Research article Special Issues

Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth

  • We study the existence of weak solutions to a Cahn-Hilliard-Darcy system coupled witha convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’slaw. The system of equations arises from a mixture model for tumour growth accounting for transportmechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, theexistence of global weak solutions in two and three dimensions, along with new regularity results forthe velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivativeshave lower regularity compared to systems without Darcy flow, but in the two dimensional case weemploy a new regularity result for the velocity to obtain better integrability and temporal regularityfor the time derivatives. Then, we deduce the global existence of weak solutions for two variantsof the model; one where the velocity is zero and another where the chemotaxis and active transportmechanisms are absent.

    Citation: Harald Garcke, Kei Fong Lam. Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth[J]. AIMS Mathematics, 2016, 1(3): 318-360. doi: 10.3934/Math.2016.3.318

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  • We study the existence of weak solutions to a Cahn-Hilliard-Darcy system coupled witha convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’slaw. The system of equations arises from a mixture model for tumour growth accounting for transportmechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, theexistence of global weak solutions in two and three dimensions, along with new regularity results forthe velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivativeshave lower regularity compared to systems without Darcy flow, but in the two dimensional case weemploy a new regularity result for the velocity to obtain better integrability and temporal regularityfor the time derivatives. Then, we deduce the global existence of weak solutions for two variantsof the model; one where the velocity is zero and another where the chemotaxis and active transportmechanisms are absent.


    Non-linear partial differential equations are extensively used in science and engineering to model real-world phenomena [1,2,3,4]. Using fractional operators like the Riemann-Liouville (RL) and the Caputo operators which have local and singular kernels, it is difficult to express many non-local dynamics systems. Thus to describe complex physical problems, fractional operators with non-local and non-singular kernels [5,6] were defined. The Atangana-Baleanu (AB) fractional derivative operator is one of these type of fractional operators which is introduced by Atangana and Baleanu[7].

    The time fractional Kolmogorov equations (TF-KEs) are defined as

    ABDγtg(s,t)=ϑ1(s)Dsg(s,t)+ϑ2(s)Dssg(s,t)+ω(s,t),0<γ1, (1.1)

    with the initial and boundary conditions

    g(s,0)=d0(s),g(0,t)=d1(t),g(1,t)=d2(t),

    where (s,t)[0,1]×[0,1], ABDγt denotes the Atangana-Baleanu (AB) derivative operator, Dsg(s,t)=sg(s,t) and Dssg(s,t)=2s2g(s,t). If ϑ1(s) and ϑ2(s) are constants, then Eq (1.1) is presenting the time fractional advection-diffusion equations (TF-ADEs).

    Many researchers are developing methods to find the solution of partial differential equations of fractional order. Analytical solutions or formal solutions of such type of equations are difficult; therefore, numerical simulations of these equations inspire a large amount of attentions. High accuracy methods can illustrate the anomalous diffusion phenomenon more precisely. Some of the efficient techniques are Adomian decomposition [8,9], a two-grid temporal second-order scheme [10], the Galerkin finite element method [11], finite difference [12], a differential transform [13], the orthogonal spline collocation method [14], the optimal homotopy asymptotic method [15], an operational matrix (OM) [16,17,18,19,20,21,22,23,24], etc.

    The OM is one of the numerical tools to find the solution of a variety of differential equations. OMs of fractional derivatives and integration were derived using polynomials like the Chebyshev [16], Legendre [17,18], Bernstein [19], clique [20], Genocchi [21], Bernoulli [22], etc. In this work, with the help of the Hosoya polynomial (HS) of simple paths and OMs, we reduce problem (1.1) to the solution of a system of nonlinear algebraic equations, which greatly simplifies the problem under study.

    The sections are arranged as follows. In Section 2, we review some basic preliminaries in fractional calculus and interesting properties of the HP. Section 3 presents a new technique to solve the TF-KEs. The efficiency and simplicity of the proposed method using examples are discussed in Section 5. In Section 6, the conclusion is given.

