusion–reaction equations with linear [27] and nonlinear sources terms [28], time-fractional and space-fractional advection-diffusion-reaction equations with linear and nonlinear sources terms [29], two dimensional time-fractional reaction-diffusion equations [30].

    As well known, classical Lie symmetries allow to reduce the partial differential equations to ordinary differential equations, for this reason, recently the Lie symmetries theory has been extended to the FDES [31,32,33,34]. By using this extension of Lie symmetries approach to FPDEs, we find the Lie symmetries admitted by the model (1.1), then Lie transformations that map the coupled nonlinear FT-PDEs to a system of nonlinear fractional ordinary differential equations (FODEs). We find solutions of a system of two nonlinear time-fractional reaction-diffusion equations by solving the reduced FODEs such that the solutions of the reduced equations lead to obtain solutions of the original equations. The original model is written in terms of the Riemann-Liouville fractional derivative in order to determine the Lie symmetries by applying the MAPLE package, that automates the method of finding symmetries for FDEs as proposed in [31,32,33,34]. After the reduction of the model to a fractional ordinary differential equation, we introduce the Caputo derivative, and rewrite the FODE in terms of this derivative so that we are able to compute the numerical solutions. Classical finite difference methods are proposed with the aim of showing the simplicity of application of the procedure and to obtain, at the same time, highly accurate solutions and a low computational cost.

    The plan of the paper is the follows: In Section 2, the Lie symmetries theory for FDEs is presented; in Section 3, we describe the reduction of the original model to FODEs and report some examples of mathematical models obtained by suitable choices of involved parameters and functions; the Section 4 is devoted to describe the numerical method used to obtain solutions of the reduced FODEs and, then, of the assigned original model. Finally, we end with the concluding remarks.

    In general, for a 2×2 system of (integer order) partial differential equations

    Δi(t,x,u,v,,u(r1),v(r2))=0,i=1,2 (2.1)

    where t,x are the independent variables, u and v are the dependent variables, and u(r1), v(r2) are all partial derivatives of the u and v with respect to t and x up to the maximum order r1 and r2 (integer order) respectively, we recall that the invertible transformations of the variables t,x,u,v

    T=T(t,x,u,v,a),X=X(t,x,u,v,a), (2.2)
    U=U(t,x,u,v,a),V=V(t,x,u,v,a) (2.3)

    depending on a continuous parameter a, are said to be one-parameter (a) Lie point symmetry transformations of the Eq (2.1) if the Eq (2.1) has the same form in the new variables T,X,U,V.

    The set G of all such transformations forms a continuous group, also known as the group admitted by the Eq (2.1).

    According to the Lie theory, by expanding (2.3) in Taylor's series around a=0, we get the infinitesimal transformations

    T=t+aξ1(t,x,u,v)+o(a),X=x+aξ2(t,x,u,v)+o(a), (2.4)
    U=u+aη1(t,x,u,v)+o(a),V=v+aη2(t,x,u,v)+o(a), (2.5)

    where their infinitesimals ξ1, ξ2, η1 and η2 are given by

    ξ1(t,x,u,v)=Ta|a=0,ξ2(t,x,u,v)=Xa|a=0
    η1(t,x,u,v)=Ua|a=0,η2(t,x,u,v)=Va|a=0.

    The corresponding operator

    Ξ=ξ1(t,x,u,v)t+ξ2(t,x,u,v)x+η1(t,x,u,v)u+η2(t,x,u,v)v (2.6)

    is known in the literature as the infinitesimal operator or generator of the group G.

    The point transformations leaving a differential equation (2.1) invariant are found by means of the straightforward Lie's algorithm, requiring that the k-order prolongation of the operator (2.6) acting on (2.1) is zero along the solutions, i.e.:

    ΞkΔi=0|Δi=0, (2.7)

    where k is the maximum order of derivations.

    The invariance condition (2.7) provides an overdetermined set of linear differential equations (determining equations) for the infinitesimals whose integration gives the generators of Lie point symmetries admitted by the Eq (2.1).

