Loading [MathJax]/extensions/TeX/boldsymbol.js
Research article Special Issues

Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation

  • In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.

    Citation: Leqiang Zou, Yanzi Zhang. Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation[J]. Networks and Heterogeneous Media, 2025, 20(2): 387-405. doi: 10.3934/nhm.2025018

    Related Papers:

    [1] Ting Liu, Guo-Bao Zhang . Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, 2021, 29(4): 2599-2618. doi: 10.3934/era.2021003
    [2] Cui-Ping Cheng, Ruo-Fan An . Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29(5): 3535-3550. doi: 10.3934/era.2021051
    [3] Bin Wang . Random periodic sequence of globally mean-square exponentially stable discrete-time stochastic genetic regulatory networks with discrete spatial diffusions. Electronic Research Archive, 2023, 31(6): 3097-3122. doi: 10.3934/era.2023157
    [4] Yijun Chen, Yaning Xie . A kernel-free boundary integral method for reaction-diffusion equations. Electronic Research Archive, 2025, 33(2): 556-581. doi: 10.3934/era.2025026
    [5] Yongwei Yang, Yang Yu, Chunyun Xu, Chengye Zou . Passivity analysis of discrete-time genetic regulatory networks with reaction-diffusion coupling and delay-dependent stability criteria. Electronic Research Archive, 2025, 33(5): 3111-3134. doi: 10.3934/era.2025136
    [6] Meng Wang, Naiwei Liu . Qualitative analysis and traveling wave solutions of a predator-prey model with time delay and stage structure. Electronic Research Archive, 2024, 32(4): 2665-2698. doi: 10.3934/era.2024121
    [7] Minzhi Wei . Existence of traveling waves in a delayed convecting shallow water fluid model. Electronic Research Archive, 2023, 31(11): 6803-6819. doi: 10.3934/era.2023343
    [8] Hongquan Wang, Yancai Liu, Xiujun Cheng . An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations. Electronic Research Archive, 2025, 33(1): 263-276. doi: 10.3934/era.2025014
    [9] Cheng Wang . Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29(5): 2915-2944. doi: 10.3934/era.2021019
    [10] Shao-Xia Qiao, Li-Jun Du . Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, 2021, 29(3): 2269-2291. doi: 10.3934/era.2020116
  • In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.



    In this article, we consider the following spatially discrete diffusion system with time delay

    {tv1(x,t)=d1D[v1](x,t)αv1(x,t)+h(v2(x,tτ1)),tv2(x,t)=d2D[v2](x,t)βv2(x,t)+g(v1(x,tτ2)) (1)

    with the initial data

    vi(x,s)=vi0(x,s), xR, s[τi,0], i=1,2, (2)

    where t>0, xR, di0 and

    D[vi](x,t)=vi(x+1,t)2vi(x,t)+vi(x1,t), i=1,2.

    Here v1(x,t) and v2(x,t) biologically stand for the spatial density of the bacterial population and the infective human population at point xR and time t0, respectively. Both bacteria and humans are assumed to diffuse, d1 and d2 are diffusion coefficients; the term αv1 is the natural death rate of the bacterial population and the nonlinearity h(v2) is the contribution of the infective humans to the growth rate of the bacterial; βv2 is the natural diminishing rate of the infective population due to the finite mean duration of the infectious population and the nonlinearity g(v1) is the infection rate of the human population under the assumption that the total susceptible human population is constant during the evolution of the epidemic, and τ1, τ2 are time delays. The nonlinearities g and h satisfy the following hypothesis:

    (H1) gC2([0,K1],R), g(0)=h(0)=0, K2=g(K1)/β>0, hC2([0,K2],R), h(g(K1)/β)=αK1, h(g(v)/β)>αv for v(0,K1), where K1 is a positive constant.

    According to (H1), the spatially homogeneous system of (1) admits two constant equilibria

    (v1,v2)=0:=(0,0)and(v1+,v2+)=K:=(K1,K2).

    It is clear that (H1) is a basic assumption to ensure that system (1) is monostable on [0,K]. When g(u)0 for u[0,K1] and h(v)0 for v[0,K2], system (1) is a quasi-monotone system. Otherwise, if g(u)0 for u[0,K1] or h(v)0 for v[0,K2] does not hold, system (1) is a non-quasi-monotone system. In this article, we are interested in the existence and stability of traveling wave solutions of (1) connecting two constant equilibria (0,0) and (K1,K2). A traveling wave solution (in short, traveling wave) of (1) is a special translation invariant solution of the form (v1(x,t),v2(x,t))=(ϕ1(x+ct),ϕ2(x+ct)), where c>0 is the wave speed. If ϕ1 and ϕ2 are monotone, then (ϕ1,ϕ2) is called a traveling wavefront. Substituting (ϕ1(x+ct),ϕ2(x+ct)) into (1), we obtain the following wave profile system with the boundary conditions

    {cϕ1(ξ)=d1D[ϕ1](ξ)αϕ1(ξ)+h(ϕ2(ξcτ1)),cϕ2(ξ)=d2D[ϕ2](ξ)βϕ2(ξ)+g(ϕ1(ξcτ2)),(ϕ1,ϕ2)()=(v1,v2),(ϕ1,ϕ2)(+)=(v1+,v2+), (3)

    where ξ=x+ct, =ddξ, D[ϕi](ξ)=ϕi(ξ+1)2ϕi(ξ)+ϕi(ξ1), i=1,2.

    System (1) is a discrete version of classical epidemic model

    {tv1(x,t)=d1xxv1(x,t)a1v1(x,t)+h(v2(x,tτ1)),tv2(x,t)=d2xxv2(x,t)a2v2(x,t)+g(v1(x,tτ2)). (4)

    The existence and stability of traveling waves of (4) have been extensively studied, see [7,19,21,24] and references therein. Note that system (1) is also a delay version of the following system

    {tv1(x,t)=d1D[v1](x,t)a1v1(x,t)+h(v2(x,t)),tv2(x,t)=d2D[v2](x,t)a2v2(x,t)+g(v1(x,t)). (5)

    When system (5) is a quasi-monotone system, Yu, Wan and Hsu [27] established the existence and stability of traveling waves of (5). To the best of our knowledge, when systems (1) and (5) are non-quasi-monotone systems, no result on the existence and stability of traveling waves has been reported. We should point out that the existence of traveling waves of (1) can be easily obtained. Hence, the main purpose of the current paper is to establish the stability of traveling waves of (1).

