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Research article

Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials

  • Received: 11 April 2022 Revised: 11 June 2022 Accepted: 16 June 2022 Published: 23 June 2022
  • MSC : 65N35, 34A08

  • The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.

    Citation: Yumei Chen, Jiajie Zhang, Chao Pan. Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials[J]. AIMS Mathematics, 2022, 7(8): 15612-15632. doi: 10.3934/math.2022855

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  • The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.



    With the development of fractional derivative, operators without singular kernel function has become a research topic for the fact that it can better describe nonlocal dynamics systems. Some new classes of them are available including the Caputo-Fabrizio derivative [1], the Atangana-Baleanu derivative [2], etc. See [3,4,5,6,7,8,9] and references therein for further details. On the other hand, it is worth noting that the fractional calculus has extended to study the variable-order (VO) models [10,11], which are generalizations of fixed-order fractional derivatives. For example, some VO fractional differential diffusion processes can better simulate the temperature change than integer order classical models.

    Many numerical treatments have recently been proposed to approximate VO fractional differential equations. VO fractional derivatives are global operators in essence, it is more convenient to use global technology such as spectral methods to deal with VO operators[12]. Spectral methods can obtain the desired solutions with a small degree of freedom, which makes improved accuracy with a significant reduction in the computational cost[13]. Here, we focus on the polynomial-based spectral collocation methods. Lagrange polynomials were applied to simulate two chaotic practice problems modeled by VO fractional differential equations [14]. A collocation technique based on a reproducing kernel function was developed for VO fractional initial value problems and terminal value problems [15]. Shifted Jacobi polynomials were used to approximate VO time fractional Emder-Fowler equation [16], two-dimensional VO time fractional cable equations[17]. Gegenbauer polynomials are a special of Jacobi polynomials. As a significant kind of orthogonal polynomials, Gegenbauer polynomials are the generalization of Legendre polynomials and Chebyshev polynomials, which have been widely used in mathematical physics, engineering technology, scientific computing and other fields. Shifted Gegenbauer operational matrices jointly with the Tau method were developed for solving fractional differential equations [18]. Shifted Gegenbauer polynomials were applied to solve time-fractional delay differential equations [19]. Fractional optimal control problems were solved via Gegenbauer cardinal functions [20]. A numerical scheme in [21] was proposed to solve the nonlinear VO fractional Schr¨odinger equation by using the shifted Legendre polynomials.

    Diffusion process is one of the most famous processes in nature, such as the numerical simulation of urban air quality [22], transport process of pollutants in groundwater [23], numerical determination the pollution sources in a river [24], transport process of bimolecular reaction in porous media [25]. The fractional form of diffusion equation can model the random collision of solute molecules with fluid molecules causes diffusion and produces fluxes from high concentrated to low concentration regions. Numerical solutions of space fractional diffusion equation defined by Caputo derivative were obtained via shifted Gegenbauer polynomials [26]. A compact implicit difference scheme was proposed to approximate time-fractional diffusion-wave equation[27]. An operational matrix was derived for multidimensional VO time fractional anomalous diffusion equations based on the shifted Chebyshev collocation methods[28]. Gegenbauer spectral method was used to approximate convection-diffusion equation with time fractional derivative [29]. By using the shifted Chebyshev polynomial of the second kind as the cardinal function, an operational matrix scheme was given in [30] for approximating VO time fractional nonlinear reaction-diffusion equation. A homotopy perturbation method was applied for time fractional advection-reaction-diffusion equation with Liouville-Caputo derivative [31]. A numerical method was proposed to approximate time fractional advection-reaction-diffusion equation by using shifted Legendre polynomials [32].

    Given the following VO time-fractional derivative model for advection-reaction-diffusion equation

    {θ(σ,τ)u(σ,τ)τθ(σ,τ)=κ2u(σ,τ)σ2μu(σ,τ)σ+δ(σ,τ,u(σ,τ))+f(σ,τ),(σ,τ)[0,X]×[0,T],u(σ,0)=g(σ),u(0,τ)=h1(τ),u(X,τ)=h2(τ). (1.1)

    Here, σ and τ represent spatial and time nodes respectively, u(σ,τ) is concentration of the solute in fluid at finite distance, parameters κ and μ are the constant coefficients on R. Variable order θ(σ,τ)C(0,1), δ(σ,τ,u(σ,τ)) and f(σ,τ) are continuous nonlinear source terms.

    This paper aims to investigate numerically problem (1.1) with Atangana-Baleanu-Caputo defined time-fractional derivative without singular kernel [2]. For solving Eq (1.1), the Gegenbauer spectral method will be used. Firstly, the considered model is approximated by the shifted Gegenbauer polynomials (SGPs) with undetermined coefficients. A nonlinear algebraic equations are derived via the SGPs operational matrices by choosing suitable collocation points. Finally, the desired problem is transformed into solving the obtained nonlinear system.