    Recently, in a series of papers of Buckwar, Luchko, Gazizov et al. [31,32,33,34,35], Lie symmetry methods have been extended to to FDEs. By extending transformations (2.5) to the operator of Riemann-Liouville fractional differentiation on derivatives of u with respect to t, Dαtuj, and v with respect to t, Dαtv, we have

    Dαˉtu(ˉt,ˉx)=Dαtu(t,x)+aζ1α+o(a), (2.8)
    Dαˉtv(ˉt,ˉx)=Dαtv(t,x)+aζ2α+o(a) (2.9)

    where ζ1α and ζ2α the infinitesimals of fractional derivatives, are given by prolongation formula i.e. [32,36],

    ζ1α=Dαt(η1)+ξ2Dαt(ux)Dαt(ξ2ux)+Dαt(Dt(ξ1)u)Dα+1t(ξ1u)+ξ1Dα+1t(u),ζ2α=Dαt(η2)+ξ2Dαt(vx)Dαt(ξ2vx)+Dαt(Dt(ξ1)v)Dα+1t(ξ1v)+ξ1Dα+1t(v) (2.10)

    where Dt denotes the total derivative. Moreover, in order to conserve the structure of the fractional derivative operator, the following invariance condition is also required

    ξ1(t,x,u)|t=0=0.

    Finally, the coefficients of the determining equation depend on all derivatives of variable u, v, Dαtu and Dαtv.

    For the determination of the Lie point symmetries of (1.1), the authors apply an algorithm that has been implemented in the MAPLE package FracSym [37,38]; this algorithm uses some routines of the MAPLE symmetry packages DESOLVII [37] and ASP [38]; the routines in FracSym automate the method of finding symmetries for FDEs as proposed in [31,32,33,34]; these are the first routines to automate the search of symmetries for FDEs in MAPLE. A limit of the algorithm FracSym is that only the fractional derivatives of Riemann-Liouville type are considered. Moreover, it is important to note that the generator of the dependent variables u and v are assumed linear in u and v respectively, as usually done in the literature [33,34,35,36,37,38,39,40].

    The extend Lie symmetries method leads to get the following infinitesimals for Eq (1.1)

    ξ1=4a3t,ξ2=a0+2a3αx,η1=χ1(t,x)+(a1+2a3(α1))u,η2=χ2(t,x)+a2v, (3.1)

    where the function χ1=χ1(t,x) and χ2=χ2(t,x) satisfy the constraints

    2a3(2ttf+αxxf+(α1)uuf+(1+α)f)+αtχ1k1xxχ1+a0xf+(χ1+a1u)uf+(χ2+a2v)vfa1f=0,2a3(2ttg+αxxg+(α1)uug+2g)+αtχ2k2xxχ2+a0xg+(χ1+a1u)ug+(χ2+a2v)vga2g=0, (3.2)

    that link the arbitrary functions χ1=χ1(t,x) and χ2=χ2(t,x) to the nonlinear reaction terms f=f(t,x,u,v) and g=g(t,x,u,v), where ai, i=0,,3, are the arbitrary parameters.

    In the following, the stretching symmetry (i.e. a3) is neglected; then the constraints (3.2) reduce to

    αtχ1k1xxχ1+a0xf+(χ1+a1u)uf+(χ2+a2v)vfa1f=0αtχ2k2xxχ2+a0xg+(χ1+a1u)ug+(χ2+a2v)vga2g=0 (3.3)

    and the infinitesimals read

    ξ1=0,ξ2=a0,η1=χ1+a1u,η2=χ2+a2v. (3.4)

    We obtain the transformation

    T=t,U=u(t,x)ea1xea1xχ1dx,V=v(t,x)ea2xea2xχ2dx , (3.5)

    where we choose a0=1. By integrating the constraints (3.3) with respect to x, and using the properties of the Riemann-Liouville fractional derivatives, the following forms of f and g are determined

    f=ea1x(ϕ1ea1x(αtχ1k1xxχ1)dx) (3.6)
    g=ea2x(ϕ2ea2x(αtχ2k2xxχ2)dx) (3.7)

    where ϕ1=ϕ1(T,U,V) and ϕ2=ϕ2(T,U,V) are arbitrary functions of their arguments.