    In this section we discuss some basic preliminaries of fractional calculus and the main properties of the HP. We also compute an error bound for the numerical solution.

    Definition 2.1. (See [25]) Let 0<γ1. The RL integral of order γ is defined as

    RLIγsg(s)=1Γ(γ)s0(sξ)γ1g(ξ) dξ.

    One of the properties of the fractional order of RL integral is

    RLIγssυ=Γ(υ+1)Γ(υ+1+γ)sυ+γ,υ0.

    Definition 2.2. (See [7]) Let 0<γ1, gH1(0,1) and Φ(γ) be a normalization function such that Φ(0)=Φ(1)=1 and Φ(γ)=1γ+γΓ(γ). Then, the following holds

    1) The AB derivative is defined as

    ABDγsg(s)=Φ(γ)1γs0Eγ(γ1γ(sξ)γ)g(ξ) dξ,0<γ<1,ABDγsg(s)=g(s),γ=1,

    where Eγ(s)=j=0sjΓ(γj+1) is the Mittag-Leffler function.

    2) The AB integral is given as

    ABIγsg(s)=1γΦ(γ)g(s)+γΦ(γ)Γ(γ)s0(sξ)γ1g(ξ)dξ. (2.1)

    Let vγ=1γΦ(γ) and wγ=1Φ(γ)Γ(γ); then, we can rewrite (2.1) as

    ABIγsg(s)=vγg(s)+wγΓ(γ+1)RLIγsg(s).

    The AB integral satisfies the following property [26]:

    ABIγs(ABDγsg(s))=g(s)g(0).

    In 1988, Haruo Hosoya introduced the concept of the HP [27,28]. This polynomial is used to calculate distance between vertices of a graph [29]. In [30,31], the HP of path graphs is obtained. The HP of the path graphs is described as

    ˜H(G,s)=l0d(G,l)sl,

    where d(G,l) denotes the distance between vertex pairs in the path graph [32,33]. Here we consider path graph with vertices n where nN. Based on n vertex values the Hosoya polynomials are calculated [34]. Let us consider the path Pn with n vertices; then the HP of the Pi,i=1,2,,n are computed as

    ˜H(P1,s)=l0d(P1,l)sl=1,˜H(P2,s)=1l=0d(P2,l)sl=s+2,˜H(P3,s)=2l=0d(P3,l)sl=s2+2s+3,˜H(Pn,s)=n+(n1)s+(n2)s2++(n(n2))sn2+(n(n1))sn1.

    Consider any function g(s) in L2(0,1); we can approximate it using the HP as follows:

    g(s)˜g(s)=N+1i=1hi ˜H(Pi,s)=hTH(s), (2.2)

    where

    h=[h1,h2,,hN+1]T,

    and

    H(s)=[˜H(P1,s),˜H(P2,s),,˜H(PN+1,s)]T. (2.3)

    From (2.2), we have

    h=Q1g(s),H(s),

    where Q=H(s),H(s) and , denotes the inner product of two arbitrary functions.

    Now, consider the function g(s,t)L2([0,1]×[0,1]); then, it can be expanded in terms of the HP by using the infinite series,

    g(s,t)=i=1j=1hij˜H(Pi,s)˜H(Pj,t). (2.4)

    If we consider the first (N+1)2 terms in (2.4), an approximation of the function g(s,t) is obtained as

    g(s,t)N+1i=1N+1j=1hij˜H(Pi,s)˜H(Pj,t)=HT(s)˜hH(t), (2.5)

    where

    ˜h=Q1H(s),g(s,t),H(t)Q1.

    Theorem 2.1. The integral of the vector H(s) given by (2.3) can be approximated as

    s0H(ξ)dξRH(s), (2.6)

    where R is called the OM of integration for the HP.