    This work proceeds as follows. Preliminaries on variable-order fractional derivatives with Mittage-Leffler kernel are provided in Section 2. Review of some useful properties of the SGPs is presented in Section 3. The operational matrices of cardinal functions for the SGPs are obtained in Section 4. Based on the Gegenbauer spectral method, the proposed numerical scheme is formulated in Section 5. A few illustrative examples with different initial conditions and boundary conditions are shown in Section 6. The last section is devoted to a brief conclusion.

    This section devotes to some useful preliminaries of Atangana-Baleanu-Caputo (ABC) derivatives with fractional order.

    To begin with we introduce the Mittag-Leffler function Eα1,α2(τ) that will commonly be encountered in fractional calculus as follows [33].

    Eα1,α2(τ)=j=0τjΓ(jα1+α2),τC,α1,α2R+. (2.1)

    Notation Eα1(τ) is used for α2=1. Where Γ() is the Gamma function and Γ(x)=(x1)! if xZ+. The relationship between the Gamma function and the Beta function is

    B(p,q)=10tp1(1t)q1dt=Γ(p)Γ(q)Γ(p+q),p,qR+. (2.2)

    The definition of the ABC fractional derivative for the order ˉθ is given by

    ˉθu(σ,τ)τˉθ=¯C(ˉθ)1ˉθτ0u(σ,s)sEˉθ(ˉθ(τs)ˉθ1ˉθ)ds,0<ˉθ<1, (2.3)

    where C(ˉθ) is a normalization function taking the form

    ¯C(ˉθ)=1ˉθ+ˉθΓ(ˉθ).

    The definition of the ABC fractional derivative in the sense of variable order θ(σ,τ) is given by

    θ(σ,τ)u(σ,τ)τθ(σ,τ)=C(θ(σ,τ))1θ(σ,τ)τ0u(σ,s)sEθ(σ,τ)(θ(σ,τ)(τs)θ(σ,τ)1θ(σ,τ))ds, (2.4)

    where C(θ(σ,τ))=1θ(σ,τ)+θ(σ,τ)Γ(θ(σ,τ)) is a normalization function, and u(σ,τ)C1([0,X]×[0,T]) is a real function.

    Corollary 2.1. ([34]) Let kN{0}, there is

    θ(σ,τ)τkτθ(σ,τ)={0,k=0,C(θ(σ,τ))k!τk1θ(σ,τ)Eθ(σ,τ),k+1(θ(σ,τ)τθ(σ,τ)1θ(σ,τ)),k=1,2,. (2.5)

    Given a closed interval [0,T], we use the shifted Gegenbauer polynomials (SGPs) of order m as the cardinal function [35], which is denoted by GλT,m(τ). It is defined by

    GλT,m(τ)=Γ(λ+0.5)Γ(2λ)mk=0(1)k+mΓ(2λ+k+m)k!(mk)!Γ(k+λ+0.5)(τT)k,mZ+,λ is a constant, (3.1)

    with the orthogonal property

    T0GλT,m(τ)GλT,n(τ)wλT(τ)dτ=hλT,mδmn. (3.2)

    In Eq (3.2), the weight function is wλT(τ)=(Tτ)λ0.5τλ0.5 and

    hλT,m=214λT2λΓ(m+2λ)π(m+λ)m!Γ2(λ). (3.3)

    If u(x)L2[0,T], then it can be approximated by a polynomial up of order p in terms of the SGPs

    u(τ)up(τ)=pm=0cmGλT,m(τ), (3.4)

    with the coefficients cm are determined by the orthogonality condition

    cm=1hλT,mT0wλT(τ)u(τ)GλT,m(τ)dτ,m=0,1,,p. (3.5)

    Let

    C=[c0,c1,,cp]T,ΨT,p(τ)=[GλT,0(τ),GλT,1(τ),,GλT,p(τ)], (3.6)

    then Eq (3.4) is expressed in matrix form as

    u(τ)up(τ)CTΨT,p(τ). (3.7)

    Likewise, the function u(σ,τ)L2([0,T]×[0,T]) with two variables can be approximated by the double SGPs of degrees q and p as

    u(σ,τ)uq,p(σ,τ)=qm=0pn=0umnGλX,m(σ)GλT,n(τ)ΨX,q(σ)TUΨT,p(τ), (3.8)

    where U=[umn] is a (q+1)×(p+1) unknown coefficients matrix with

    umn=1h(a,b)X,m1h(a,b)T,n1X0T0wλX(σ)wλT(τ)u(σ,τ)GλX,m1(σ)GλT,n1(τ)dτdσ,

    for m=1,2,,q+1,n=1,2,,p+1.