    When the transformation (3.5) and the previous forms of f and g are inserted into the system (1.1), it is reduced into the following system of fractional nonlinear ordinary differential equations

    DαTU(T)k1a21U(T)+ϕ1(T,U,V)=0 (3.8)
    DαTV(T)k2a22V(T)+ϕ2(T,U,V)=0. (3.9)

    The choice of the arbitrary functions ϕ1 and ϕ2 characterize the solutions of the Eqs (3.8) and (3.9) that, with a suitable choice of χ1 and χ2, specialize the source terms f and g and, then, the classes of solutions given by (3.5).

    In order to perform some example of solutions of the system under study, we decide to choose ϕ1 and ϕ2 as nonlinear functions of two field variables U and V

    ϕ1=c1U+c0UV+h1(T) (3.10)
    ϕ2=c2V+c0UV+h2(T), (3.11)

    that lead to get a reaction-diffusion model describes the interaction between two filed variables. The functions ϕ1 and ϕ2 characterize the source terms (3.6) and (3.7) of mathematical models that can be used in order to describe several natural phenomena where the fractional order temporal variation of the field variables involves both the reaction and diffusion processes, simultaneously. They have been applied in several contexts such as chemical reactions [41], genes behaviour [42], populations evolution [43], biological pattern formation [44], epidemics [45] and computer virus spreading [46]. Moreover, they are strictly valid to describe spatially distributed chemical and biochemical system, and can also be applied with considerable success to model non-molecular ensembles of interacting and diffusing objects [47]. The functions h1=h1(T) and h2=h2(T) are arbitrary ones of the time variable.

    By these assumptions, the FODEs (3.8) and (3.9) read

    DαTU(T)+(c1k1a21)U(T)+c0U(T)V(T)+h1(T)=0 (3.12)
    DαTV(T)+(c2k2a22)V(T)+c0U(T)V(T)+h2(T)=0. (3.13)

    The system (3.12)–(3.13) reduces to the following FODE

    DαT(V(T)U(T))+(c2k2a22)(V(T)U(T))+h2(T)h1(T)=0 (3.14)

    if we impose that

    c1k1a21=c2k2a22.

    Under non-vanishing initial condition

    limT0Dα1T(V(T)U(T))=b0, (3.15)

    and by the Laplace transform, it demonstrates that the FODE (3.14) admits the following exact solution (see [18] for more details)

    V(T)=U(T)+b0Tα1Eα,α((k1a21c1)Tα)T0(TS)α1Eα,α((k1a21c1)(TS)α)(h2(S)h1(S))dS (3.16)

    where Eα,α(t)=k=0tkΓ(α(k+1)) is the Mittag-Leffler function. Substituting in (3.12), we get the following FODE

    DαTU(T)+c0U(T)2+((c1k1a21)+c0b0Tα1Eα,α((k1a21c1)Tα)c0T0(TS)α1Eα,α((k1a21c1)(TS)α)(h2(S)h1(S))dS)U(T)+h1(T)=0. (3.17)

    By solving the (3.17), we find the solution U(T) that leads to compute the solution V(T) by the (3.16) and, then, by the inverse transformations (3.5), the solutions of the proposed system of FPDEs (1.1) with source terms (3.6) and (3.7) characterized by (3.10) and (3.11). In general, Eq (3.17) could be not analytically resolvable, except for suitable choices of parameters and arbitrary functions, then numerical methods are proposed in order to compute the approximation of the exact solutions, as shown in the following Sections.

    Starting from Eq (3.17) and setting c1=k1a21 and b0=0, we get the following FODE

    DαTU(T)+c0U2(T)c0Γ(α)(T0(TS)α1(h2(S)h1(S))dS)U(T)+h1(T)=0. (3.18)

    Suitable choices of the functions h1(T) and h2(T), involved in (3.18), are proposed with the aim to perform examples of mathematical models of interest in many fields of the applied sciences.