    The stability of traveling waves for the classical reaction-diffusion equations with and without time delay has been extensively investigated, see e.g., [4,9,10,12,13,14,16,22,24]. Compared to the rich results for the classical reaction-diffusion equations, limited results exist for the spatial discrete diffusion equations. Chen and Guo [1] took the squeezing technique to prove the asymptotic stability of traveling waves for discrete quasilinear monostable equations without time delay. Guo and Zimmer [5] proved the global stability of traveling wavefronts for spatially discrete equations with nonlocal delay effects by using a combination of the weighted energy method and the Green function technique. Tian and Zhang [19] investigated the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with two species by the weighted energy method together with the comparison principle. Later on, Chen, Wu and Hsu [2] employed the similar method to show the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with three species. We should point out that the methods for the above stability results heavily depend on the monotonicity of equations and the comparison principle. However, the most interesting cases are the equations without monotonicity. It is known that when the evolution equations are non-monotone, the comparison principle is not applicable. Thus, the methods, such as the squeezing technique, the weighted energy method combining with the comparison principle are not valid for the stability of traveling waves of the spatial discrete diffusion equations without monotonicity.

    Recently, the technical weighted energy method without the comparison principle has been used to prove the stability of traveling waves of nonmonotone equations, see Chern et al. [3], Lin et al. [10], Wu et al. [22], Yang et al. [24]. In particular, Tian et al. [20] and Yang et al. [26], respectively, applied this method to prove the local stability of traveling waves for nonmonotone traveling waves for spatially discrete reaction-diffusion equations with time delay. Later, Yang and Zhang [25] established the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Unfortunately, the local stability (the initial perturbation around the traveling wave is properly small in a weighted norm) of traveling waves can only be obtained. Very recently, Mei et al. [15] developed a new method to prove the global stability of the oscillatory traveling waves of local Nicholson's blowflies equations. This method is based on some key observations for the structure of the govern equations and the anti-weighted energy method together with the Fourier transform. Later on, Zhang [28] and Xu et al. [23], respectively, applied this method successfully to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. More recently, Su and Zhang [17] further studied a discrete diffusion equation with a monostable convolution type nonlinearity and established the global stability of traveling waves with large speed. Motivated by the works [15,28,23,17,18], in this paper, we shall extend this method to study the global stability of traveling waves of spatial discrete diffusion system (1) without quasi-monotonicity.

    The rest of this paper is organized as follows. In Section 2, we present some preliminaries and summarize our main results. Section 3 is dedicated to the global stability of traveling waves of (1) by the Fourier transform and the weighted energy method, when h(u) and g(u) are not monotone.

    In this section, we first give the equivalent integral form of the initial value problem of (1) with (2), then recall the existence of traveling waves of (1), and finally state the main result on the global stability of traveling waves of (1). Throughout this paper, we assume τ1=τ2=τ.

    First of all, we consider the initial value problem (1) with (2), i.e.,

    {tv1(x,t)=d1D[v1](x,t)αv1(x,t)+h(v2(x,tτ)),tv2(x,t)=d2D[v2](x,t)βv2(x,t)+g(v1(x,tτ)),vi(x,s)=vi0(x,s), xR, s[τ,0], i=1,2. (6)

    According to [8], with aid of modified Bessel functions, the solution to the initial value problem

    {tu(x,t)=d[u(x+1,t)2u(x,t)+u(x1,t)], xR, t>0,u(x,0)=u0(x), xR,

    can be expressed by

    u(x,t)=(S(t)u0)(x)=e2dtm=Im(2dt)u0(xm),

    where u0()L(R), Im(), m0 are defined as

    Im(t)=k=0(t/2)m+2kk!(m+k)!,

    and Im(t)=Im(t) for m<0. Moreover,

    Im(t)=12[Im+1(t)+Im1(t)], t>0,mZ, (7)

    and Im(0)=0 for m0 while I0=1, and Im(t)0 for any t0. In addition, one has

    etm=Im(t)=et[I0(t)+2I1(t)+2I2(t)+I3(t)+]=1. (8)

    Thus, the solution (v1(x,t),v2(x,t)) of (6) can be expressed as

    {v1(x,t)=e(2d1+α)tm=Im(2d1t)v10(xm,0)+m=t0e(2d1+α)(ts)Im(2d1(ts))(h(v2(xm,sτ)))ds,v2(x,t)=e(2d2+β)tm=Im(2d2t)v20(xm,0)+m=t0e(2d2+β)(ts)Im(2d2(ts))(g(v1(xm,sτ)))ds. (9)

    In fact, by [8,Lemma 2.1], we can differentiate the series on t variable in (9). Using the recurrence relation (7), we obtain

    tv1(x,t)=(2d1+α)e(2d1+α)tm=Im(2d1t)v10(xm,0)   +e(2d1+α)tm=2d1Im(2d1t)v10(xm,0)   +m=Im(0)(h(v2(xm,tτ)))   (2d1+α)m=t0e(2d1+α)(ts)Im(2d1(ts))(h(v2(xm,sτ)))ds   +m=t0e(2d1+α)(ts)2d1Im(2d1(ts))(h(v2(xm,sτ)))ds=d1[v1(x+1,t)2v1(x,t)+v1(x1,t)]αv1(x,t)+h(v2(x,tτ))

    and

    tv2(x,t)=(2d2+β)e(2d2+β)tm=Im(2d2t)v20(xm,0)   +e(2d2+β)tm=2d2Im(2d2t)v20(xm,0)   +m=Im(0)(g(v1(xm,tτ)))   (2d2+β)m=t0e(2d2+β)(ts)Im(2d2(ts))(g(v1(xm,sτ)))ds   +m=t0e(2d2+β)(ts)2d2Im(2d2(ts))(g(v1(xm,sτ)))ds=d2[v2(x+1,t)2v2(x,t)+v2(x1,t)]βv2(x,t)+g(v1(x,tτ)).