    In this section, some novel operator matrices about the SGPs are derived. Let

    Φp(τ)=[φp,0(τ),φp,1(τ),,φp,p(τ)]T,withφp,m(τ)=τm(m=0,1,,p). (4.1)

    Lemma 4.1. The basis vector ΨT,p(τ) in (3.6) satisfies

    Φp(τ)=ΘT,pΨT,p(τ), (4.2)

    where the transform matrix ΘT,p=[ξT,mn] is an (p+1)-order square matrix with

    ξT,mn=24λ1Tm3(n+λ1)(n1)!Γ(λ+0.5)Γ2(λ)πΓ(n+2λ1)×n1k=0(1)k+n1Γ(2λ+k+n1)Γ(λ+k+m0.5)k!(nk1)!Γ(λ+k+0.5)Γ(2λ+k+m),1m,np+1.

    Proof. By expressing the element φp,ˆm(τ)(ˆm=0,1,,p) of Φp(τ) in terms of the SGPs, we have

    φp,ˆm(τ)=pˆn=0ˆξT,ˆmˆnGλT,ˆn(τ)ˆΘTT,ˆmΨT,p(τ), (4.3)

    where

    ˆΘTT,ˆm=[ˆξT,ˆm0,ˆξT,ˆm1,,ˆξT,ˆmp].

    Eq (3.5) yields

    ˆξT,ˆmˆn=1hλT,ˆnT0wλT(τ)φp,ˆm(τ)GλT,ˆn(τ)dτ,0ˆnp. (4.4)

    By recalling that wλT(τ)=(Tτ)λ0.5τλ0.5 and using Eqs (3.1), (3.3) and (4.1), we get

    ˆξT,ˆmˆn=(ˆn+λ)ˆn!Γ2(λ)214λT2λΓ(ˆn+2λ)πT0(Tτ)λ0.5τλ0.5τˆmdτ×Γ(λ+0.5)Γ(2λ)ˆnk=0(1)k+ˆnΓ(2λ+k+ˆn)k!(ˆnk)!Γ(k+λ+0.5)(τT)k=(ˆn+λ)ˆn!Γ2(λ)214λT2λΓ(ˆn+2λ)πˆnk=0(1)k+ˆnΓ(2λ+k+ˆn)k!(ˆnk)!Γ(k+λ+0.5)Tk Iτ, (4.5)

    where Iτ=T0(Tτ)λ0.5τλ+ˆm+k0.5dτ.

    Let τ=Tx and from (2.2), it is

    Iτ=T2λ+ˆm+k210xλ+ˆm+k0.5(1x)λ0.5dx=T2λ+ˆm+k2B(λ+ˆm+k+0.5,λ+0.5)=T2λ+ˆm+k2Γ(λ+ˆm+k+0.5)Γ(λ+0.5)(2λ+ˆm+k+1). (4.6)

    Substituting Eq (4.6) in (4.5) gives

    ˆξT,ˆmˆn=Tˆm224λ1ˆn!(ˆn+λ)Γ2(λ)Γ(λ+0.5)πΓ(ˆn+2λ)׈nk=0(1)k+ˆnΓ(k+2λ+ˆn)Γ(k+λ+ˆm0.5)k!(ˆnk1)!Γ(k+λ+0.5)Γ(k+2λ+ˆm). (4.7)

    Finally, the desired result can be obtained by changing the indices m=ˆm+1,n=ˆn+1 and replacing ˆΘTT,m1 and ˆξT,(m1)(n1) by ΘTT,m and ξT,mn, respectively.

    We give an example to illustrate for λ=1, T=1 and m=4.

    Θ1,4=(167/1480000167//296167/592000707/2005167/592167/236800643/2605643/2605437/4131135/76570453/2447373/1763437/3672270/7657135/30628).

    Lemma 4.2. The derivatives of the monomial function vector Φp(τ) in (4.1) satisfy

    dΦp(τ)dτ=D(1)pΦp(τ), (4.8)

    and

    drΦp(τ)dτr=D(r)pΦp(τ), (4.9)

    in which D(r)p represents the rth power of the strictly lower triangular matrix D(1)p with

    [D(1)p]mn={0,m=1,1np+1,m1,2mp+1,1np+1,mn=1.

    Proof. The result of the lemma can be implied by the definitions of vectors Φp(τ) and D(1)p.