    Example 1: By following choice of the functions h1(T) and h2(T)

    h1(T)=h0c0(1Γ(α)+1)eλT,h2(T)=h0c0(1Γ(α)1)eλT,

    with h0 and λ arbitrary constants, the Eq (3.18) reads

    DαTU(T)+c0U2(T)2h0Γ(α)Γ(α,λT)λαΓ(α)2eλTU(T)h0c0(1Γ(α)+1)eλT=0. (3.19)

    Computed U(T), the solution V(T), given in (3.16), reads

    V(T)=U(T)2h0c0Γ(α)Γ(α,λT)λαΓ(α)2eλT. (3.20)

    Finally, (3.10) and (3.11) assume the following form

    ϕ1(T,U,V)=k1a21U+c0UVh0c0(1Γ(α)+1)eλTϕ2(T,U,V)=k2a22V+c0UV+h0c0(1Γ(α)1)eλT.

    Found U(T) and V(T), by inverse transformations (3.5), we get the solutions u(t,x) and v(t,x) of target model (1.1) with source terms obtained by inserting ϕ1 and ϕ2 in (3.6) and (3.7).

    Remark 1: For α=1, we are able to solve the FODE (3.19) that assumes the following form

    U(T)+c0U2(T)+2h0λ(1eλT)U(T)2h0c0eλT=0,

    whose exact solution, when U(0)=0, is

    U(T)=2h0eλT1c0λ (3.21)

    and substituting in (3.16), we obtain

    V(T)=0.

    By inverse transformation (2.3), assumed χ1(x,t)=χ2(x,t)=0, we get the solutions

    u(t,x)=2h0ea1xeλt1c0λ,v(t,x)=0,

    of the system of PDEs with the source terms

    f(t,x,u,v)=c1u+c0ea2xuv2h0c0eλt+a1x,g(t,x,u,v)=c2v+c0ea1xuv.

    The exact solution of the model with α=1 is considered in the following numerical examples as a reference for testing the qualitative behavior of the numerical solution of the model (3.19) for values of the α parameter increasing towards 1.

    Example 2: By following choice of the functions h1(T) and h2(T)

    h1(T)=h0Γ(α1)Tα+λ24c0eλT(eλT+21Γ(α)Γ(α)),
    h2(T)=h0Γ(α1)Tα+λ24c0eλT(eλT21+Γ(α)Γ(α)),

    with h0 and λ arbitrary constants, the Eq (3.18) reads

    DαTU(T)+c0U2(T)+λ2αΓ(α)Γ(α,λt)Γ(α)2eλtU(T)+h0Γ(α1)Tα+λ24c0eλt(eλt+21Γ(α)Γ(α))=0. (3.22)

    Computed the solution U(T), the solution V(T), given in (3.16), reads

    V(T)=U(T)+λ2αΓ(α)Γ(α,λt)c0Γ(α)2eλt (3.23)

    and (3.10) and (3.11) assume the following form

    ϕ1(T,U,V)=k1a21U+c0UV+h0Γ(α1)Tα+λ24c0eλT(eλT+21Γ(α)Γ(α))ϕ2(T,U,V)=k2a22V+c0UV+h0Γ(α1)Tα+λ24c0eλT(eλT21+Γ(α)Γ(α)).

    Found U(T) and V(T), by inverse transformations (3.5), we get the solutions u(t,x) and v(t,x) of target model (1.1) with source terms obtained by inserting ϕ1 and ϕ2 in (3.6) and (3.7).