    Next we investigate the characteristic roots of the linearized system for the wave profile system (3) at the trivial equilibrium 0. Clearly, the characteristic function of (3) at 0 is

    P1(c,λ):=f1(c,λ)f2(c,λ)

    for c0 and λC, where

    f1(c,λ):=Δ1(c,λ)Δ2(c,λ),f2(c,λ):=h(0)g(0)e2cλτ,

    with

    Δ1(c,λ)=d1(eλ+eλ2)cλα,Δ2(c,λ)=d2(eλ+eλ2)cλβ.

    It is easy to see that Δ1(c,λ)=0 admits two roots λ1<0<λ+1, and Δ2(c,λ)=0 has two roots λ2<0<λ+2. We denote λ+m=min{λ+1,λ+2}.

    Similar to [27,Lemma 3.1], we can obtain the following result.

    Lemma 2.1. There exists a positive constant c such that if c>c, then P1(c,λ)=0 has two distinct positive real roots λ1:=λ1(c) and λ2:=λ2(c) with λ1(c)<λ2(c)<λ+m, i.e. P1(c,λ1)=P1(c,λ2)=0, and P(c,λ)>0 for λ(λ1(c),λ2(c)). In addition, limccλ1(c)=limccλ2(c)=λ>0, i.e., P1(c,λ)=0.

    Furthermore, we show the existence of traveling wave of (1). When system (1) is a quasi-monotone system, the existence of traveling wavefronts follows from [6,Theorem 1.1]. When system (1) is a non-quasi-monotone system, the existence of traveling waves can also be obtained by using auxiliary equations and Schauder's fixed point theorem [21,24], if we assume the following assumptions:

    (H2) There exist K±=(K±1,K±2)0 with K<K<K+ and four continuous and twice piecewise continuous differentiable functions g±:[0,K+1]R and h±:[0,K+2]R such that

    (i) K±2=g±(K±1)/β, h±(1βg±(K±1))=αK±1, and h±(1βg±(v))>αv for v(0,K±1);

    (ii) g±(u) and h±(v) are non-decreasing on [0,K+1] and [0,K+2], respectively;

    (iii) (g±)(0)=g(0), (h±)(0)=h(0) and

    0<g(u)g(u)g+(u)g(0)u for u[0,K+1],0<h(v)h(v)h+(v)h(0)v for v[0,K+2].

    Proposition 1. Assume that (H1) and (H2) hold, τ0, and let c be defined as in Lemma 2.1. Then for every c>c, system (1) has a traveling wave (ϕ1(ξ),ϕ2(ξ)) satisfying (ϕ1(),ϕ2())=(0,0) and

    K1lim infξ+ϕ1(ξ)lim supξ+ϕ1(ξ)K+1,0lim infξ+ϕ2(ξ)lim supξ+ϕ2(ξ)K+2.

    Finally, we shall state the stability result of traveling waves derived in Proposition 1. Before that, let us introduce the following notations.

    Notations. C>0 denotes a generic constant, while Ci(i=1,2,) represents a specific constant. Let and denote 1-norm and -norm of the matrix (or vector), respectively. Let I be an interval, typically I=R. Denote by L1(I) the space of integrable functions defined on I, and Wk,1(I)(k0) the Sobolev space of the L1-functions f(x) defined on the interval I whose derivatives dndxnf(n=1,,k) also belong to L1(I). Let L1w(I) be the weighted L1-space with a weight function w(x)>0 and its norm is defined by

    ||f||L1w(I)=Iw(x)|f(x)|dx,

    Wk,1w(I) be the weighted Sobolev space with the norm given by

    ||f||Wk,1w(I)=ki=0Iw(x)|dif(x)dxi|dx.

    Let T>0 be a number and B be a Banach space. We denote by C([0,T];B) the space of the B-valued continuous functions on [0,T], and by L1([0,T];B) the space of the B-valued L1-functions on [0,T]. The corresponding spaces of the B-valued functions on [0,) are defined similarly. For any function f(x), its Fourier transform is defined by

    F[f](η)=ˆf(η)=Reixηf(x)dx

    and the inverse Fourier transform is given by

    F1[ˆf](x)=12πReixηˆf(η)dη,

    where i is the imaginary unit, i2=1.

    To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.

    (H3) |g(u)|g(0) and |h(v)|h(0) for u,v[0,+).

    (H4) d2>d1, α>β, d2d1<αβ2 and max{h(0),g(0)}>β.

    (H5) The initial data (v10(x,s),v20(x,s))(0,0) satisfies

    limx±(v10(x,s),v20(x,s))=(v1±,v2±) uniformly in  s[τ,0].

    Consider the following function

    P2(λ,c)=d2(eλ+eλ2)cλβ+max{h(0),g(0)}eλcτ.

    Since max{h(0),g(0)}>β, it then follows from [20,Lemma 2.1] that there exists λ>0 and c>0, such that P2(λ,c)=0 and P2(λ,c)λ|(λ,c)=0. When c>c, the equation P2(λ,c)=0 has two positive real roots λ1(c) and λ2(c) with 0<λ1(c)<λ<λ2(c). When λ(λ1(c),λ2(c)), P2(λ,c)<0. Moreover, (λ1)(c)<0 and (λ2)(c)>0.

    We select the weight function w(ξ)>0 as the form

    w(ξ)=e2λξ,

    where λ>0 satisfies λ1(c)<λ<λ2(c). Now we are ready to present the main result of this paper.

    Theorem 2.2. (Global stability of traveling waves). Assume that (H1), (H3)-(H5) hold. For any given traveling wave (ϕ1(x+ct),ϕ2(x+ct)) of (1) with speed c>max{c,c} connecting (0,0) and (K1,K2), whether it is monotone or non-monotone, if the initial data satisfy

    vi0(x,s)ϕi(x+cs)Cunif[τ,0]C([τ,0];W1,1w(R)), i=1,2,s(vi0ϕi)L1([τ,0];L1w(R)), i=1,2,

    then there exists τ0>0 such that for any ττ0, the solution (v1(x,t),v2(x,t)) of (1)-(2) converges to the traveling wave (ϕ1(x+ct),ϕ2(x+ct)) as follows:

    supxR|vi(x,t)ϕi(x+ct)|Ceμt,t>0,

    where C and μ are two positive constants, and Cunif[r,T] is the uniformly continuous space, for 0<T, defined by

    Cunif[r,T]={uC([r,T]×R)such thatlimx+v(x,t)exists uniformly int[r,T]}.