    Lemma 4.3. Φp(τ) denotes the monomial function vector in (4.1), then

    θ(σ,τ)Φp(τ)τθ(σ,τ)=Z(θ(σ,τ))pΦp(τ), (4.10)

    with

    [Z(θ(σ,τ))p]mn={0,m=1,1np+1,(m1)!C(θ(σ,τ))1θ(σ,τ)Eθ(σ,τ),m(θ(σ,τ)τθ(σ,τ)1θ(σ,τ)),2mp+1,1np+1,m=n.

    Proof. It is easy to complete the proof by using Corollary 2.1.

    Theorem 4.1. The derivative of SGPs vector ΨT,p(τ) in (3.6) satisfies

    dΨT,p(τ)dτ=D(1)T,pΨT,p(τ), (4.11)

    where D(1)T,p represents the (p+1)-order operational matrix of the SGPs given by

    D(1)T,p=Θ1T,pD(1)pΘT,p.

    Generally, the r-order derivative of ΨT,p(τ) satisfies

    drΨT,p(τ)dτr=D(r)T,pΨT,p(τ), (4.12)

    in which D(r)T,p represents the rth power of the matrix D(1)T,p.

    Proof. It can be proved simply from Eq (4.1) and Lemma 4.2.

    Theorem 4.2. The SGPs vector ΨT,p(τ) in (3.6) satisfies

    θ(σ,τ)ΨT,p(τ)τθ(σ,τ)=Z(θ(σ,τ))T,pΨT,p(τ) (4.13)

    with

    Z(θ(σ,τ))T,p=Θ1T,pZ(θ(σ,τ))pΘT,p.

    Here the (p+1)-order matrix Z(θ(σ,τ))T,p is the operator matrix of SGPs in the sense of variable order θ(σ,τ).

    Proof. The proof is completed by using Lemma 4.2 and Eq (4.10).

    The unknown solution u(σ,τ) of Eq (1.1) is approximated by the SGPs as follows

    u(σ,τ)uq,p(σ,τ)=qm=0pn=0umnGλX,m(σ)GλT,n(τ)ΨX,q(ξ)TUΨT,p(τ), (5.1)

    where U=[umn](q+1)×(p+1) is an unknown matrix, ΨX,q(ξ) and ΨT,p(τ) represent the vectors mentioned in Eq (3.8).

    Using Eqs (4.11), (4.12) and (5.1), one has

    u(σ,τ)σΨX,q(σ)T(D(1)X,q)TUΨT,p(τ), (5.2)

    and

    2u(σ,τ)σ2ΨX,q(σ)T(D(2)X,q)TUΨT,p(τ). (5.3)

    Moreover, Theorem 4.2 results in

    θ(σ,τ)u(σ,τ)τθ(σ,τ)ΨX,q(ξ)TUZ(θ(σ,τ))T,pΨT,p(τ). (5.4)

    Combining Eqs (5.1)–(5.4) and (1.1), we have

    R(σ,τ)≜=ΨX,q(σ)T[UZ(θ(σ,τ))T,pκ(D(2)X,q)U+λ(D(1)X,q)U]ΨT,p(τ)δ(σ,τ,ΨX,q(σ)TUΨT,p(τ))f(σ,τ). (5.5)

    From the initial and boundary conditions in Eq (1.1) and (5.1), we have

    M1(σ)ΨX,q(σ)TUΨT,p(0)g(σ), (5.6)

    and

    M2(τ)ΨX,q(0)TUΨT,p(τ)h1(τ),M3(τ)ΨX,q(X)TUΨT,p(τ)h2(τ). (5.7)

    Finally, Eqs (5.5)–(5.7) are combined to the following system

    {R(σm,τn)=0,2mq,2np+1,M1(σm)=0,1mq+1,M2(τn)=0,2np+1,M3(τn)=0,2np+1. (5.8)

    For solving the unknown matrix U in the above (q+1)×(p+1) nonlinear algebraic equations, we can choose Gaussian nodes

    σm=X2(1cos((2m1)π2(q+1))),m=1,2,,q+1

    and

    τn=T2(1cos((2n1)π2(p+1))),n=1,2,,p+1.

    Then by substituting U into Eq (5.1), an approximate solution of Eq (1.1) can be obtained.

    The effectiveness of the proposed scheme is shown through some numerical simulations. All computations are carried out in MATLAB software. For i=1,2, let εi=max(|uiEuiN|) be the i-th maximum absolute error (MAE), with uiE and uiN are the analytical solution and numerical solution, respectively. ηi=(qi+1)(pi+1) represents the number of the SGPs used in the i-th approximations. Then the order of the convergence (CO) corresponding to the approximate solutions is defined by

    CO=logη1η2(ε2ε1).