    Remark 2: For α=1, the FODE (3.22) reduces to

    U(T)+c0U2(T)+λ(eλt1)U(T)+λ24c0e2λt=0,

    whose the exact solution, when U(0)=b1, is

    U(T)=λeλT2c0(2(2b1c0+λ)(2b1c0+λ)eλT2b1c0+λ1). (3.24)

    Substituting in (3.16), it has

    V(T)=λ2c0(2(2b1c0λ)(2b1c0+λ)eλT2b1c0+λ+eλT),

    and, by the inverse transformation, we obtain the solutions

    u(t,x)=λeλt+a1x2c0(2(2b1c0+λ)(2b1c0+λ)eλt2b1c0+λ1),
    v(t,x)=ea2xλ2c0(2(2b1c0λ)(2b1c0+λ)eλt2b1c0+λ+eλt),

    of the system of PDEs with the following source terms

    f(t,x,u,v)=k1a21u+c0ea2xuv+λ24c0e2λt+a1x,g(t,x,u,v)=k2a22v+c0ea1xuv+λ24c0e2λt+a2x.

    The exact solution of the model with α=1 is considered in the following numerical examples as a reference for testing the qualitative behavior of the numerical solution of the model (3.22) for values of the α parameter increasing towards 1.

    In this Section, we find the numerical solution of the system of FPDEs (1.1) computed by solving the following FODE

    DαTU(T)+c0U2(T)c0Γ(α)(T0(TS)α1(h2(S)h1(S))dS)U(T)+h1(T)=0,

    obtained in the previous Section by the suitable choice of the parameters. Computed the numerical solution U(T), we obtain the solution V(T) by (3.16) and, then the solutions u(t,x) and v(t,x) of the target model (1.1) by the inverse transformations (3.5).

    We introduce the α-order Caputo fractional derivative of the solution U(T)

    CDαTU(T)=1Γ(1α)T0(TS)αU(S)dS

    and its connection with the α-order Riemann-Liouville fractional derivative

    CDαTU(T)=DαT(U(T)U(0)),

    with U(0) initial condition. We remark that the Caputo definition of the fractional derivative allows to define a FIVP whose the initial conditions are given in terms of the field variable and its integer order derivatives in agreement with the clear physical meaning of most of the processes arising in the real world. In terms of the Caputo derivative, the following fractional initial value problem (FIVP) is obtained

    CDαTU(T)=F(T,U),U(0)=U0, (4.1)

    where

    F(T,U)=c0U2(T)+(c0Γ(α)T0(TS)α1(h2(S)h1(S))dS)U(T)h1(T)U0Γ(α1)Tα.

    In order to numerically solve the FIVP (4.1), the classical implicit trapezoidal method (PI2 Im) is used. We built a computational uniform mesh of grid points denoted by Tn, with Tn=T0+nΔT and integration step sizes ΔT and N positive integer. We denote by Un the numerical approximation provided by the numerical method of the exact solution U(Tn) at the mesh points Tn, for n=0,,N. The numerical method reads

    Un+1=U0+1Γ(α)(β0F(T0,U0)+n+1k=1βkF(Tk,Uk)), (4.2)

    where the coefficient values βk, for k=0,,n+1, are computed as follows

    β0=1α(α+1)(Tn+1)α((T1T0)(α+1)Tn+1)+(Tn+1T1)α+1T1T0,βk=1α(α+1)×k=1,,n,×((Tn+1Tk1)α+1(Tn+1Tk)α+1TkTk1(Tn+1Tk)α+1(Tn+1Tk+1)α+1Tk+1Tk),βn+1=1α(α+1)(Tn+1Tn)α .

    The convergence order of the scheme is O((ΔT)min(1+α,2)). Note that, the convergence order of the trapezoidal method usually is 1+α when 0<α<1, and only when α>1 or when the solution is sufficiently smooth we obtain the reached order two [48]. In general, the numerical method (4.2) leads to obtain a nonlinear equation for whose resolution a root-finding solver is needed. The classical Newton method is proposed.

    In the follows, we report some test problems with the aim to validate the proposed approach. Note that the proposed mathematical models represent a wide class of mathematical ones that can be used in order to describe several natural phenomena arising in many field of the applied sciences. Different simulation parameters, initial conditions and fractional α parameter values are considered as the input features of the proposed models. The exact solutions, reported in Remarks 1 and 2, are considered as references for testing qualitative behavior of the numerical solutions of the models (3.19) and (3.22) for values of the α parameter increasing towards 1. All numerical simulations are performed on Intel Core i7 by using Matlab 2020 software.