    Proof. Firstly, we express the basis vector of the HP, H(s), in terms of the Taylor basis functions,

    H(s)=AˆS(s), (2.7)

    where

    ˆS(s)=[1,s,,sN]T,

    and

    A=[aq,r],q,r=1,2,,N+1,

    with

    aq,r={q(r1),qr,0,q<r.

    Now, we can write

    s0H(ξ)dξ=As0ˆS(ξ)dξ=ABS(s),

    where B=[bq,r],q,r=1,2,,N+1 is an (N+1)×(N+1) matrix with the following elements

    bq,r={1q,q=r,0,qr,

    and

    S(s)=[s,s2,,sN+1]T.

    Now, by approximating sk,k=1,2,,N+1 in terms of the HP and by (2.7), we have

    {sk=A1k+1H(s),k=1,2,,N,sN+1=LTH(s),

    where A1r, r=2,3,,N+1 is the r-th row of the matrix A1 and L=Q1sN+1,H(s). Then, we get

    S(s)=EH(s),

    where E=[A12,A13,,A1N+1,LT]T. Therefore, by taking R=ABE, the proof is completed.

    Theorem 2.2. The OM of the product based on the HP is given by (2.3) can be approximated as

    CTH(s)HT(s)HT(s)ˆC,

    where ˆC is called the OM of product for the HP.

    Proof. Multiplying the vector C=[c1,c2,,cN+1]T by H(s) and HT(s) gives

    CTH(s)HT(s)=CTH(s)(ˆST(s)AT)=[CTH(s),s(CTH(s)),,sN(CTH(s))]AT=[N+1i=1ci˜H(Pi,s),N+1i=1cis˜H(Pi,s),,N+1i=1cisN˜H(Pi,s)]AT. (2.8)

    Taking ek,i=[e1k,i,e2k,i,,eN+1k,i]T and expanding sk1˜H(Pi,s)eTk,iH(s),i,k=1,2,,N+1 using the HP, we can write

    ek,i=Q110sk1˜H(Pi,s)H(s)ds=Q1[10sk1˜H(Pi,s)˜H(P1,s)ds,10sk1˜H(Pi,s)˜H(P2,s)ds,,10sk1˜H(Pi,s)˜H(PN+1,s)ds]T.

    Therefore,

    N+1i=1cisk1˜H(Pi,s)N+1i=1ci(N+1j=1ejk,i˜H(Pj,s))=N+1j=1˜H(Pj,s)(N+1i=1ciejk,i)=HT(s)[N+1i=1cie1k,i,N+1i=1cie2k,i,,N+1i=1cieN+1k,i]T=HT(s)[ek,1,ek,2,,ek,N+1]C=HT(s)EkC, (2.9)

    where Ek is an (N+1)×(N+1) matrix and the vectors ek,i for k=1,2,,N+1 are the columns of Ek. Let ¯Ek=EkC,k=1,2,,N+1. Setting ¯C=[¯E1,¯E2,,¯EN+1] as an (N+1)×(N+1) matrix and using (2.8) and (2.9), we have

    CTH(s)HT(s)=[N+1i=1ci˜H(Pi,s),N+1i=1cis˜H(Pi,s),,N+1i=1cisN˜H(Pi,s)]ATHT(s)ˆC,

    where by taking ˆC=¯CAT, the proof is completed.

    Theorem 2.3. Consider the given vector H(s) in (2.3); the fractional RL integral of this vector is approximated as

    RLIγsH(s)PγH(s),

    where Pγ is named the OM based on the HP which is given by

    Pγ=[σ1,1,1σ1,2,1σ1,N+1,12k=1σ2,1,k2k=1σ2,2,k2k=1σ2,N+1,kN+1k=1σN+1,1,kN+1k=1σN+1,2,kN+1k=1σN+1,N+1,k],

    with

    σi,j,k=(i(k1))Γ(k)ek,jΓ(k+γ).