    This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let (ϕ1(x+ct),ϕ2(x+ct))=(ϕ1(ξ),ϕ2(ξ)) be a given traveling wave solution with speed cc and define

    {Vi(ξ,t):=vi(x,t)ϕi(x+ct)=vi(ξct,t)ϕi(ξ), i=1,2,Vi0(ξ,s):=vi0(x,s)ϕi(x+cs)=vi0(ξcs,s)ϕ(ξ), i=1,2.

    Then it follows from (1) and (3) that Vi(ξ,t) satisfies

    {V1t+cV1ξd1D[V1]+αV1=Q1(V2(ξcτ,tτ)),V2t+cV2ξd2D[V2]+βV2=Q2(V1(ξcτ,tτ)),Vi(ξ,s)=Vi0(ξ,s), (ξ,s)R×[τ,0], i=1,2. (10)

    The nonlinear terms Q1 and Q2 are given by

    {Q1(V2):=h(ϕ2+V2)h(ϕ2)=h(˜ϕ2)V2,Q2(V1):=g(ϕ1+V1)g(ϕ1)=g(˜ϕ1)V1, (11)

    for some ˜ϕi between ϕi and ϕi+Vi, with ϕi=ϕi(ξcτi) and Vi=Vi(ξcτi,tτi).

    We first prove the existence and uniqueness of solution (V1(ξ,t),V2(ξ,t)) to the initial value problem (10) in the uniformly continuous space Cunif[τ,+)×Cunif[τ,+).

    Lemma 3.1. Assume that (H1)and(H3) hold. If the initial perturbation (V10,V20)Cunif[τ,0]×Cunif[τ,0] for cc, then the solution (V1,V2) of the perturbed equation (10) is unique and time-globally exists in Cunif[τ,+)×Cunif[τ,+).

    Proof. Let Ui(x,t)=vi(x,t)ϕi(x+ct), i=1,2. It is clear that Ui(x,t)=Vi(ξ,t), i=1,2, and satisfies

    {U1td1D[U1]+αU1=Q1(U2(x,tτ)),U2td2D[U2]+βU2=Q2(U1(x,tτ)),Ui(x,s)=vi0(x,s)ϕi(x+cs):=Ui0(x,s), (x,s)R×[τ,0], i=1,2. (12)

    Thus, the global existence and uniqueness of solution of (10) are transformed into that of (12).

    When t[0,τ], we have tτ[τ,0] and Ui(x,tτ)=Ui0(x,tτ), i=1,2, which imply that (12) is linear. Thus, the solution of (12) can be explicitly and uniquely solved by

    {U1(x,t)=e(2d1+α)tm=Im(2d1t)U10(xm,0)             +m=t0e(2d1+α)(ts)Im(2d1(ts))Q1(U20(xm,sτ))ds,U2(x,t)=e(2d2+β)tm=Im(2d2t)U20(xm,0)             +m=t0e(2d2+β)(ts)Im(2d2(ts))Q2(U10(xm,sτ))ds (13)

    for t[0,τ].

    Since Vi0(ξ,t)Cunif[τ,0], i=1,2, namely, limξ+Vi0(ξ,t) exist uniformly in t[τ,0], which implies limx+Ui0(x,t) exist uniformly in t[τ,0]. Denote Ui0(,t)=limx+Ui0(x,t), i=1,2. Taking the limit x+ to (13) yields

    limx+U1(x,t)=e(2d1+α)tm=Im(2d1t)limx+U10(xm,0)+m=t0e(2d1+α)(ts)Im(2d1(ts))limx+Q1(U20(xm,sτ))ds=eαtU10(,0)+t0eα(ts)Q1(U20(,sτ))m=e2d1(ts)Im(2d1(ts))ds=:U1(t)  uniformly in t[0,τ] (14)

    and

    limx+U2(x,t)=e(2d2+β)tm=Im(2d2t)limx+U20(xm,0)
    +m=t0e(2d2+β)(ts)Im(2d2(ts))limx+Q2(U10(xm,sτ))ds=eβtU20(,0)+t0eβ(ts)Q2(U10(,sτ))m=e2d2(ts)Im(2d2(ts))ds=:U2(t)  uniformly in t[0,τ], (15)

    where we have used (8). Thus, we obtain that (U1,U2)Cunif[τ,τ)×Cunif[τ,τ).

    When t[τ,2τ], system (12) with the initial data Ui(x,s) for s[0,τ] is still linear, because the source term Q1(U2(x,tτ)) and Q2(U1(x,tτ)) is known due to tτ[0,τ] and Ui(s,tτ) is solved in (13). Hence, the solution Ui(x,t) for t[τ,2τ] is uniquely and explicitly given by

    U1(x,t)=e(2d1+α)(tτ)m=Im(2d1(tτ))U1(xm,τ)+m=tτe(2d1+α)(ts)Im(2d1(ts))Q1(U2(xm,sτ))ds,U2(x,t)=e(2d2+β)(tτ)m=Im(2d2(tτ))U2(xm,τ)+m=tτe(2d2+β)(ts)Im(2d2(ts))Q2(U1(xm,sτ))ds.

    Similarly, by (14) and (15), we have

    limx+U1(x,t)=e(2d1+α)(tτ)m=Im(2d1(tτ))limx+U1(xm,τ)+m=tτe(2d1+α)(ts)Im(2d1(ts))limx+Q1(U2(xm,sτ))ds=eα(tτ)U1(τ)+tτeα(ts)Q1(U1(sτ))m=e2d1(ts)Im(2d1(ts))ds=:ˉU1(t)  uniformly in t[τ,2τ],

    and

    limx+U2(x,t)=e(2d2+β)(tτ)m=Im(2d2(tτ))limx+U2(xm,τ)+m=tτe(2d2+β)(ts)Im(2d2(ts))limx+Q2(U1(xm,sτ))ds=eβ(tτ)U2(τ)+tτeβ(ts)Q2(U2(sτ))m=e2d2(ts)Im(2d2(ts))ds
    =:ˉU2(t)  uniformly in t[τ,2τ].