    Example 1: In this example, we recall the FIVP (3.19), obtained in the previous Section, and we rewrite it in term of the Caputo fractional derivative choosing U0=0

    CDαTU(T)+c0U2(T)2h0Γ(α)Γ(α,λT)λαΓ(α)2eλTU(T)h0c0(1Γ(α)+1)eλT=0,U(0)=0. (4.3)

    We set the parameters values as follows: c0=5, h0=1 and λ=1 and choose a computational domain [0,Tmax] with Tmax=10 and N=100 grid points. In Figure 1, we report the numerical results obtained for different values of the α parameter: in the left frame, the numerical solution Un obtained by solving the FIVP (4.3) by using the PI2 Im numerical method; in the right frame, the numerical solution Vn (3.20) obtained by applying the (3.16). The black lines represent the exact solution U(T) given by (3.21) of the model with α=1 and the exact solution V(T) computed by the (3.16). They are reported in order to test the qualitative behavior of the numerical solutions as the α parameter increases towards 1. The solutions reveal a similar behaviour: Both starts from the value 0, both increase at the beginning of the integration process and, after, decreases as the time evolves. Note that, as the value of α increases, the solution U(T) increases unlike the solution V(T) decreases.

    Figure 1.  Numerical solutions for different values of α. Left frame: Numerical solution Un of the FIVP (4.3). Right frame: Numerical solution Vn.

    In Figure 2, we report the numerical solutions unj and vnj, approximations of the exact solutions obtained by the inverse transformations (3.5),

    u(t,x)=U(t)ea1x,v(t,x)=V(t)ea2x,
    Figure 2.  Numerical solutions of the system of FPDE (1.1) for α=0.5. Left frame: Numerical solution unj. Right frame: Numerical solution vnj.

    solutions of the model (1.1) with source terms

    f(t,x,u,v)=k1a21u+c0ea2xuvh0c0(1Γ(α)+1)eλt+a1x,g(t,x,u,v)=k2a22v+c0ea1xuv+h0c0(1Γ(α)1)eλt+a2x,

    computed by setting k1=k2=0.5, for α=0.5, with c1=k1a21 and c2=k2a22 with a1=a2=1, on a computational domain [0,Tmax]×[0,Xmax] with Tmax=10, Xmax=1 and N=J=100 grid points.

    Example 2: In this example, we recall the FIVP (3.22), obtained in the previous Section, and we rewrite it in term of the Caputo fractional derivative choosing U0=b1=h0

    CDαTU(T)+c0U2(T)+λ2αΓ(α)Γ(α,λt)Γ(α)2eλtU(T)+λ24c0eλt(eλt+21Γ(α)Γ(α))=0,U(0)=U0. (4.4)

    We choose the initial condition U0=1 and set the parameters values as follows: c0=0.5 and λ=1 and consider a computational domain [0,Tmax] with Tmax=10 and N=100 grid points. In Figure 3, we report the numerical results obtained for different values of α parameter: in the left frame, the numerical solution Un obtained by solving the FIVP (4.4) by using the PI2 Im numerical method; in the right frame, the numerical solution Vn (3.23) obtained by the (3.16). The black lines represent the exact solution U(T) given by (3.24) of the model with α=1 and the exact solution V(T) computed by the (3.16). They are reported in order to test the qualitative behavior of the numerical solutions as the α parameter increases towards 1. The solutions reveal a different behaviour: both starts from the value 1 but, the solution U(T) immediately decreases as the time evolves, unlike the solution V(T) increases at the beginning of the integration process and, after, decreases. As the value of α increases, the solution U(T) decreases unlike the solution V(T) increases.

    Figure 3.  Numerical solutions for different values of α. Left frame: Numerical solution Un of the FIVP (4.4). Right frame: Numerical solution Vn.