    Proof. First, we rewrite ˜H(Pi,s) in the following form:

    ˜H(Pi,s)=ik=1(i(k1))sk1.

    Let us apply, the RL integral operator, RLIγs, on ˜H(Pi,s),i=1,,N+1; this yields

    RLIγs˜H(Pi,s)=RLIγs(ik=1(i(k1))sk1)=ik=1(i(k1))(RLIγssk1)=ik=1(i(k1))Γ(k)Γ(k+γ)sk+γ1. (2.10)

    Now, using the HP, the function sk+γ1 is approximated as:

    sk+γ1N+1j=1ek,j˜H(Pj,s). (2.11)

    By substituting (2.11) into (2.10), we have,

    RLIγs˜H(Pi,s)=ik=1(i(k1))Γ(k)Γ(k+γ)(N+1j=1ek,j˜H(Pj,s))=N+1j=1(ik=1(i(k1))Γ(k)ek,jΓ(k+γ))˜H(Pj,s)=N+1j=1(ik=1σi,j,k)˜H(Pj,s).

    Theorem 2.4. Suppose that 0<γ1 and ˜H(Pi,x) is the HP vector; then,

    ABIγtH(s)IγH(s),

    where Iγ=vγI+wγΓ(γ+1)Pγ is called the OM of the AB-integral based on the HP and I is an (N+1)×(N+1) identity matrix.

    Proof. Applying the AB integral operator, ABIγs, on H(s) yields

    ABIγsH(s)=vγH(s)+wγΓ(γ+1)RLIγsH(s).

    According to Theorem 2.3, we have that RLIγsH(s)PγH(s). Therefore

    ABIγsH(s)=vγH(s)+wγΓ(γ+1)PγH(s)=(vγI+wγΓ(γ+1)Pγ)H(s).

    Setting Iγ=vγI+wγΓ(γ+1)Pγ, the proof is complete.

    The main aim of this section is to introduce a technique based on the HP of simple paths to find the solution of the TF-KEs. To do this, we first expand Dssg(s,t) as

    Dss g(s,t)N+1i=1N+1j=1hij˜H(Pi,s)˜H(Pj,t)=HT(s)˜hH(t). (3.1)

    Integrating (3.1) with respect to s gives

    Dsg(s,t)Dsg(0,t)+HT(s)RT˜hH(t). (3.2)

    Again integrating the above equation with respect to s gives

    g(s,t)d1(t)+sDsg(0,t)+HT(s)(R2)T˜hH(t). (3.3)

    By putting s=1 into (3.3), we have

    Dsg(0,t)=d2(t)d1(t)HT(1)(R2)T˜hH(t). (3.4)

    By substituting (3.4) into (3.3), we get

    g(s,t)d1(t)+s(d2(t)d1(t)HT(1)(R2)T˜hH(t))+HT(s)(R2)T˜hH(t). (3.5)

    Now, we approximate that d1(t)=ST0H(t),d2(t)=ST1H(t) and s=HT(s)S and putting in (3.5), we get

    g(s,t)ST0H(t)+HT(s)S(ST1H(t)ST0H(t)HT(1)(R2)T˜hH(t))+HT(s)(R2)T˜hH(t).

    The above relation can be written as

    g(s,t)1×ST0H(t)+HT(s)S(ST1H(t)ST0H(t)HT(1)(R2)T˜hH(t))+HT(s)(R2)T˜hH(t).

    Approximating 1=ˆSTH(s)=HT(s)ˆS, the above relation is rewritten as

    g(s,t)HT(s)ˆSST0H(t)+HT(s)S(ST1H(t)ST0H(t)HT(1)(R2)T˜hH(t))+HT(x)(R2)T˜hH(t)=HT(s)(ˆSST0+SST1SST0SHT(1)(R2)T˜h+(R2)T˜h)H(t). (3.6)

    Setting ρ1=ˆSST0+SST1SST0SHT(1)(R2)T˜h+(R2)T˜h, we have

    g(s,t)HT(s)ρ1H(t). (3.7)

    According to (1.1), we need to obtain Ds g(s,t). Putting the approximations d1(t),d2(t) and the relation (3.4) into (3.2) yields

    Dsg(s,t)ST1H(t)ST0H(t)HT(1)(R2)T˜hH(t)+HT(s)RT˜hH(t). (3.8)

    The above relation can be written as

    Dsg(s,t)1×ST1H(t)1×ST0H(t)1×HT(1)(R2)T˜hH(t)+HT(s)RT˜hH(t). (3.9)

    Putting 1=HT(s)ˆS into the above relation, we get

    Dsg(s,t)HT(s)ˆSST1H(t)HT(s)ˆSST0H(t)HT(s)ˆSHT(1)(R2)T˜hH(t)+HT(s)RT˜hH(t)=HT(s)(ˆSST1ˆSST0ˆSHT(1)(R2)T˜h+RT˜h)H(t). (3.10)

    Setting ρ2=ˆSST1ˆSST0ˆSHT(1)(R2)T˜h+RT˜h, we have

    Dsg(s,t)HT(s)ρ2H(t). (3.11)

    Applying ABIγt to (1.1), putting g(s,t)HT(s)ρ1H(t),Dsg(s,t)HT(s)ρ2H(t), Dss g(s,t)HT(s)˜hH(t) and approximating ω(s,t)HT(s)ρ3H(t) in (1.1) yields

    HT(s)ρ1H(t)=d0(s)+ϑ1(s)HT(s)ρ2(ABIγtH(t))+ϑ2(s)HT(s)˜h(ABIγtH(t))+HT(s)ρ3(ABIγtH(t)). (3.12)

    Now approximating d0(s)HT(s)S2,ϑ1(s)ST3H(s),ϑ2(s)ST4H(s) and using Theorem 2.4, the above relation can be rewritten as

    HT(s)ρ1H(t)=HT(s)S2+ST3H(s)HT(s)ρ2IγH(t)+ST4H(s)HT(s)˜hIγH(t)+HT(s)ρ3IγH(t). (3.13)

    By Theorem 2.2, the above relation can be written as

    HT(s)ρ1H(t)=HT(s)S2×1+ST3H(s)HT(s)HT(s)^S3ρ2IγH(t)+ST4H(s)HT(s)HT(s)^S4˜hIγH(t)+HT(s)ρ3IγH(t). (3.14)

    Now approximating 1=ˆSTH(t), we have

    HT(s)ρ1H(t)=HT(s)S2ˆSTH(t)+HT(s)^S3ρ2IγH(t)+HT(s)^S4˜hIγH(t)+HT(s)ρ3IγH(t). (3.15)

    We can write the above relation as

    HT(s)(ρ1S2ˆST^S3ρ2Iγ^S4˜hIγρ3Iγ)H(t)=0. (3.16)

    Therefore we have

    ρ1S2ˆST^S3ρ2Iγ^S4˜hIγρ3Iγ=0. (3.17)

    By solving the obtained system, we find hij, i,j=1,2,,N+1. Consequently, g(s,t) can be calculated by using (3.7).

    Set I=(a,b)n,n=2,3 in Rn. The Sobolev norm is given as

    gHϵ(I)=(ϵk=0nl=0D(k)lg2L2(I))12,ϵ1,

    where D(k)lu and Hϵ(I) are the k-th derivative of g and Sobolev space, respectively. The notation |g|Hϵ;N is given as [35]

    |g|Hϵ;N(I)=(ϵk=min{ϵ,N+1}nl=0D(k)lg2L2(I))12.

    Theorem 4.1 (See [36]). Let g(s,t)Hϵ(I) with ϵ1. Considering PNg(s,t)=N+1r=1N+1n=1ar,nPr(s)Pn(t) as the best approximation of g(s,t), we have

    gPNgL2(I)CN1ϵ|g|Hϵ;N(I),

    and if 1ιϵ, then

    gPNgHι(I)CNϑ(ι)ϵ|g|Hϵ;N(I),

    with

    ϑ(ι)={0,ι=0,2ι12,ι>0.

    Lemma 4.1. The AB derivative can be written by using the fractional order RL integral as follows:

    ABDγtg(t)=Φ(γ)1γl=0ϖlRLIlγ+1tg(t),ϖ=γ1γ.

    Proof. According to the definitions of the AB derivative and the RL integral, the proof is complete.

    Theorem 4.2. Suppose that 0<γ1,|ϑ1(s)|τ1,|ϑ2(s)|τ2 and g(s,t)Hϵ(I) with ϵ1. If E(s,t) is the residual error by approximating g(s,t), then E(s,t) can be evaluated as

    E(s,t)L2(I)ϱ1(|g|Hϵ;N(I)+|sg|Hϵ;N(I)),

    where 1ιϵ and ϱ1 is a constant number.

    Proof. According to (1.1),

    ABDγt g(s,t)=ϑ1(s)Ds g(s,t)+ϑ2(s)Dss g(s,t)+ω(s,t), (4.1)

    and

    ABDγt gN(s,t)=ϑ1(s)Ds gN(s,t)+ϑ2(s)Dss gN(s,t)+ω(s,t). (4.2)

    Substituting Eqs (4.1) and (4.2) in E(s,t) yields

    E(s,t)=ABDγt(g(s,t)gN(s,t))+ϑ1(s)Ds(gN(s,t)g(s,t))+ϑ2(s)Dss(gN(s,t)g(s,t)).

    and then

    E(s,t)2L2(I)ABDγt(g(s,t)gN(s,t))2L2(I)+τ1Ds(g(s,t)gN(s,t))2L2(I)+τ2Dss(g(s,t)gN(s,t))2L2(I). (4.3)

    Now, we must find a bound for ABDγt(g(s,t)gN(s,t))L2(I). In view of [26], and by using Lemma 4.1, in a similar way, we write

    ABDγt(g(s,t)gN(s,t))2L2(I)=Φ(γ)1γl=0ϖlRLIlγ+1t(Dtg(s,t)DtgN(s,t))2L2(I)(Φ(γ)1γl=0ϖlΓ(lγ+2))2Dtg(s,t)DtgN(s,t)2L2(I)(Φ(γ)1γEγ,2(ϖ))2g(s,t)gN(s,t)2Hι(I).

    Therefore,

    ABDγt(g(s,t)gN(s,t))L2(I)δ1CNϑ(ι)ϵ|g|Hϵ;N(I), (4.4)

    where Φ(γ)1γEγ,2(ϖ)δ1. Thus, from (4.4), we can write

    ABDγt(g(s,t)gN(s,t))2L2(I)δ1|g|Hϵ;N(I), (4.5)

    where |g|Hϵ;N(I)=CNϑ(ι)ϵ|g|Hϵ;N(I). By Theorem 4.1,

    Ds(g(s,t)gN(s,t))L2(I)CNϑ(ι)ϵ|g|Hϵ;N(I)=|g|Hϵ;N(I), (4.6)

    and

    Dss(g(s,t)gN(s,t))L2(I)=Ds(Ds(g(s,t)gN(s,t)))L2(I)Dsg(s,t)DsgN(s,t)Hι(I)|Dsg|Hϵ;N(I), (4.7)

    where |Dsg|Hϵ;N(I)=CNϑ(ι)ϵ|Dsg|Hϵ;N(I). Taking ϱ1=max{δ1+τ1,τ2} and substituting (4.5)–(4.7) into (4.3); then, the desired result is obtained.

    In this section, the proposed technique which is described in Section 3 is shown to be tested using some numerical examples. The codes are written in Mathematica software.

    Example 5.1. Consider (1.1) with ϑ1(s)=1,ϑ2(s)=0.1 and ω(s,t)=0. The initial and boundary conditions can be extracted from the analytical solution g(s,t)=τ0eτ1tτ2s when γ=1. Setting τ0=1,τ1=0.2,τ2=ϑ1(s)+ϑ21(s)+4ϑ2(s)τ12ϑ2(s), considering N=3 and using the proposed technique, the numerical results of the TF-ADE are reported in Tables 1 and 2, and in Figures 13.

    Table 1.  (Example 5.1) Numerical results of the absolute error when γ=0.99,N=3, t=1.
    s Method of [21] The presented method
    0.1 1.05799e2 3.86477e4
    0.2 1.21467e2 1.33870e4
    0.3 4.94776e3 4.08507e5
    0.4 2.35280e4 1.48842e4
    0.5 2.36604e3 2.01089e4
    0.6 1.08676e2 2.08410e4
    0.7 2.18851e2 1.81459e4
    0.8 2.91950e2 1.30730e4
    0.9 2.49148e2 6.65580e5

     | Show Table
    DownLoad: CSV
    Table 2.  (Example 5.1) Numerical results of the absolute error when γ=0.99,N=3, s=0.75.
    t Method of [19] The presented method
    0.1 1.13874e3 2.15272e3
    0.2 1.41664e3 2.32350e3
    0.3 1.62234e3 2.30934e3
    0.4 1.76917e3 2.14768e3
    0.5 1.87045e3 1.87583e3
    0.6 1.93953e3 1.53092e3
    0.7 1.98971e3 1.14997e3
    0.8 2.03434e3 7.69801e4
    0.9 2.08671e3 4.27112e4

     | Show Table
    DownLoad: CSV
    Figure 1.  (Example 5.1) The absolute error at some selected points when (a) γ=0.8, (b) γ=0.9, (c) γ=0.99, (d) γ=1.
    Figure 2.  (Example 5.1) Error contour plots when (a) γ=0.99, (b) γ=1, (c) γ=0.8, (d) γ=0.9.
    Figure 3.  (Example 5.1) The absolute error at some selected points when (a) γ=0.8, (b) γ=0.9, (c) γ=0.99, (d) γ=1.

    Example 5.2. Consider (1.1) with ϑ1(s)=s,ϑ2(s)=s22 and ω(s,t)=0. The initial and boundary conditions can be extracted from the analytical solution g(s,t)=sEα(tα). By setting N=5 and using the proposed technique, the numerical results of the TF–KE are as reported in Figures 46.

    Figure 4.  (Example 5.2) The absolute error at some selected points when (a) γ=0.7, (b) γ=0.8, (c) γ=0.9, (d) γ=1.
    Figure 5.  (Example 5.2) Error contour plots when (a) γ=0.7, (b) γ=0.8, (c) γ=0.9, (d) γ=1.
    Figure 6.  (Example 5.2) The absolute error at some selected points when (a) γ=0.7, (b) γ=0.8, (c) γ=0.9, (d) γ=1.

    Time fractional Kolmogorov equations and time fractional advection-diffusion equations have been used to model many problems in mathematical physics and many scientific applications. Developing efficient methods for solving such equations plays an important role. In this paper, a proposed technique is used to solve TF-ADEs and TF-KEs. This technique reduces the problems under study to a set of algebraic equations. Then, solving the system of equations will give the numerical solution. An error estimate is provided. This method was tested on a few examples of TF-ADEs and TF-KEs to check the accuracy and applicability. This method might be applied for system of fractional order integro-differential equations and partial differential equations as well.

    The authors declare that they have not used artificial intelligence tools in the creation of this article.

    The authors would like to thank for the support from Scientific Research Fund Project of Yunnan Provincial Department of Education, No. 2022J0949. The authors also would like to thank the anonymous reviewers for their valuable and constructive comments to improve our paper.

    The authors declare there is no conflicts of interest.

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