    By repeating this procedure for t[nτ,(n+1)τ] with nZ+, we prove that there exists a unique solution (V1,V2)Cunif[τ,(n+1)τ]×Cunif[τ,(n+1)τ] for (10), and step by step, we finally prove the uniqueness and time-global existence of the solution (V1,V2)Cunif[τ,)×Cunif[τ,) for (10). The proof is complete.

    Now we state the stability result for the perturbed system (10), which automatically implies Theorem 2.2.

    Proposition 2. Assume that (H1), (H3)-(H5) hold. If

    Vi0Cunif[τ,0]C([τ,0];W1,1w(R)), i=1,2,

    and

    sVi0L1([τ,0];L1w(R)), i=1,2,

    then there exists τ0>0 such that for any ττ0, when c>max{c,c}, it holds

    supξR|Vi(ξ,t)|Ceμt,t>0, i=1,2, (16)

    for some μ>0 and C>0.

    In order to prove Proposition 2, we first investigate the decay estimate of Vi(ξ,t) at ξ=+, i=1,2.

    Lemma 3.2. Assume that Vi0Cunif[τ,0], i=1,2. Then, there exist τ0>0 and a large number x01 such that when ττ0, the solution Vi(ξ,t) of (10) satisfies

    supξ[x0,+)|Vi(ξ,t)|Ceμ1t, t>0, i=1,2,

    for some μ1>0 and C>0.

    Proof. Denote

    z+i(t):=Vi(,t), z+i0(s):=Vi0(,s), s[τ,0], i=1,2.

    Since Vi0Cunif[τ,0], i=1,2, by Lemma 3.1, we have ViCunif[τ,+), which implies

    limξ+Vi(ξ,t)=z+i(t)

    exists uniformly for t[τ,+). Taking the limit ξ+ to (10), we obtain

    {dz+1dt+αz+1h(v2+)z+2(tτ)=P1(z+2(tτ)),dz+2dt+βz+2g(v1+)z+1(tτ)=P2(z+1(tτ)),z+i(s)=z+i0(s), s[τ,0], i=1,2,

    where

    {P1(z+2)=h(v2++z+2)h(v2+)h(v2+)z+2,P2(z+1)=g(v1++z+1)g(v1+)g(v1+)z+1.

    Then by [9,Lemma 3.8], there exist positive constants τ0, μ1 and C such that when ττ0,

    |Vi(,t)|=|z+i(t)|Ceμ1t, t>0, i=1,2, (17)

    provided that |z+i0|1, i=1,2.

    By the continuity and the uniform convergence of Vi(ξ,t) as ξ+, there exists a large x01 such that (17) implies

    supξ[x0,+)|Vi(ξ,t)|Ceμ1t, t>0, i=1,2,

    provided that supξ[x0,+)|Vi0(ξ,s)|1 for s[τ,0]. Such a smallness for the initial perturbation (V10,V20) near ξ+ can be easily verified, since

    limx+(v10(x,s),v20(x,s))=(K1,K2) uniformly in s[τ,0],

    which implies

    limξ+Vi0(ξ,s)=limξ+[vi0(ξ,s)ϕi(ξ)]=KiKi=0

    uniformly for s[τ,0], i=1,2. The proof is complete.

    Next we are going to establish the a priori decay estimate of supξ(,x0]|Vi(ξ,t)| by using the anti-weighted technique [3] together with the Fourier transform. First of all, we shift Vi(ξ,t) to Vi(ξ+x0,t) by the constant x0 given in Lemma 3.2, and then introduce the following transformation

    ˜Vi(ξ,t)=w(ξ)Vi(ξ+x0,t)=eλξVi(ξ+x0,t),i=1,2.

    Substituting Vi=w1/2˜Vi to (10) yields

    {˜V1t+c˜V1ξ+c1˜V1(ξ,t)d1eλ˜V1(ξ+1,t)d1eλ˜V1(ξ1,t)=˜Q1(˜V2(ξcτ,tτ)),˜V2t+c˜V2ξ+c2˜V2(ξ,t)d2eλ˜V2(ξ+1,t)d2eλ˜V2(ξ1,t)=˜Q2(˜V1(ξcτ,tτ)),˜Vi(ξ,s)=w(ξ)Vi0(ξ+x0,s)=:˜Vi0(ξ,s), ξR,s[τ,0], i=1,2, (18)

    where

    c1=cλ+2d1+α,c2=cλ+2d2+β

    and

    ˜Q1(˜V2)=eλξQ1(V2),˜Q2(˜V1)=eλξQ2(V1).

    By (11), ˜Q1(˜V2) satisfies

    ˜Q1(˜V2(ξcτ,tτ))=eλξQ1(V2(ξcτ+x0,tτ))=eλξh(˜ϕ2)V2(ξcτ+x0,tτ)=eλcτh(˜ϕ2)˜V2(ξcτ,tτ) (19)

    and ˜Q2(˜V1) satisfies

    ˜Q2(˜V1(ξcτ,tτ))=eλcτg(˜ϕ1)˜V1(ξcτ,tτ). (20)

    By (H3), we further obtain

    |˜Q1(˜V2(ξcτ,tτ))|h(0)eλcτ|˜V2(ξcτ,tτ)|,|˜Q2(˜V1(ξcτ,tτ))|g(0)eλcτ|˜V1(ξcτ,tτ)|.

    Taking (19) and (20) into (18), one can see that the coefficients h(˜ϕ2) and g(˜ϕ1) on the right side of (18) are variable and can be negative. Thus, the classical methods, such as the monotone technique and the Fourier transform cannot be applied directly to establish the decay estimate for (˜V1,˜V2). Motivated by [15,28,17,23], we introduce a new method which can be described as follows.