    In Figure 4, we report the numerical solutions unj and vnj, approximations of the exact solutions obtained by the inverse transformations (3.5), u(t,x)=U(t)ea1x and v(t,x)=V(t)ea2x, solutions of the model (1.1) with source terms

    f(t,x,u,v)=k1a21u+c0ea2xuv+h0Γ(α1)Tα+λ24c0eλt+a1x(eλt+21Γ(α)Γ(α)),g(t,x,u,v)=k2a22v+c0ea1xuv+h0Γ(α1)Tα+λ24c0eλt+a2x(eλt21+Γ(α)Γ(α)), (4.5)
    Figure 4.  Numerical solutions of the system of FPDE (1.1) for α=0.5. Left frame: Numerical solution unj. Right frame: Numerical solution vnj.

    computed by setting k1=k2=0.5, for α=0.5, with c1=k1a21 and c2=k2a22 with a1=a2=1, on a computational domain [0,Tmax]×[0,Xmax] with Tmax=10, Xmax=1 and N=J=100 grid points.

    For the second test, we set the parameters values as before and c0=5 in order to provide an example of how this parameter, that represent the rate of interaction between the two solutions, affects their behaviour as the time evolves. In Figure 5, we report the numerical results obtained for different values of α parameter: in the left frame, the numerical solution Un obtained by solving the FIVP (4.4) by using the PI2 Im numerical method, in the right frame, the numerical solution Vn (3.23) obtained by the (3.16). The black lines represent the exact solution U(T) given by (3.24) of the model with α=1 and the exact solution V(T) computed by the (3.16). The solutions reveal a very similar behaviour: both starts from the value 1 and both immediately decrease as the time evolves. As the values of α increases, the solution U(T) decreases. A slightly different behaviour appears for the solution V(T) in the last part of the domain.

    Figure 5.  Numerical solutions for different values of α. Left frame: Numerical solution Un of the FIVP (4.4). Right frame: Numerical solution Vn.

    In Figure 6, we report the numerical solutions unj and vnj, approximations of the exact solutions obtained by the inverse transformations (3.5), solutions of the model (1.1) with source terms given by (4.5), computed by setting k1=k2=0.5, for α=0.5, with c1=k1a21 and c2=k2a22 with a1=a2=1, on a computational domain [0,Tmax]×[0,Xmax] with Tmax=10, Xmax=1 and N=J=100 grid points.

    Figure 6.  Numerical solutions of the system of FPDE (1.1) for α=0.5. Left frame: Numerical solution unj. Right frame: Numerical solution vnj.

    This paper deals with the solutions of coupled nonlinear time-fractional reaction–diffusion equations, where the fractional derivatives are defined in terms of the Riemann-Liouville ones. We apply the strategy that combines an analytical and numerical approach in order to solve the problem under study, used recently in [25,26,27,28,29,30]. By Lie symmetries, the assigned system of two FPDEs is reduced into a system of two nonlinear FODEs and, under suitable assumptions of the involved parameters and functions, it is reduced into a nonlinear FODE. By introducing the Caputo derivative, we are able to properly handle the obtained FIVP and, then, by the trapezoidal numerical method, to find the solutions. Finally, the inverse transformations deal to compute the solutions of the original model. We remark that the Caputo definition of the fractional derivative allows to define a FIVP whose the initial condition is given in terms of the field variable in agreement with most of the processes with a clear physical meaning. Numerical examples are performed in order to show how the proposed approach good works with nonlinear systems of FPDEs confirming the clear advantage, from numerical point of view, to use a combined strategy: we numerically solve only a FODE whose solution allows to compute the solution of the original model of FPDEs. We remark that the presented test problems, performed in order to confirm the efficiency of the proposed approach, represent a wide class of mathematical models that can be used in order to describe several natural phenomena arising in many field of the applied sciences.

    A. J. acknowledges the financial support by G.N.C.S. of I.N.d.A.M.

    M.P.S. acknowledges the financial support by G.N.F.M. of I.N.d.A.M.

    The authors declare no conflict of interest.



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