    Example 1. If δ(σ,τ,u(σ,τ))+f(σ,τ)=0, then the model (1.1) reduces to [36]

    {θ(σ,τ)u(σ,τ)τθ(σ,τ)=κ2u(σ,τ)σ2μu(σ,τ)σ,(σ,τ)[0,1]×[0,1],u(σ,0)=exp{(σμ)24κ},u(0,τ)=11+τexp{((1+τ)μ)24κ(1+τ)},u(1,τ)=11+τexp{(1(1+τ)μ)24κ(1+τ)}. (6.1)

    If θ(σ,τ)=1, the solution of Eq (6.1) reads

    u(σ,τ)=11+τexp{(σ(1+τ)μ)24κ(1+τ)}.

    Set κ=0.1,μ=0.25,λ=1 and q=7,p=7. Figure 1 shows the numerical results for u(σ,0.4), u(0.4,τ) for different constant functions θ(σ,τ). Numerical simulations for u(σ,0.6), u(0.6,τ) with some values of θ(σ,τ) are displayed in Figure 2. Comparison of the approximation solution with θ(σ,τ)=0.75+0.15sin(2πστ) and the exact solution with θ(σ,τ)=1 are depicted in Figure 3. It can be seen that the numerical solutions are convergent. Table 1 compares the absolute errors of the method proposed in this paper (abbreviated as SGPs method) with those obtained in [36] for various choices of θ(σ,τ) and q=p=4 at t=0.8. It is observed that we achieved an excellent approximation for the exact solution. In Table 2, the L2 errors are presented for q=p=6 and various values θ(σ,τ) at x=0.6.

    Figure 1.  Approximate solutions for various constant functions θ(σ,τ) when τ=0.4 (left) and σ=0.4 (right) in Example 1.
    Figure 2.  Approximate solutions for various sin functions θ(σ,τ) when τ=0.6 (left) and σ=0.6 (right) in Example 1.
    Figure 3.  Comparison of the numerical solution (θ(σ,τ)=0.75+0.15sin(2πστ), left) and exact solution (θ(σ,τ)=1, right) in Example 1.
    Table 1.  Comparison of the absolute errors between the SGPs method and [36] for various constant functions θ(σ,τ) and q=p=4 at t=0.8 for Example 1.
    x θ(σ,τ)=0.9 θ(σ,τ)=0.97 θ(σ,τ)=0.99
    SGPs method [36] SGPs method [36] SGPs method [36]
    0.0 7.7543e-06 9.1048e-05 7.7543e-06 2.9324e-04 7.7543e-06 2.7498e-04
    0.1 4.1764e-03 2.8503e-03 2.1942e-03 1.4968e-03 7.9833e-04 6.3345e-04
    0.2 4.5583e-03 2.1194e-03 3.0129e-03 1.4223e-03 1.2540e-03 6.0235e-04
    0.3 1.7086e-03 1.4002e-03 2.3432e-03 1.2294e-04 1.0646e-03 2.0999e-05
    0.4 3.2240e-03 6.5033e-03 6.2026e-04 1.8748e-03 4.2900e-04 8.2568e-04
    0.5 8.9240e-03 1.1885e-02 1.5920e-03 3.9751e-03 3.7131e-04 1.6113e-03
    0.6 1.4130e-02 1.6362e-02 3.8215e-03 5.7354e-03 1.1935e-03 2.1911e-03
    0.7 1.7624e-02 1.8878e-02 5.6763e-03 6.8670e-03 2.0213e-03 2.6014e-03
    0.8 1.7986e-02 1.8271e-02 6.6020e-03 7.0116e-03 2.7225e-03 2.8375e-03
    0.9 1.3170e-02 1.2881e-02 5.4550e-03 5.3375e-03 2.6220e-03 2.4489e-03

     | Show Table
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    Table 2.  L2 errors for various choices of θ(σ,τ) and q=p=6 at x=0.6 for Example 1.
    t θ(σ,τ)=0.65 θ(σ,τ)=0.75 θ(σ,τ)=0.85 θ(σ,τ)=0.95
    0.0 1.7024e-04 1.7024e-04 1.7024e-04 1.7024e-04
    0.1 2.1400e-02 1.3067e-02 5.6521e-03 1.6086e-03
    0.2 3.5167e-02 2.4885e-02 1.4133e-02 4.4922e-03
    0.3 4.2944e-02 3.2616e-02 2.0529e-02 6.8666e-03
    0.4 4.6426e-02 3.6283e-02 2.3783e-02 8.2178e-03
    0.5 4.6979e-02 3.7017e-02 2.4574e-02 8.6250e-03
    0.6 4.5540e-02 3.5991e-02 2.3948e-02 8.3791e-03
    0.7 4.2631e-02 3.3851e-02 2.2541e-02 7.7379e-03
    0.8 3.8505e-02 3.0675e-02 2.0401e-02 6.8207e-03
    0.9 3.3413e-02 2.6451e-02 1.7405e-02 5.6513e-03

     | Show Table
    DownLoad: CSV

    Example 2. If κ=1,μ=0 and f(σ,τ)=0, then the model (1.1) becomes [30]