    By replacing h(˜ϕ2) in the first equation of (18) with a constant h(0), and g(˜ϕ1) in the second equation of (18) with a constant g(0), we can obtain a linear delayed reaction-diffusion system

    {V+1t+cV+1ξ+c1V+1(ξ,t)d1eλV+1(ξ+1,t)d1eλV+1(ξ1,t) =h(0)eλcτV+2(ξcτ,tτ),V+2t+cV+2ξ+c2V+2(ξ,t)d2eλV+2(ξ+1,t)d2eλV+2(ξ1,t) =g(0)eλcτV+1(ξcτ,tτ), (21)

    with

    V+i(ξ,s)=w(ξ)Vi0(ξ+x0,s)=:V+i0(ξ,s), i=1,2,

    where ξR, t(0,+] and s[τ,0]. Then we investigate the decay estimate of (V+1,V+2) by applying the Fourier transform to (21);

    We prove that the solution (˜V1,˜V2) of (18) can be bounded by the solution (V+1,V+2) of (21).

    Now we are in a position to derive the decay estimate of (V+1,V+2) for the linear system (21). We first recall some properties of the solutions to the delayed ODE system.

    Lemma 3.3. ([11,Lemma 3.1]) Let z(t) be the solution to the following scalar differential equation with delay

    {ddtz(t)=Az(t)+Bz(tτ),t0,τ>0,z(s)=z0(s),s[τ,0]. (22)

    where A,BCN×N, N2, and z0(s)C1([τ,0],CN). Then

    z(t)=eA(t+τ)eB1tτz0(τ)+0τeA(ts)eB1(tτs)τ[z0(s)Az0(s)]ds,

    where B1=BeAτ and eB1tτ is the so-called delayed exponential function in the form

    eB1tτ={0,<t<τ,I,τt<0,I+B1t1!,0t<τ,I+B1t1!+B21(tτ)22!,τt<2τ,I+B1t1!+B21(tτ)22!++Bm1[t(m1)τ]mm!,(m1)τt<mτ,

    where 0,ICN×N, and 0 is zero matrix and I is unit matrix.

    Lemma 3.4. ([11,Theorem 3.1]) Suppose μ(A):=μ1(A)+μ(A)2<0, where μ1(A) and μ(A) denote the matrix measure of A induced by the matrix 1-norm 1 and -norm , respectively. If ν(B):=B+B2μ(A), then there exists a decreasing function ετ=ε(τ)(0,1) for τ>0 such that any solution of system (22) satisfies

    z(t)C0eετσt,t>0,

    where C0 is a positive constant depending on initial data z0(s),s[τ,0] and σ=|μ(A)|ν(B). In particular,

    eAteB1tτC0eετσt,t>0,

    where eB1tτ is defined in Lemma 3.3.

    From the proof of [11,Theome 3.1], one can see that

    μ1(A)=limθ0+I+θA1θ=max1jN[Re(ajj)+Nji|aij|]

    and

    μ(A)=limθ0+I+θA1θ=max1iN[Re(aii)+Nij|aij|].

    Taking the Fourier transform to (21) and denoting the Fourier transform of V+(ξ,t):=(V+1(ξ,t),V+2(ξ,t))T by ˆV+(η,t):=(ˆV+1(η,t),ˆV+2(η,t))T, we obtain

    {tˆV+1(η,t)=(c1+d1(eλ+iη+e(λ+iη))icη)ˆV+1(η,t)                 +h(0)ecτ(λ+iη)ˆV+2(η,tτ),tˆV+2(η,t)=(c2+d2(eλ+iη+e(λ+iη))icη)ˆV+2(η,t)                 +g(0)ecτ(λ+iη)ˆV+1(η,tτ),ˆV+i(η,s)=ˆV+i0(η,s), ηR, s[τ,0], i=1,2. (23)

    Let

    A(η)=(c1+d1(eλ+iη+e(λ+iη))icη00c2+d2(eλ+iη+e(λ+iη))icη)

    and

    B(η)=(0h(0)ecτ(λ+iη)g(0)ecτ(λ+iη)0).

    Then system (23) can be rewritten as

    ˆV+t(η,t)=A(η)ˆV+(η,t)+B(η)ˆV+(η,tτ). (24)

    By Lemma 3.3, the linear delayed system (24) can be solved by

    ˆV+(η,t)=eA(η)(t+τ)eB1(η)tτˆV+0(η,τ)+0τeA(η)(ts)eB1(η)(tsτ)τ[sˆV+0(η,s)A(η)ˆV+0(η,s)]ds:=I1(η,t)+0τI2(η,ts)ds, (25)

    where . Then by taking the inverse Fourier transform to (25), one has

    (26)
    (27)

    Lemma 3.5. Let the initial data , , be such that

    Then

    where and .

    Proof. According to (26), we shall estimate and , respectively. By the definition of and , we have

    where and

    since , and , and

    By considering , we get and

    Furthermore, we obtain

    where for . It then follows from Lemma 3.4 that there exists a decreasing function such that

    (28)

    where is a positive constant and with . By the definition of Fourier's transform, we have

    Applying (28), we derive

    (29)

    with .

    Note that

    Similarly, we can obtain

    It then follows that

    (30)

    Substituting (29) and (30) to (26), we obtain the following the decay rate

    This proof is complete.

    The following maximum principle is needed to obtain the crucial boundedness estimate of , which has been proved in [17,Lemma 3.4].

    Lemma 3.6. Let . For any and , if the bounded function satisfies

    (31)

    then for all .

    Lemma 3.7. When for , then for .

    Proof. When , we have and

    (32)

    Applying (32) to the first equation of (21), we get

    By Lemma 3.6, we derive

    (33)

    Similarly, we obtain

    Using Lemma 3.6 again, we obtain

    (34)

    When , , repeating the above procedure step by step, we can similarly prove

    (35)

    Combining (33), (34) and (31), we obtain for . The proof is complete.

    Now we establish the following crucial boundedness estimate for .

    Lemma 3.8. Let and be the solutions of (18) and (21), respectively. When

    (36)

    then

    Proof. First of all, we prove for In fact, when , namely, , it follows from (36) that

    (37)

    Then by and and (37), we get

    (38)

    and

    (39)

    Let

    We are going to estimate respectively.

    From (18), (19), (21) and (38), we see that satisfies

    By Lemma 3.6, we obtain

    namely,

    (40)

    Similarly, one has

    Applying Lemma 3.6 again, we have

    i.e.,

    (41)

    On the other hand, satisfies

    Then Lemma 3.6 implies that

    that is,

    (42)

    Similarly, satisfies

    Therefore, we can prove that

    namely

    (43)

    Combining (40) and (42), we obtain

    (44)

    and combining (41) and (43), we prove

    (45)

    Next, when , namely, , based on (44) and (45), we can similarly prove

    Repeating this procedure, we then further prove

    which implies

    The proof is complete.

    Let us choose such that

    and

    Combining Lemmas 3.5 and 3.8, we can get the convergence rates for

    Lemma 3.9. When and , then

    for some , .

    Lemma 3.10. It holds that

    for some .

    Proof. Since and for , then we obtain

    which implies

    Thus, the estimate for the unshifted is obtained. The proof is complete.

    Proof of Proposition 3.2. By Lemmas 3.2 and 3.10, we immediately obtain (16) for

    We are grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.



    [1] I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
    [2] W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1768–1777. https://doi.org/10.1016/j.na.2009.09.018 doi: 10.1016/j.na.2009.09.018
    [3] Q. Li, Y. Chen, Y. Huang, Y. Wang, Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM, Math. Comput. Simul., 185 (2021), 436–451. https://doi.org/10.1016/j.matcom.2020.12.033 doi: 10.1016/j.matcom.2020.12.033
    [4] X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522–1545. https://doi.org/10.1093/imanum/draa013 doi: 10.1093/imanum/draa013
    [5] Y. Zhao, C. Shen, M. Qu, W. P. Bu, Y. F. Tang, Finite element methods for fractional diffusion equations, Int. J. Model., Simul., Sci. Comput., 11 (2020), 2030001. https://doi.org/10.1142/S1793962320300010 doi: 10.1142/S1793962320300010
    [6] X. Li, H. Rui, Two temporal second-order -Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations, Numerical Algorithms, 79 (2018), 1107–1130. https://doi.org/10.1007/s11075-018-0476-4 doi: 10.1007/s11075-018-0476-4
    [7] W. Bu, A. Xiao, W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72 (2017), 422–441. https://doi.org/10.1007/s10915-017-0360-8 doi: 10.1007/s10915-017-0360-8
    [8] L. Feng, P. Zhuang, F. Liu, I. Turner, Y. Gu, Finite element method for space-time fractional diffusion equation, Numerical Algorithms, 72 (2016), 749–767. https://doi.org/10.1007/s11075-015-0065-8 doi: 10.1007/s11075-015-0065-8
    [9] A. J. Cheng, H. Wang, K. X. Wang, A Eulerian-Lagrangian control volume method for solute transport with anomalous diffusion, Numer. Methods Partial Differ. Equ., 31 (2015), 253–267. https://doi.org/10.1002/num.21901 doi: 10.1002/num.21901
    [10] M. Badr, A. Yazdani, H. Jafari, Stability of a finite volume element method for the time-fractional advection-diffusion equation, Numer. Methods Partial Differ. Equ., 34 (2018), 1459–1471. https://doi.org/10.1002/num.22243 doi: 10.1002/num.22243
    [11] F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Modell., 38 (2014), 3871–3878. https://doi.org/10.1016/j.apm.2013.10.007 doi: 10.1016/j.apm.2013.10.007
    [12] M. Stynes, E. O'Riordan, J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
    [13] R. Choudhary, S. Singh, P. Das, D. Kumar, A higher-order stable numerical approximation for time-fractional non-linear Kuramoto-Sivashinsky equation based on quintic B-spline, Math. Methods Appl. Sci., 47 (2024), 1–23. https://doi.org/10.1002/mma.9778 doi: 10.1002/mma.9778
    [14] X. M. Gu, H. W. Sun, Y. L. Zhao, X. C. Zheng, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Appl. Math. Lett., 120 (2021), 107270. https://doi.org/10.1016/j.aml.2021.107270 doi: 10.1016/j.aml.2021.107270
    [15] X. D. Zhang, Y. L. Feng, Z. Y. Luo, J. Liu, A spatial sixth-order numerical scheme for solving fractional partial differential equation, Appl. Math. Lett., 159 (2025), 109265. https://doi.org/10.1016/j.aml.2024.109265 doi: 10.1016/j.aml.2024.109265
    [16] Y. Feng, X. Zhang, Y. Chen, L. Wei, A compact finite difference scheme for solving fractional Black-Scholes option pricing model, J. Inequal. Appl., 36 (2025), 36. https://doi.org/10.1186/s13660-025-03261-2 doi: 10.1186/s13660-025-03261-2
    [17] C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15 (2012), 383–406. doi.org/10.2478/s13540-012-0028-x doi: 10.2478/s13540-012-0028-x
    [18] S. Guo, L. Mei, Z. Zhang, Y. Jiang, Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation, Appl. Math. Lett., 85 (2018), 157–163. https://doi.org/10.1016/j.aml.2018.06.005 doi: 10.1016/j.aml.2018.06.005
    [19] X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
    [20] A. Bhardwaj, A. Kumar, A meshless method for time fractional nonlinear mixed diffusion and diffusion-wave equation, Appl. Numer. Math., 160 (2021), 146–165. https://doi.org/10.1016/j.apnum.2020.09.019 doi: 10.1016/j.apnum.2020.09.019
    [21] Y. Gu, H. G. Sun, A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives, Appl. Math. Modell., 78 (2020), 539–549. https://doi.org/10.1016/j.apm.2019.09.055 doi: 10.1016/j.apm.2019.09.055
    [22] V. R. Hosseini, E. Shivanian, W. Chen, Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation, Eur. Phys. J. Plus, 130 (2015), 33. https://doi.org/10.1140/epjp/i2015-15033-5 doi: 10.1140/epjp/i2015-15033-5
    [23] Z. Avazzadeh, W. Chen, V. R. Hosseini, The coupling of RBF and FDM for solving higher order fractional partial differential equations, Appl. Mech. Mater., 598 (2014), 409–413. https://doi.org/10.4028/www.scientific.net/AMM.598.409 doi: 10.4028/www.scientific.net/AMM.598.409
    [24] P. Das, S. Rana, H. Ramos, A perturbation-based approach for solving fractional-order Volterra-Fredholm integro-differential equations and its convergence analysis, Int. J. Comput. Math., 97 (2020), 1994–2014. https://doi.org/10.1080/00207160.2019.1673892 doi: 10.1080/00207160.2019.1673892