    {θ(σ,τ)u(σ,τ)τθ(σ,τ)=2u(σ,τ)σ2+u(σ,τ)(1u(σ,τ))(u(σ,τ)γ),(σ,τ)[0,20]×[0,1],0<γ<1,u(σ,0)=(1+eσ2)1,u(0,τ)=(1+e12((12γ)2τ))1,u(20,τ)=(1+e12(20+(12γ)2τ))1. (6.2)

    One finds that the solution of Eq (6.2) is u(σ,τ)=(1+e12(σ+(12γ)2τ))1 in the case θ(σ,τ)=1. For λ=0.5,q=9,p=12, the obtained numerical results of u(σ,0.2) and u(3,τ) with different constant functions θ(σ,τ) are shown in Figure 4. Results of some functions θ(σ,τ) with λ=1,q=9,p=9 are shown in Figure 5. Comparison of the numerical solution with λ=1,q=9,p=9, θ(σ,τ)=0.75+0.2sin(στ) and analytical solution with λ=1,q=9,p=9,θ(σ,τ)=1 are depicted in Figure 6. It can be seen that the numerical solutions are convergent.

    Figure 4.  Exact solution and the numerical solutions for some constant functions θ(σ,τ) when τ=0.2 (left) and σ=3 (right) in Example 2.
    Figure 5.  Exact solution and the numerical solutions for various sin functions θ(σ,τ) when τ=0.2 (left) and σ=3 (right) in Example 2.
    Figure 6.  Comparison of the numerical solutions (θ(σ,τ)=0.75+0.2sin(στ), left) and the exact solution (θ(σ,τ)=1, right) in Example 2.

    In order to compare the results with the works in [30], we discuss the following case

    {κ=2,δ(σ,τ,u(σ,τ))=sin(u(σ,τ)),g(σ)=(1+eσ2)1,h1(τ)=(1+e12((12γ)2τ))1,h2(τ)=(1+e12(20+(12γ)2τ))1,(σ,τ)[0,2]×[0,2],f(σ,τ)=(C(θ(σ,τ))3!τ31θ(σ,τ)Eθ(σ,τ),4(θ(σ,τ)τθ(σ,τ)1θ(σ,τ))+2τ3)sin(σ)sin(τ3sin(σ)). (6.3)

    The analytical solution of this case is given by u(σ,τ)=τ3sin(σ). This example has been solved for two different functions θ(σ,τ) by utilizing the presented method (abbreviated as SGPs method) with p=3 and some values of q. In Tables 3 and 4, the MAE errors are listed for the present method (abbreviated as SGPs method) and the method proposed in [30]. It can be seen that the SGPs method is able to achieve smaller errors.

    Table 3.  Comparison of the MAEs of the SGPs method and [30] for θ(σ,τ)=0.35+0.20sin(στ) with q=6 and q=8 for p=3 in (6.3).
    (σ,τ) θ(σ,τ)=0.35+0.2sin(στ)
    q=6 q=8
    SGPs method [15] SGPs method [15]
    (0.4, 0.4) 7.5142e-09 4.0307e-06 2.0855e-10 6.0803e-08
    (0.8, 0.8) 1.3277e-06 2.1531e-05 4.0701e-10 1.1765e-07
    (1.2, 1.2) 1.4326e-06 2.5412e-05 8.6563e-09 1.0783e-07
    (1.6, 1.6) 1.2779e-07 2.6078e-05 2.8258e-08 1.1608e-07
    (2.0, 2.0) 1.5072e-05 0.0000e-00 8.0406e-08 0.0000e-00

     | Show Table
    DownLoad: CSV
    Table 4.  Comparison of the MAEs of the SGPs method and [30] for θ(σ,τ)=0.75+0.20sin(στ) with q=6 and q=8 for p=3 in (6.3).
    (σ,τ) θ(σ,τ)=0.75+0.2sin(στ)
    q=6 q=8
    SGPs method [30] SGPs method [30]
    (0.4, 0.4) 1.0110e-08 2.3337e-06 1.3449e-10 3.2104e-10
    (0.8, 0.8) 1.2079e-06 1.0804e-05 6.3519e-12 4.8010e-08
    (1.2, 1.2) 1.5555e-06 2.8356e-05 8.3785e-09 1.0948e-07
    (1.6, 1.6) 4.4606e-07 3.2703e-05 2.7956e-08 1.2767e-07
    (2.0, 2.0) 1.5071e-05 0.0000e-00 8.0337e-08 0.0000e-00

     | Show Table
    DownLoad: CSV

    Example 3. If κ=μ=1 and f(σ,τ)=0, then the model (1.1) is

    {θ(σ,τ)u(σ,τ)τθ(σ,τ)=2u(σ,τ)σ2u(σ,τ)σ+δ(σ,τ,u(σ,τ)),0σX,0τT,u(σ,0)=eσ,u(0,τ)=eτ,u(X,τ)=eτ+X. (6.4)

    Let δ(σ,τ,u(σ,τ))=u(σ,τ)2u(σ,τ)σ2u(σ,τ)2+u(σ,τ) and θ(σ,τ)=1, then the solution of Eq (6.4) reads u(σ,τ)=eσ+τ. When λ=0.5,X=0.5,T=0.5,q=10,p=8, numerical results of u(σ,0.2) and u(0.2,τ) with constant function θ(σ,τ) are shown in Figure 7. Comparison of the numerical solution with λ=1,X=0.5,T=1,q=9,p=9, θ(σ,τ)=0.75+0.15sin(2πστ) and the analytical solution with θ(σ,τ)=1 are depicted in Figure 8. It can be seen that the numerical solutions are convergent.

    Figure 7.  Exact solution and the numerical solutions for some constants when τ=0.2 (left) and σ=0.2 (right) in Example 3.
    Figure 8.  Comparison of the numerical solution (θ(σ,τ)=0.75+0.15sin(2πστ), left) and exact solution (θ(σ,τ)=1, right) in Example 3.

    Example 4. If κ=μ=1 and δ(σ,τ,u(σ,τ))=2u(σ,τ), then the model (1.1) reads like

    {θ(σ,τ)u(σ,τ)τθ(σ,τ)=2u(σ,τ)σ2u(σ,τ)σ+2u(σ,τ)+f(σ,τ),0σX,0τT,u(σ,0)=sin(σ),u(0,τ)=0,u(X,τ)=sin(X)eτ. (6.5)

    Given

    f(σ,τ)=C(θ(σ,τ))τsin(σ)1θ(σ,τ)k=0(τ)kEθ(σ,τ),k+2(θ(σ,τ)τθ(σ,τ)1θ(σ,τ))sin(σ)eτ+0.5cos(σ)eτ,

    the solution of Eq (6.5) is u(\sigma, \tau) = {\rm{sin(}}\sigma{\rm{)}}{{\rm{e}}^{{\rm{(- \mathsf{ τ})}}}} . Comparison of the numerical solution when X = 0.5, T = 1, \lambda = 1, q = 6, p = 6, \theta (\sigma, \tau) = {\rm{0}}{\rm{.7 + 0}}{\rm{.2sin(}}\sigma \tau {\rm{)}} and the exact solution are depicted in Figure 9. Table 5 displays the results of the MAEs and CO for different parameters in Example 4. It is clear that the SGPs method has a good convergence rate.

    Figure 9.  Comparison of the approximate solution ( \theta (\sigma, \tau) = 0.7+ 0.2\sin (\sigma \tau) , left) and the exact solution (right) in Example 4.
    Table 5.  The MAEs, L_2 errors and CO of the SGPs method with some different parameters in Example 4.
    q p \lambda =0.5 \lambda =1
    L_2 MAEs CO L_2 MAEs CO
    5 5 1.8456e-05 1.1892E-06 - 1.8456e-05 1.1892E-06 -
    6 6 2.4800e-07 5.5076E-08 8.4255 2.4790e-07 5.5079E-08 8.4254
    7 7 2.3590e-08 1.3617E-09 12.0013 2.3477e-08 1.3629E-09 11.9984
    8 8 1.6077e-10 5.4460E-11 12.0533 9.5183e-10 8.9108E-11 10.2131
    q p \lambda =1.5 \lambda =2
    L_2 MAEs CO L_2 MAEs CO
    5 5 1.8456e-05 1.1892E-06 - 1.8456e-05 1.1892E-06 -
    6 6 2.4802e-07 5.5075E-08 8.4256 2.4798e-07 5.5077E-08 8.4255
    7 7 2.3628e-08 1.3595E-09 12.0065 2.3620e-08 1.3403E-09 12.0526
    8 8 8.7781e-11 5.0231E-11 12.3499 3.0124e-09 1.4625E-10 8.2952

     | Show Table
    DownLoad: CSV

    Example 5. If \kappa = \mu = 1 and \delta (\sigma, \tau, u(\sigma, \tau)) = 0 , then the model (1.1) is

    \begin{equation} \left\{ \begin{array}{l} \frac{{{\partial ^{\theta (\sigma , \tau )}}u(\sigma , \tau )}}{{\partial {\tau ^{\theta (\sigma , \tau )}}}} = \frac{{{\partial ^2}u(\sigma , \tau )}}{{\partial {\sigma ^2}}} - \frac{{\partial u(\sigma , \tau )}}{{\partial \sigma }} + f(\sigma , \tau )\;, (\sigma , \tau ) \in [0, 1] \times [0, 1], \\ u(\sigma , 0) = 0\;, \;\;u(0, \tau ) = u(1, \tau ) = 0.\\ \end{array} \right. \end{equation} (6.6)

    For force function

    f(\sigma , \tau ) = 120\frac{{C(\theta (\sigma , \tau ))}}{{1 - \theta (\sigma , \tau )}}{\tau ^5}\sin \pi \sigma \times {E_{\theta (\sigma , \tau ), 6}}\left[ {\frac{{ - \theta (\sigma , \tau )}}{{1 - \theta (\sigma , \tau )}}{\tau ^{\theta (\sigma , \tau )}}} \right] + (\pi \sin (\pi \theta (\sigma , \tau )) + \cos (\pi \theta (\sigma , \tau ))),

    the solution of Eq (6.6) reads u(\sigma, \tau) = {\tau ^5}\sin \pi \sigma . Comparison of the numerical solution when \lambda = 1.2, q = 14, \; p = 9 , \theta (\sigma, \tau) = 0.8 - 0.2{e^{ - 2\sigma \tau }} and the exact solution are displayed in Figure 10. It can be seen that the numerical solution is convergent.

    Figure 10.  Comparison of the approximate solution ( \theta (\sigma, \tau) = 0.8-0.2{e^{ -2\sigma \tau}} , left) and the exact solution (right) in Example 4.

    Finally, we try to use the proposed computational method to solve another type problem.

    Example 6. Consider the VO fractional burgers equation[37]

    \begin{equation} \left\{ \begin{array}{l} \frac{{{\partial ^{\theta (\sigma , \tau )}}u(\sigma , \tau )}}{{\partial {\tau ^{\theta (\sigma , \tau )}}}} = V\frac{{{\partial ^2}u(\sigma , \tau )}}{{\partial {\sigma ^2}}} - u(\sigma , \tau )\frac{{\partial u(\sigma , \tau )}}{{\partial \sigma }} + f(\sigma , \tau ), \, \, 0 < \sigma < X, \, \, 0 < \tau < T, \\ u(\sigma , 0) = \sin (2\pi \sigma )\;, \;\;u(0, \tau ) = u(1, \tau ) = 0. \end{array} \right. \end{equation} (6.7)

    It is hard to find the analytical solution of Eq (6.7) with ABC fractional derivative, but we are able to predict by the proposed method simultaneously the solutions with ordinary and variable fractional derivative with some initial parameters. Choose X = 1, \; T = 5 , numerical results of some spatial and time nodes with different \theta (\sigma, \tau) are shown in Figures 11 and 12 respectively. Choose \lambda = 0.5 , the 3D diagrams of the solutions predicted by the proposed algorithm when \theta (\sigma, \tau) = 0.4 and \theta (\sigma, \tau) = {\text{0}}{\text{.75 + 0}}{\text{.2sin(2}}\sigma \tau) respectively are shown in Figure 13. It is clear that the numerical solution is convergent.

    Figure 11.  Exact solution and the numerical solutions for various constant functions \theta (\sigma, \tau) when \tau = 2 (left) and \sigma = 0.2 (right) in Example 6.
    Figure 12.  Exact solution and the numerical solutions for some \theta (\sigma, \tau) when \tau = 4 (left) and \sigma = 0.4 (right) in Example 6.
    Figure 13.  Numerical solutions when \theta(\sigma, \tau) = 0.4 ( \lambda = 0.5 , left) and \theta(\sigma, \tau) = {\text{0}}{\text{.75 + 0}}{\text{.2sin(2}}\sigma \tau) ( \lambda = 0.5 , right) in Example 6.

    Shifted orthogonal polynomials combined with spectral methods (such as Galerkin method, collocation method and tau method) can transform complex variable fractional differential equations into algebraic equations, so as to reduce the complexity and improve the accuracy. We propose a numerical approach to solve a VO time fractional model for advection-reaction-diffusion equation with Atangana-Baleanu-Caputo derivative via the shifted Gegenbauer polynomial. By using a collocation approach and obtaining the operational matrix, all that remains is to solve a nonlinear equations. Some examples are given to illustrate the effectiveness of the proposed numerical algorithm.

    This work was supported by National Natural Science Foundation of China (No. 11971094).

    The authors declare no conflicts of interest.



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