    [25] P. Das, S. Rana, H. Ramos, Homotopy perturbation method for solving Caputo-type fractional-order Volterra-Fredholm integro-differential equations, Comput. Math. Methods, 1 (2019), e1047. https://doi.org/10.1002/cmm4.1047 doi: 10.1002/cmm4.1047
    [26] P. Das, S. Rana, H. Ramos, On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis, J. Comput. Appl. Math., 404 (2022), 113116. https://doi.org/10.1016/j.cam.2020.113116 doi: 10.1016/j.cam.2020.113116
    [27] L. Wei, Y. F. Yang, Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation, J. Comput. Appl. Math., 383 (2021), 113129. https://doi.org/10.1016/j.cam.2020.113129 doi: 10.1016/j.cam.2020.113129
    [28] L. Wei, W. Li, Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo-Fabrizio fractional derivative, Math. Comput. Simul., 188 (2021), 280–290. https://doi.org/10.1016/j.matcom.2021.04.001 doi: 10.1016/j.matcom.2021.04.001
    [29] W. Li, L. Wei, Analysis of Local Discontinuous Galerkin Method for the Variable-order Subdiffusion Equation with the Caputo-Hadamard Derivative, Taiwanese J. Math., 28 (2024), 1095–1110. https://doi.org/10.11650/tjm/240801 doi: 10.11650/tjm/240801
    [30] Y. Liu, M. Zhang, H. Li, J. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional sub-diffusion equation, Comput. Math. Appl., 73 (2017), 1298–1314. https://doi.org/10.1016/j.camwa.2016.08.015 doi: 10.1016/j.camwa.2016.08.015
    [31] B. P. Moghaddam, J. A. T. Machado, Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput., 71 (2016), 1351–1374. https://doi.org/10.1007/s10915-016-0343-1 doi: 10.1007/s10915-016-0343-1
    [32] L. Ramirez, C. Coimbra, On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equ., 2010 (2010), 1–16. https://doi.org/10.1155/2010/846107 doi: 10.1155/2010/846107
    [33] Z. Chen, J. Z. Qian, H. B. Zhan, L. W. Chen, S. H. Luo, Mobile-immobile model of solute transport through porous and fractured media, IAHS Publ., 341 (2011), 154–158.
    [34] R. Schumer, D. A. Benson, M. M. Meerschaert, B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res., 39 (2003), 1–12. https://doi.org/10.1029/2003WR002141 doi: 10.1029/2003WR002141
    [35] Y. Zhang, D. A. Benson, D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561–581.
    [36] K. Sadri, H. Aminikhah, An efficient numerical method for solving a class of variable-order fractional mobile-immobile advection-dispersion equations and its convergence analysis, Chaos, Solitons Fractals, 146 (2021), 110896. https://doi.org/10.1016/j.chaos.2021.110896 doi: 10.1016/j.chaos.2021.110896
    [37] H. Ma, Y. Yang, Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model, East Asian J. Appl. Math., 6 (2016), 337–352. https://doi.org/10.4208/eajam.141115.060616a doi: 10.4208/eajam.141115.060616a
    [38] Z. G. Liu, A. J. Cheng, X. L. Li, A second order finite dfference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math., 95 (2017), 396–411. https://doi.org/10.1080/00207160.2017.1290434 doi: 10.1080/00207160.2017.1290434
    [39] A. Golbabai, O. Nikan, T. Nikazad, Numerical investigation of the time fractional mobile-immobile advection-dispersion model arising from solute transport in porous media, Int. J. Appl. Comput. Math., 5 (2019), 1–22. https://doi.org/10.1007/s40819-019-0635-x doi: 10.1007/s40819-019-0635-x
    [40] H. Zhang, F. Liu, M. S. Phanikumar, M. M. Meerschaert, A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput. Math. Appl., 66 (2013), 693–701. https://doi.org/10.1016/j.camwa.2013.01.031 doi: 10.1016/j.camwa.2013.01.031
    [41] W. Jiang, N. Liu, A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Appl. Numer. Math., 119 (2017), 18–32. https://doi.org/10.1016/j.apnum.2017.03.014 doi: 10.1016/j.apnum.2017.03.014
    [42] M. Saffarian, A. Mohebbi, An efficient numerical method for the solution of 2D variable order time fractional mobile-immobile advection-dispersion model, Math. Methods Appl. Sci., 44 (2021), 5908–5929. https://doi.org/10.1002/mma.7158 doi: 10.1002/mma.7158
    [43] C. F. M. Coimbra, Mechanica with variable-order differential operators, Ann. Phys., 12 (2003), 692–703. https://doi.org/10.1002/andp.200351511-1203 doi: 10.1002/andp.200351511-1203
    [44] Y. Cheng, X. Meng, Q. Zhang, Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp., 86 (2017), 1233–1267. https://doi.org/10.1090/mcom/3141 doi: 10.1090/mcom/3141
    [45] L. Wei, H. Wang, Y. Chen, Local discontinuous Galerkin method for a hidden-memory variable order reaction-diffusion equation, J. Appl. Math. Comput., 69 (2023), 2857–2872. https://doi.org/10.1007/s12190-023-01865-9 doi: 10.1007/s12190-023-01865-9
    [46] W. Wang, E. Barkai, Fractional advection-diffusion-asymmetry equation, Phys. Rev. Lett., 125 (2020), 240606. https://doi.org/10.1103/PhysRevLett.125.240606 doi: 10.1103/PhysRevLett.125.240606
  • This article has been cited by:

    1. Xing-Xing Yang, Guo-Bao Zhang, Yu-Cai Hao, Existence and stability of traveling wavefronts for a discrete diffusion system with nonlocal delay effects, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023160
    2. Zhi-Jiao Yang, Guo-Bao Zhang, Ge Tian, Global stability of traveling wave solutions for a discrete diffusion epidemic model with nonlocal delay effects, 2025, 66, 0022-2488, 10.1063/5.0202813
    3. Jiao Dang, Guo-Bao Zhang, Ge Tian, Wave Propagation for a Discrete Diffusive Mosquito-Borne Epidemic Model, 2024, 23, 1575-5460, 10.1007/s12346-024-00964-7
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(402) PDF downloads(28) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog