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Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives

  • The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.

    Citation: M. Mossa Al-Sawalha, Rasool Shah, Adnan Khan, Osama Y. Ababneh, Thongchai Botmart. Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives[J]. AIMS Mathematics, 2022, 7(10): 18334-18359. doi: 10.3934/math.20221010

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  • The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.



    Fractional calculus is as old as ordinary calculus, i. e. at three centuries; however, it is not widely used in research and engineering. In a letter to Leibniz dated September 30, 1695, L'Hopital introduced the concept of fractional-order derivatives. In Lacroix's writings, P. S. Laplace defined a fractional derivative of arbitrary order in 1812. Several researchers have developed excellent literature on fractional differentiation and integration operators for the purposes of extending scientific and technical areas, including Caputo [1], Oldham and Spanier [2], Carpinteri and Mainardi [3], Samko et al. [4], Ahmed et al. [5], Podlubny [6], Atangana and Alabaraoye [7], Kumar et al. [8], Yin et al. [9] and Arife et al. [10]. The beauty of fractional derivatives is that they are not local point properties. The genetic and nonlocal dispersed effects are taken into account in fractional calculus. This property makes it more accurate than the integer-order derivative description.

    Due to their proven applications in science and engineering, fractional differential equations have grown in prominence and popularity. These equations, for example, are increasingly being utilized to simulate problems in signal processing, biology, fluid mechanics, acoustics, diffusion, electromagnetism and a wide range of other physical phenomena. The nonlocal quality of fractional differential equations is the most essential advantage of utilizing them in these and other applications. The integer order differential operator is well-known to be a local operator, whereas the fractional order differential operator is not. This indicates that a system's next state is determined not just by its current state, but also by all of its previous states. The theory of fractional differential equations better and more systematically describes natural occurrences [11,12,13,14,15,16,17,18,19].

    Fractional coupled systems are commonly used to investigate the complex behavior of plasma that contains several components such as atoms, free electrons and ions. Many scholars have tried to assess this behavior. Recently, Paul Kersten and Joseph Krasil'shchik investigated the Korteweg-de Vries (KdV) equation and modified KdV (mKdV) equation, proposing absolute complexity between coupled KdV-mKdV nonlinear systems for the investigation of nonlinear system behavior [20,21,22,23]. Many scholars have proposed numerous variants of this Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system [24,25,26,27,28]. Among these variants, the nonlinear fractional Kersten-Krasil'shchik linked KdV-mKdV system provides a mathematical model for understanding the behavior of multi-component plasma for waves travelling along the positive zeta axis:

    DαχF+F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ=0,  χ>0,  ϑR,  0<α1,DαχG+G3ϑ3G2Gϑ3FGϑ+3FϑG=0,  χ>0,  ϑR,  0<α1, (1.1)

    where ϑ is a spatial coordinate and χ is a time coordinate. The fractional operator's order is represented by the factor α. The Caputo form is used to study this operator. When α=1, the fractional coupled system becomes a classical system, as follows:

    Fχ+F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ=0,  χ>0,  ϑR,Gχ+G3ϑ3G2Gϑ3FGϑ+3FϑG=0,  χ>0,  ϑR. (1.2)

    If G=0, the Kersten-Krasil'shchik linked KdV-mKdV system is converted to the well-known KdV system as follows:

    Fχ+F3ϑ6FFϑ=0,  χ>0,  ϑR. (1.3)

    When F=0, the Kersten-Krasil'shchik coupled KdV-mKdV system becomes the well-known mKdV system as follows:

    Gχ+G3ϑ3G2Gϑ=0,  χ>0,  ϑR. (1.4)

    As a result, the Kersten-Krasil'shchik linked KdV-mKdV system can be considered of as a combination of the KdV system and the mKdV system, which are represented by (1.2) to (1.4). We also investigate the following fractional nonlinear two component homogeneous time fractional coupled third order KdV system in this work as follows:

    DαχFF3ϑFFϑGGϑ=0,χ>0,ϑR,0<α1,DαχG+2G3ϑFGϑ=0,χ>0,ϑR,0<α1, (1.5)

    where χ is the temporal coordinate, ϑ is the spatial coordinate and α is the fractional operator's order factor. The Caputo form is used to study this operator. When α=1, the fractional coupled system becomes a classical system, as follows:

    FχF3ϑFFϑGGϑ=0,  χ>0,  ϑR,Gχ+2G3ϑFGϑ=0,  χ>0,ϑR. (1.6)

    To solve the differential equations, we employ the natural decomposition method (NDM), which combines the natural transform (NT) and Adomian decomposition method and offers the approximate solution in series form. The proposed method has been implemented with the aid of two different fractional derivatives to solve two nonlinear systems. Many researchers have employed the NDM to obtain approximate analytical solutions, and it has provided reliable and closely convergent results. The calculations were done in Maple. The convergence of the proposed technique was also achieved by extending the concept discussed in [29,30].

    The organization of the present paper is as follows: Section 2 gives some basic definitions and the properties of the natural transform that is used in our present work. Section 3 handles the methodology of the proposed technique. In Section 4, we presented the convergence analysis of the suggested technique. Section 5 gives the implementation of the suggested approach to approximate the solution of the above systems. Finally, we discuss the obtained results and conclusion.

    In this section, we recall some basic definitions and results from fractional calculus.

    Definition 2.1. The Riemann-Liouville integral of a function jCμ, μ1, having fractional-order is defined as [31]

    Iαj(ρ)=1Γ(α)ρ0(ρμ)α1j(μ)dμ,  α>0,  ρ>0,and  I0j(ρ)=j(ρ). (2.1)

    Definition 2.2. The derivative of j(ρ) with fractional-order in the Caputo sense is given as [31]

    Dαρj(ρ)=ImαDmj(ρ)=1Γ(mα)ρ0(ρμ)mα1D(j(μ))dμ, (2.2)

    for m1<αm,  mN,  ρ>0,jCmμ,μ1.

    Definition 2.3. The derivative of j(ρ) with fractional-order in the Caputo-Fabrizio (CF) manner is given as [31]

    Dαρj(ρ)=F(α)1αρ0exp(α(ρμ)1α)D(j(μ))dμ, (2.3)

    where 0<α<1 and F(α) is a normalization function with F(0)=F(1)=1.

    Definition 2.4. The derivative of j(ρ) with fractional-order in the Atangana-Baleanu-Caputo operator (ABC) manner is given as [31]

    Dαρj(ρ)=B(α)1αρ0Eα(α(ρμ)α1α)D(j(μ))dμ, (2.4)

    where 0<α<1, B(α) is normalization function and Eα(z)=m=0zmΓ(αm+1) is the Mittag-Leffler function.

    Definition 2.5. For the function F(χ), the natural transformation is given as

    N(F(χ))=U(ζ,τ)=eζχF(τχ)dχ,  ζ,τ(,), (2.5)

    and for χ(0,), the natural transformation of F(χ) is given as

    N(F(χ)H(χ))=N+F(χ)=U+(ζ,τ)=0eζχF(τχ)dχ,  ζ,τ(0,), (2.6)

    where H(χ) is the Heaviside function.

    Definition 2.6. For the function F(ζ,τ), the inverse natural transformation is given as

    N1[U(ζ,τ)]=F(χ),  χ0. (2.7)

    Lemma 2.1. Let the F1(χ) and F2(χ) natural transformations be U1(ζ,τ) and U2(ζ,τ); so,

    N[c1F1(χ)+c2F2(χ)]=c1N[F1(χ)]+c2N[F2(χ)]=c1U1(ζ,τ)+c2U2(ζ,τ), (2.8)

    where c1 and c2 constants.

    Lemma 2.2. Let the F1(χ) and F2(χ) inverse natural transformations be F1(ζ,τ) and F2(ζ,τ); so,

    {N}1[c1U1(ζ,τ)+c2U2(ζ,τ)]=c1N1[U1(ζ,τ)]+c2N1[U2(ζ,τ)]=c1F1(χ)+c2F2(χ), (2.9)

    where c1 and c2 constants.

    Definition 2.7. The NT of DαχF(χ) in the Caputo sense is stated as [31]

    N[DαχF(χ)]=(ζτ)α(N[F(χ)](1ζ)F(0)). (2.10)

    Definition 2.8. The NT of DαχF(χ) in the CF sense is defined as [31]

    N[DαχF(χ)]=11α+α(τζ)(N[F(χ)](1ζ)F(0)). (2.11)

    Definition 2.9. The NT of DαχF(χ) in the ABC manner is given as [31]

    N[DαχF(χ)]=M[α]1α+α(τζ)α(N[F(χ)](1ζ)F(0)), (2.12)

    with M[α] denoting a normalization function.

    This section is concerned with the general procedure for numerical treatment of the below equation.

    DαχF(ϑ,χ)=L(F(ϑ,χ))+N(F(ϑ,χ))+h(ϑ,χ)=M(ϑ,χ), (3.1)

    where the initial condition

    F(ϑ,0)=ϕ(ϑ) (3.2)

    has L and N linear and nonlinear terms, and where h(ϑ,χ) represents the source term.

    By means of the NT and CF fractional derivative, Eq (3.1) can be stated as

    1p(α,τ,ζ)(N[F(ϑ,χ)]ϕ(ϑ)ζ)=N[M(ϑ,χ)], (3.3)

    with

    p(α,τ,ζ)=1α+α(τζ). (3.4)

    By taking the natural inverse transform, we get

    F(ϑ,χ)=N1(ϕ(ϑ)ζ+p(α,τ,ζ)N[M(ϑ,χ)]). (3.5)

    Assume that for F(ϑ,χ), the series form solution is determined as

    F(ϑ,χ)=i=0Fi(ϑ,χ), (3.6)

    and the N(F(ϑ,χ)) decomposition is given as

    N(F(ϑ,χ))=i=0Ai(F0,...,Fi), (3.7)

    with Ai representing the Adomian polynomials, which is illustrated as

    An=1n!dndεnN(ε,Σnk=0εkFk)|ε=0.

    By putting Eqs (3.6) and (3.7) into Eq (3.5), we have

    i=0Fi(ϑ,χ)=N1(ϕ(ϑ)ζ+p(α,τ,ζ)N[h(ϑ,χ)])+N1(p(α,τ,ζ)N[i=0L(Fi(ϑ,χ))+Ai]). (3.8)

    From Eq (3.8), we obtain

    FCF0(ϑ,χ)=N1(ϕ(ϑ)ζ+p(α,τ,ζ)N[h(ϑ,χ)]),FCF1(ϑ,χ)=N1(p(α,τ,ζ)N[L(F0(ϑ,χ))+A0]),FCFl+1(ϑ,χ)=N1(p(α,τ,ζ)N[L(Fl(ϑ,χ))+Al]),  l=1,2,3,. (3.9)

    In this way, the solution of Eq (3.1) is obtained by putting Eq (3.9) into Eq (3.6) to solve for the NTDMCF,

    FCF(ϑ,χ)=FCF0(ϑ,χ)+FCF1(ϑ,χ)+FCF2(ϑ,χ)+. (3.10)

    By means of the NT and ABC fractional derivative, Eq (3.1) can be stated as

    1q(α,τ,ζ)(N[F(ϑ,χ)]ϕ(ϑ)ζ)=N[M(ϑ,χ)], (3.11)

    with

    q(α,τ,ζ)=1α+α(τζ)αB(α). (3.12)

    By taking the natural inverse transform, we get

    F(ϑ,χ)=N1(ϕ(ϑ)ζ+q(α,τ,ζ)N[M(ϑ,χ)]). (3.13)

    In terms of the Adomain decomposition, we have

    i=0Fi(ϑ,χ)=N1(ϕ(ϑ)ζ+q(α,τ,ζ)N[h(ϑ,χ)])+N1(q(α,τ,ζ)N[i=0L(Fi(ϑ,χ))+Ai]). (3.14)

    From Eq (3.8), we have the following:

    FABC0(ϑ,χ)=N1(ϕ(ϑ)ζ+q(α,τ,ζ)N[h(ϑ,χ)]),FABC1(ϑ,χ)=N1(q(α,τ,ζ)N[L(F0(ϑ,χ))+A0]),FABCl+1(ϑ,χ)=N1(q(α,τ,ζ)N[L(Fl(ϑ,χ))+Al]),  l=1,2,3,. (3.15)

    In this way, the solution of Eq (3.1) is obtained to solve for the NTDMABC:

    FABC(ϑ,χ)=FABC0(ϑ,χ)+FABC1(ϑ,χ)+FABC2(ϑ,χ)+. (3.16)

    This section is concerned with the NTDMABC and NTDMCF convergence and uniqueness.

    Theorem 4.1. Let |L(F)L(F)|<γ1|FF| and |N(F)N(F)|<γ2|FF|, where F:=F(μ,χ) and F:=F(μ,χ) are two variable functions values, and γ1, γ2 are Lipschitz constants.

    The operators L and N are given in Eq (3.1). Thus, the solution for the NTDMCF is unique for Eq (3.1) when 0<(γ1+γ2)(1α+αχ)<1 for all χ.

    Proof. Let K=(C[J],||.||), where norm ||ϕ(χ)||=maxχJ|ϕ(χ)| is the Banach space and continuous on the J=[0,T] interval. Let I:KK be a nonlinear mapping with

    FCl+1=FC0+N1[p(α,τ,ζ)N[L(Fl(μ,χ))+N(Fl(μ,χ))]],  l0.
    ||I(F)I(F)||maxχJ|N1[p(α,τ,ζ)N[L(F)L(F)]+p(α,τ,ζ)N[N(F)N(F)]|]maxχJ[γ1N1[p(α,τ,ζ)N[|FF|]]+γ2N1[p(α,τ,ζ)N[|FF|]]]maxχJ(γ1+γ2)[N1[p(α,τ,ζ)N|FF|]](γ1+γ2)[N1[p(α,τ,ζ)N||FF||]]=(γ1+γ2)(1α+αχ)||FF||. (4.1)

    So, I is a contraction as 0<(γ1+γ2)(1α+αχ)<1. Thus, by means of the Banach fixed point theorem, the solution of Eq (3.1) is unique.

    Theorem 4.2. According to the above theorem, the solution of Eq (3.1) is unique for the NTDMABC when 0<(γ1+γ2)(1α+αχαΓ(α+1))<1 for all χ.

    Proof. Now, from the theorem above, let K=(C[J],||.||) be the Banach space that is continuous on the J=[0,T] interval. Let I:KK be the nonlinear mapping with

    FCl+1=FC0+N1[p(α,τ,ζ)N[L(Fl(μ,χ))+N(Fl(μ,χ))]],  l0.
    ||I(F)I(F)||maxχJ|N1[q(α,τ,ζ)N[L(F)L(F)]+q(α,τ,ζ)N[N(F)N(F)]|]maxχJ[γ1N1[q(α,τ,ζ)N[|FF|]]+γ2N1[q(α,τ,ζ)N[|FF|]]]maxχJ(γ1+γ2)[N1[q(α,τ,ζ)N|FF|]](γ1+γ2)[N1[q(α,τ,ζ)N||FF||]]=(γ1+γ2)(1α+αχαΓ(α+1))||FF||. (4.2)

    So, I is a contraction as 0<(γ1+γ2)(1α+αχαΓ(α+1))<1. Thus, by means of the Banach fixed point theorem, the solution of Eq (3.1) is unique.

    Theorem 4.3. Let L and N be Lipschitz functions as given in the above theorems; then, the solution for the NTDMCF is convergent for Eq (3.1).

    Proof. Let us consider K to be the Banach space as stated above and let Fm=mr=0Fr(μ,χ). To prove that Fm is a Cauchy sequence in H, let

    ||FmFn||=maxχJ|mr=n+1Fr|,  n=1,2,3,maxχJ|N1[p(α,τ,ζ)N[mr=n+1(L(Fr1)+N(Fr1))]]|=maxχJ|N1[p(α,τ,ζ)N[m1r=n+1(L(Fr)+N(Fr))]]|maxχJ|N1[p(α,τ,ζ)N[(L(Fm1)L(Fn1)+N(Fm1)N(Fn1))]]|γ1maxχJ|N1[p(α,τ,ζ)N[(L(Fm1)L(Fn1))]]|+γ2maxχJ|N1[p(α,τ,ζ)N[(N(Fm1)N(Fn1))]]|=(γ1+γ2)(1α+αχ)||Fm1Fn1||. (4.3)

    Let m=n+1; then,

    ||Fn+1Fn||γ||FnFn1||γ2||Fn1Fn2||γn||F1F0||, (4.4)

    where γ=(γ1+γ2)(1α+αχ). Thus, we have that

    ||FmFn||||Fn+1Fn||+||Fn+2Fn+1||++||FmFm1||,(γn+γn+1++γm1)||F1F0||γn(1γmn1γ)||F1||. (4.5)

    As 0<γ<1, we have that 1γmn<1. Thus,

    ||FmFn||γn1γmaxχJ||F1||. (4.6)

    Since ||F1||<,  ||FmFn||0 when n. In this way, Fm is a Cauchy sequence in K and is convergent.

    Theorem 4.4. Let L and N be Lipschitz functions as given in the above theorems; then, the solution for the NTDMABC is convergent for Eq (3.1).

    Proof. Suppose Fm=mr=0Fr(μ,χ). To prove that Fm is a Cauchy sequence in K, let

    ||FmFn||=maxχJ|mr=n+1Fr|,  n=1,2,3,maxχJ|N1[q(α,τ,ζ)N[mr=n+1(L(Fr1)+N(Fr1))]]|=maxχJ|N1[q(α,τ,ζ)N[m1r=n+1(L(Fr)+N(ur))]]|maxχJ|N1[q(α,τ,ζ)N[(L(Fm1)L(Fn1)+N(Fm1)N(Fn1))]]|
    γ1maxχJ|N1[q(α,τ,ζ)N[(L(Fm1)L(Fn1))]]|+γ2maxχJ|N1[p(α,τ,ζ)N[(N(Fm1)N(Fn1))]]|=(γ1+γ2)(1α+αχαΓ(α+1))||Fm1Fn1||. (4.7)

    Suppose m=n+1; thus,

    ||Fn+1Fn||γ||FnFn1||γ2||Fn1Fn2||γn||F1F0||, (4.8)

    where γ=(γ1+γ2)(1α+αχαΓ(α+1)). Thus, we have that

    ||FmFn||||Fn+1Fn||+||Fn+2Fn+1||++||FmFm1||,(γn+γn+1++γm1)||F1F0||γn(1γmn1γ)||F1||. (4.9)

    As 0<γ<1, we have that 1γmn<1. Thus,

    ||FmFn||γn1γmaxχJ||F1||. (4.10)

    Since ||F1||<,  ||FmFn||0 when n. In this way, Fm is a Cauchy sequence in K and convergent.

    Example 5.1. Let us consider the fractional Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system as

    DαχF+F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ=0,  χ>0,  ϑR,  0<α1,DαχG+G3ϑ3G2Gϑ3FGϑ+3FϑG=0, (5.1)

    where the initial conditions are as follows:

    F(ϑ,0)=c2csech2(cϑ),    c>0,G(ϑ,0)=2csech(cϑ). (5.2)

    By taking the NT, we get

    N[DαχF(ϑ,χ)]=N[F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ],N[DαχG(ϑ,χ)]=N[G3ϑ3G2Gϑ3FGϑ+3FϑG]. (5.3)

    Thus, we have that

    1ζαN[F(ϑ,χ)]ζ2αF(ϑ,0)=N[F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ],1ζαN[G(ϑ,χ)]ζ2αF(ϑ,0)=N[G3ϑ3G2Gϑ3FGϑ+3FϑG]. (5.4)

    By simplification, we get

    N[F(ϑ,χ)]=ζ2[c2csech2(cϑ)]α(ζα(ζα))ζ2N[F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ],N[G(ϑ,χ)]=ζ2[2csech(cϑ)]α(ζα(ζα))ζ2N[G3ϑ3G2Gϑ3FGϑ+3FϑG]. (5.5)

    By taking the inverse NT, we have that

    F(ϑ,χ)=[c2csech2(cϑ)]N1[α(ζα(ζα))ζ2N{F3ϑ6FFϑ+3GG3ϑ+3GϑG2ϑ3FϑG2+6FGGϑ}],G(ϑ,χ)=[2csech(cϑ)]N1[α(ζα(ζα))ζ2N{G3ϑ3G2Gϑ3FGϑ+3FϑG}]. (5.6)

    Solution for the NDMCF:

    Assume that for the unknown functions F(ϑ,χ) and G(ϑ,χ), the series form solutions are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ)andG(ϑ,χ)=l=0Fl(ϑ,χ). (5.7)

    The nonlinear components in terms of the Adomian polynomials are given as 6FFϑ+3GG3ϑ=m=0Am, 3GϑG2ϑ3FϑG2=m=0Bm, 6FGGϑ=m=0Cm and 3G2Gϑ3FGϑ+3FϑG=m=0Dm. With the help of these terms, Eq (5.6) can be stated as

    l=0Fl+1(ϑ,χ)=c2csech2(cϑ)N1[α(ζα(ζα))ζ2N{F3ϑ+l=0Al+l=0Bl+l=0Cl}],l=0Gl+1(ϑ,χ)=2csech(cϑ)N1[α(ζα(ζα))ζ2N{F3ϑ+l=0Dl}]. (5.8)

    By comparing both sides of Eq (5.8), we have that

    F0(ϑ,χ)=c2csech2(cϑ),G0(ϑ,χ)=2csech(cϑ),
    F1(ϑ,χ)=8c52sinh(cϑ)sech3(cϑ)(α(χ1)+1),G1(ϑ,χ)=4c2sinh(cϑ)sech2(cϑ)(α(χ1)+1), (5.9)
    F2(ϑ,χ)=16c4[2cosh2(cϑ)3]sech4(cϑ)((1α)2+2α(1α)χ+α2χ22),G2(ϑ,χ)=8c72[cosh2(cϑ)2]sech3(cϑ)((1α)2+2α(1α)χ+α2χ22). (5.10)

    In this way, given (l3), the remaining terms for Fl and Gl are easy to get. Thus, the series form solutions are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ)=F0(ϑ,χ)+F1(ϑ,χ)+F2(ϑ,χ)+,F(ϑ,χ)=c2csech2(cϑ)+8c52sinh(cϑ)sech3(cϑ)(α(χ1)+1)16c4[2cosh2(cϑ)3]sech4(cϑ)((1α)2+2α(1α)χ+α2χ22)+,G(ϑ,χ)=l=0Gl(ϑ,χ)=G0(ϑ,χ)+G1(ϑ,χ)+G2(ϑ,χ)+,G(ϑ,χ)=2csech(cϑ)4c2sinh(cϑ)sech2(cϑ)(α(χ1)+1)+8c72[cosh2(cϑ)2]sech3(cϑ)((1α)2+2α(1α)χ+α2χ22)+. (5.11)

    Solution for the NDMABC:

    Assume that for the unknown functions F(ϑ,χ) and G(ϑ,χ), the series form solutions respectively are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ),G(ϑ,χ)=l=0Fl(ϑ,χ). (5.12)

    The nonlinear components in terms of the Adomian polynomials are given as 6FFϑ+3GG3ϑ=m=0Am, 3GϑG2ϑ3FϑG2=m=0Bm, 6FGGϑ=m=0Cm and 3G2Gϑ3FGϑ+3FϑG=m=0Dm. With the help of these terms, Eq (5.6) can be stated as

    l=0Fl+1(ϑ,χ)=c2csech2(cϑ)N1[τα(ζα+α(ταζα))ζ2αN{F3ϑ+l=0Al+l=0Bl+l=0Cl}],l=0Gl+1(ϑ,χ)=2csech(cϑ)N1[τα(ζα+α(ταζα))ζ2αN{F3ϑ+l=0Dl}]. (5.13)

    By comparing both sides of Eq (5.13), we have that

    F0(ϑ,χ)=c2csech2(cϑ),G0(ϑ,χ)=2csech(cϑ),
    F1(ϑ,χ)=8c52sinh(cϑ)sech3(cϑ)(1α+αχαΓ(α+1)),G1(ϑ,χ)=4c2sinh(cϑ)sech2(cϑ)(1α+αχαΓ(α+1)), (5.14)
    F2(ϑ,χ)=16c4[2cosh2(cϑ)3]sech4(cϑ)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2],G2(ϑ,χ)=8c72[cosh2(cϑ)2]sech3(cϑ)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2]. (5.15)

    In this way, given (l3), the remaining terms for Fl and Gl are easy to get. Thus, the series form solutions are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ)=F0(ϑ,χ)+F1(ϑ,χ)+F2(ϑ,χ)+,F(ϑ,χ)=c2csech2(cϑ)+8c52sinh(cϑ)sech3(cϑ)(1α+αχαΓ(α+1))16c4[2cosh2(cϑ)3]sech4(cϑ)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2]+,G(ϑ,χ)=l=0Gl(ϑ,χ)=G0(ϑ,χ)+G1(ϑ,χ)+G2(ϑ,χ)+,G(ϑ,χ)=2csech(cϑ)4c2sinh(cϑ)sech2(cϑ)(1α+αχαΓ(α+1))+8c72[cosh2(cϑ)2]sech3(cϑ)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2]+. (5.16)

    When α=1, we get the exact solution as

    F(ϑ,χ)=c2csech2(c(ϑ+2cχ)),G(ϑ,χ)=2csech(c(ϑ+2cχ)). (5.17)

    Example 5.2. Let us consider the time-fractional homogeneous two component coupled third order KdV system as

    DαχFF3ϑFFϑGGϑ=0,  χ>0,  ϑR,  0<α1,DαχG+2G3ϑFGϑ=0, (5.18)

    where the initial conditions are as follows:

    F(ϑ,0)=36tanh2(ϑ2),G(ϑ,0)=3c2tanh(ϑ2). (5.19)

    By taking the NT, we get

    N[DαχF(ϑ,χ)]=N[F3ϑFFϑGGϑ],N[DαχG(ϑ,χ)]=N[2G3ϑFGϑ]. (5.20)

    Thus, we have

    1ζαN[F(ϑ,χ)]ζ2αF(ϑ,0)=N[F3ϑFFϑGGϑ],1ζαN[G(ϑ,χ)]ζ2αF(ϑ,0)=N[2G3ϑFGϑ]. (5.21)

    By simplification, we get

    N[F(ϑ,χ)]=ζ2[36tanh2(ϑ2)]α(ζα(ζα))ζ2N[F3ϑFFϑGGϑ],N[G(ϑ,χ)]=ζ2[3c2tanh(ϑ2)]α(ζα(ζα))ζ2N[2G3ϑFGϑ]. (5.22)

    By taking the inverse NT, we have that

    F(ϑ,χ)=36tanh2(ϑ2)N1[α(ζα(ζα))ζ2N{F3ϑFFϑGGϑ}],G(ϑ,χ)=[3c2tanh(ϑ2)]N1[α(ζα(ζα))ζ2N{2G3ϑFGϑ}]. (5.23)

    Solution for the NTDMCF:

    Assume that for the unknown functions F(ϑ,χ) and G(ϑ,χ), the series form solution are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ)andG(ϑ,χ)=l=0Fl(ϑ,χ). (5.24)

    The nonlinear components in terms of the Adomian polynomials are given as FFϑGGϑ=m=0Am and FGϑ=m=0Bm. With the help of these terms, Eq (5.23) can be stated as

    l=0Fl+1(ϑ,χ)=36tanh2(ϑ2)N1[α(ζα(ζα))ζ2N{F3ϑ+l=0Al}],l=0Gl+1(ϑ,χ)=3c2tanh(ϑ2)N1[α(ζα(ζα))ζ2N{2G3ϑl=0Bl}]. (5.25)

    By comparing both sides of Eq (5.25), we have that

    F0(ϑ,χ)=36tanh2(ϑ2),G0(ϑ,χ)=3c2tanh(ϑ2),
    F1(ϑ,χ)=6sech2(ϑ2)tanh(ϑ2)(α(χ1)+1),G1(ϑ,χ)=3c2sech2(ϑ2)tanh(ϑ2)(α(χ1)+1), (5.26)
    F2(ϑ,χ)=3[2+7sech2(ϑ2)15sech4(ϑ2)]sech2(ϑ2)((1α)2+2α(1α)χ+α2χ22),G2(ϑ,χ)=3c22[2+21sech2(ϑ2)24sech4(ϑ2)]sech2(ϑ2)((1α)2+2α(1α)χ+α2χ22). (5.27)

    In this way, given (l3), the remaining terms for Fl and Gl are easy to get. Thus, the series form solutions are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ)=F0(ϑ,χ)+F1(ϑ,χ)+F2(ϑ,χ)+,F(ϑ,χ)=36tanh2(ϑ2)+6sech2(ϑ2)tanh(ϑ2)(α(χ1)+1)+3[2+7sech2(ϑ2)15sech4(ϑ2)]sech2(ϑ2)((1α)2+2α(1α)χ+α2χ22)+,G(ϑ,χ)=l=0Gl(ϑ,χ)=G0(ϑ,χ)+G1(ϑ,χ)+G2(ϑ,χ)+,G(ϑ,χ)=3c2tanh(ϑ2)+3c2sech2(ϑ2)tanh(ϑ2)(α(χ1)+1)+3c22[2+21sech2(ϑ2)24sech4(ϑ2)]sech2(ϑ2)((1α)2+2α(1α)χ+α2χ22)+. (5.28)

    Solution for the NTDMABC:

    Assume that for the unknown functions F(ϑ,χ) and G(ϑ,χ), the series form solutions are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ),G(ϑ,χ)=l=0Fl(ϑ,χ). (5.29)

    The nonlinear components in terms of the Adomian polynomials are given as FFϑGGϑ=m=0Am and FGϑ=m=0Bm. Given these terms, Eq (5.23) can be stated as

    l=0Fl+1(ϑ,χ)=36tanh2(ϑ2)+N1[τα(ζα+α(ταζα))ζ2αN{F3ϑ+l=0Al}],l=0Gl+1(ϑ,χ)=3c2tanh(ϑ2)+N1[τα(ζα+α(ταζα))ζ2αN{2G3ϑl=0Bl}]. (5.30)

    By comparing both sides of Eq (5.30), we have that

    F0(ϑ,χ)=36tanh2(ϑ2),G0(ϑ,χ)=3c2tanh(ϑ2),
    F1(ϑ,χ)=6sech2(ϑ2)tanh(ϑ2)(1α+αχαΓ(α+1)),G1(ϑ,χ)=3c2sech2(ϑ2)tanh(ϑ2)(1α+αχαΓ(α+1)), (5.31)
    F2(ϑ,χ)=3[2+7sech2(ϑ2)15sech4(ϑ2)]sech2(ϑ2)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2],G2(ϑ,χ)=3c22[2+21sech2(ϑ2)24sech4(ϑ2)]sech2(ϑ2)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2]. (5.32)

    In this way, given (l3), the remaining terms for Fl and Gl are easy to get. Thus, the series form solutions are given as

    F(ϑ,χ)=l=0Fl(ϑ,χ)=F0(ϑ,χ)+F1(ϑ,χ)+F2(ϑ,χ)+,F(ϑ,χ)=36tanh2(ϑ2)+6sech2(ϑ2)tanh(ϑ2)(1α+αχαΓ(α+1))+3[2+7sech2(ϑ2)15sech4(ϑ2)]sech2(ϑ2)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2]+,G(ϑ,χ)=l=0Gl(ϑ,χ)=G0(ϑ,χ)+G1(ϑ,χ)+G2(ϑ,χ)+,G(ϑ,χ)=3c2tanh(ϑ2)+3c2sech2(ϑ2)tanh(ϑ2)(1α+αχαΓ(α+1))+3c22[2+21sech2(ϑ2)24sech4(ϑ2)]sech2(ϑ2)[α2χ2αΓ(2α+1)+2α(1α)χαΓ(α+1)+(1α)2]+. (5.33)

    When α=1, we get the exact solution as

    F(ϑ,χ)=36tanh2(ϑ+χ2),G(ϑ,χ)=3c2tanh(ϑ+χ2). (5.34)

    In Figure 1, the actual and suggested methods solutions of F(ϑ,χ) are calculated at α=1. Figure 2 gives the graphical layouts of F(ϑ,χ) when α=0.8 and 0.6. In Figure 3, the 2D and 3D behaviors of F(ϑ,χ) for different fractional orders are respectively given. In Figure 4, the actual and suggested methods solutions of G(ϑ,χ) are calculated at α=1. Figure 5 gives the graphical layouts of G(ϑ,χ) when α=0.8 and 0.6. In Figure 6, the 2D and 3D behaviors of G(ϑ,χ) for different fractional orders are respectively given. The graphical layout of Example 5.1 confirms that our solutions are in a strong agreement with the exact solution. Similarly, in Figures 7, the actual and suggested methods solutions of F(ϑ,χ) are respectively presented for α=1. Figures 8 gives the graphical layouts of F(ϑ,χ) when α=0.8 and 0.6, while Figures 9 shows the 2D and 3D behaviors of F(ϑ,χ) for different fractional orders, respectively. Also, in Figures 10, the actual and suggested methods solutions of G(ϑ,χ) are respectively presented for α=1. Figures 11 gives the graphical layouts of G(ϑ,χ) when α=0.8 and 0.6, while Figures 12 shows the 2D and 3D behaviors of G(ϑ,χ) for different fractional orders, respectively. In the same way, the graphical layout of Example 5.2 confirms that our solutions are in a strong agreement with the exact solution. All the figures have been drawn at c = 0.1 and χ=0.01 within the domain 5ϑ5. Furthermore, Tables 1 and 2 shows the approximate solution of Example 5.1 at different values of ϑ and χ. Tables 3 and 4 show a numerical comparison of the reduced differential transform method and proposed method in terms of the absolute error for Example 5.1. Finally, Tables 5 and 6 show the approximate solution of Example 5.2 at different values of ϑ and χ. From the figures and tables, it can be observed that our methods solution and the exact solution are very close to each other and possess a higher degree of accuracy.

    Figure 1.  Exact and proposed method solution at α=1 for Example 5.1.
    Figure 2.  Proposed method solution at α=0.8,0.6 for Example 5.1.
    Figure 3.  Proposed method solution at different values of α for Example 5.1.
    Figure 4.  Exact and proposed method solution at α=1 for Example 5.1.
    Figure 5.  Proposed method solution at α=0.8,0.6 for Example 5.1.
    Figure 6.  Proposed method solution at different values of α for Example 5.1.
    Figure 7.  Exact and proposed method solution at α=1 for Example 5.2.
    Figure 8.  Proposed method solution at α=0.8,0.6 for Example 5.2.
    Figure 9.  Proposed method solution at different values of α for Example 5.2.
    Figure 10.  Exact and proposed method solution at α=1 for Example 5.2.
    Figure 11.  Proposed method solution at α=0.8,0.6 for Example 5.2.
    Figure 12.  Proposed method solution at different values of α for Example 5.2.
    Table 1.  For Example 5.1, the suggested technique solution for F(ϑ,χ) given c = 0.1 at various fractional orders.
    (ϑ,χ) (NTDMABC) at α= 0.5 (NTDMCF) at α= 0.75 (NTDMABC) at α= 1 (NTDMCF) at α= 1 Exact result
    (0.2, 0.01) -0.099152 -0.099184 -0.099200 -0.099200 -0.099200
    (0.4, 0.02) -0.096736 -0.096799 -0.096830 -0.096830 -0.096830
    (0.6, 0.03) -0.092826 -0.092918 -0.092964 -0.092964 -0.092964
    (0.2, 0.01) -0.099150 -0.099182 -0.099198 -0.099198 -0.099198
    (0.4, 0.02) -0.096732 -0.096795 -0.096827 -0.096827 -0.096827
    (0.6, 0.03) -0.092821 -0.092913 -0.092960 -0.092960 -0.092960
    (0.2, 0.01) -0.099148 -0.099181 -0.099197 -0.099197 -0.099197
    (0.4, 0.02) -0.096728 -0.096792 -0.096824 -0.096824 -0.096824
    (0.6, 0.03) -0.092815 -0.092909 -0.092955 -0.092955 -0.092955
    (0.2, 0.01) -0.099147 -0.099179 -0.099195 -0.099195 -0.099195
    (0.4, 0.02) -0.096725 -0.096789 -0.096821 -0.096821 -0.096821
    (0.6, 0.03) -0.092810 -0.092904 -0.092951 -0.092951 -0.092951
    (0.2, 0.01) -0.099145 -0.099177 -0.099194 -0.099194 -0.099194
    (0.4, 0.02) -0.096721 -0.096786 -0.096818 -0.096818 -0.096818
    (0.6, 0.03) -0.092805 -0.092899 -0.092946 -0.092946 -0.092946

     | Show Table
    DownLoad: CSV
    Table 2.  For Example 5.1, the suggested technique solution for G(ϑ,χ) given c = 0.1 at various fractional orders.
    (ϑ,χ) (NTDMABC) at α=0.5 (NTDMCF) at α=0.75 (NTDMABC) at α= 1 (NTDMCF) at α= 1 Exact result
    (0.2, 0.01) 0.631114 0.631164 0.631190 0.631190 0.631190
    (0.4, 0.02) 0.627273 0.627374 0.627424 0.627424 0.627424
    (0.6, 0.03) 0.621009 0.621158 0.621232 0.621232 0.621232
    (0.2, 0.01) 0.631111 0.631162 0.631187 0.631187 0.631187
    (0.4, 0.02) 0.627267 0.627368 0.627419 0.627419 0.627419
    (0.6, 0.03) 0.621001 0.621150 0.621224 0.621224 0.621224
    (0.2, 0.01) 0.631108 0.631159 0.631185 0.631185 0.631185
    (0.4, 0.02) 0.627262 0.627363 0.627414 0.627414 0.627414
    (0.6, 0.03) 0.620992 0.621142 0.621217 0.621217 0.621217
    (0.2, 0.01) 0.631105 0.631156 0.631182 0.631182 0.631182
    (0.4, 0.02) 0.627256 0.627358 0.627409 0.627409 0.627409
    (0.6, 0.03) 0.620984 0.621135 0.621210 0.621210 0.621210
    (0.2, 0.01) 0.631102 0.631154 0.631180 0.631180 0.631180
    (0.4, 0.02) 0.627250 0.627353 0.627404 0.627404 0.627404
    (0.6, 0.03) 0.620976 0.621127 0.621202 0.621202 0.621202

     | Show Table
    DownLoad: CSV
    Table 3.  For Example 5.1, the suggested technique absolute error comparison with a reduced differential transform method (RDTM) for F(ϑ,χ) at c = 1.
    χ ϑ |RDTM| |NTDMCF| |NTDMCF|
    -2 9.00000000E-09 8.5920000000E-09 8.5920000000E-09
    -1 2.22440000E-06 1.3071200000E-08 1.3071200000E-08
    0.05 0 7.51000000E-07 2.0000000000E-08 2.0000000000E-08
    1 2.29510000E-06 1.2990800000E-07 1.2990800000E-07
    2 2.77000000E-08 7.9080000000E-08 7.9080000000E-08
    -2 7.14000000E-08 3.5763000000E-09 3.5763000000E-09
    -1 6.89230000E-05 5.2446300000E-08 5.2446300000E-08
    0.1 0 4.73670000E-05 7.9998000000E-07 7.9998000000E-07
    1 7.35887000E-05 5.1802700000E-08 5.1802700000E-08
    2 1.11602000E-06 3.0257000000E-07 3.0257000000E-07
    -2 3.75120000E-06 8.3575000000E-08 8.3575000000E-08
    -1 4.99186000E-04 1.1836350000E-07 1.1836350000E-07
    0.15 0 5.26076000E-04 1.7998900000E-07 1.7998900000E-07
    1 5.56398000E-04 1.1619250000E-06 1.1619250000E-06
    2 9.98978000E-06 6.5015000000E-08 6.5015000000E-08

     | Show Table
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    Table 4.  For Example 5.1, the suggested technique absolute error comparison for G(ϑ,χ) at c = 1.
    χ ϑ |RDTM| |NTDMCF| |NTDMCF|
    -2 1.1860000E-07 9.8320000000E-09 9.8320000000E-09
    -1 2.1200000E-07 2.4506000000E-08 2.4506000000E-08
    0.05 0 1.6800000E-07 3.1623000000E-08 3.1623000000E-08
    1 2.8200000E-07 2.4420000000E-09 2.4420000000E-09
    2 1.0710000E-07 9.7370000000E-08 9.7370000000E-08
    -2 4.0069000E-06 3.9524000000E-08 3.9524000000E-08
    -1 5.3900000E-06 9.8199000000E-07 9.8199000000E-07
    0.1 0 1.0671000E-05 1.2648900000E-08 1.2648900000E-08
    1 9.8860000E-06 9.7508000000E-08 9.7508000000E-08
    2 3.2565000E-06 3.8752000000E-07 3.8752000000E-07
    -2 3.2082600E-05 8.9365000000E-08 8.9365000000E-08
    -1 2.8970000E-05 2.2133100000E-07 2.2133100000E-07
    0.15 0 1.1917600E-04 2.8459400000E-07 2.8459400000E-07
    1 8.0549000E-05 2.1900500000E-07 2.1900500000E-07
    2 2.3525200E-05 8.6759000000E-06 8.6759000000E-06

     | Show Table
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    Table 5.  For Example 5.2, the suggested technique solution for F(ϑ,χ) given c = 0.1 at various fractional orders.
    (ϑ,χ) (NTDMABC) at α=0.5 (NTDMCF) at α=0.75 (NTDMABC) at α= 1 (NTDMCF) at α= 1 Exact result
    (0.2, 0.01) 2.952239 2.946318 2.940397 2.940397 2.940397
    (0.4, 0.01) 2.789020 2.777639 2.766257 2.766257 2.766257
    (0.6, 0.01) 2.522812 2.506817 2.490821 2.490821 2.490821
    (0.2, 0.02) 2.952239 2.946318 2.940397 2.940397 2.940397
    (0.4, 0.02) 2.789020 2.777639 2.766257 2.766257 2.766257
    (0.6, 0.02) 2.522812 2.506817 2.490821 2.490821 2.490821
    (0.2, 0.03) 2.952239 2.946318 2.940397 2.940397 2.940397
    (0.4, 0.03) 2.789020 2.777639 2.766257 2.766257 2.766257
    (0.6, 0.03) 2.522812 2.506817 2.490821 2.490821 2.490821
    (0.2, 0.04) 2.952239 2.946318 2.940397 2.940397 2.940397
    (0.4, 0.04) 2.789020 2.777639 2.766257 2.766257 2.766257
    (0.6, 0.04) 2.522812 2.506817 2.490821 2.490821 2.490821
    (0.2, 0.05) 2.952239 2.946318 2.940397 2.940397 2.940397
    (0.4, 0.05) 2.789020 2.777639 2.766257 2.766257 2.766257
    (0.6, 0.05) 2.522812 2.506817 2.490821 2.490821 2.490821

     | Show Table
    DownLoad: CSV
    Table 6.  For Example 5.2, the suggested technique solution for G(ϑ,χ) given c=0.1 at various fractional orders.
    (ϑ,χ) (NTDMABC) at α=0.5 (NTDMCF) at α=0.75 (NTDMABC) at α= 1 (NTDMCF) at α= 1 Exact result
    (0.2,0.01) -0.000410 -0.000418 -0.000422 -0.000422 -0.000422
    (0.4,0.01) -0.000813 -0.000829 -0.000837 -0.000837 -0.000837
    (0.6,0.01) -0.001202 -0.001224 -0.001235 -0.001235 -0.001235
    (0.2,0.02) -0.000410 -0.000418 -0.000422 -0.000422 -0.000422
    (0.4,0.02) -0.000813 -0.000829 -0.000837 -0.000837 -0.000837
    (0.6,0.02) -0.001201 -0.001224 -0.001235 -0.001235 -0.001235
    (0.2,0.03) -0.000410 -0.000418 -0.000422 -0.000422 -0.000422
    (0.4,0.03) -0.000813 -0.000829 -0.000837 -0.000837 -0.000837
    (0.6,0.03) -0.001201 -0.001224 -0.001235 -0.001235 -0.001235
    (0.2,0.04) -0.000410 -0.000418 -0.000422 -0.000422 -0.000422
    (0.4,0.04) -0.000813 -0.000829 -0.000837 -0.000837 -0.000837
    (0.6,0.04) -0.001201 -0.001224 -0.001235 -0.001235 -0.001235
    (0.2,0.05) -0.000410 -0.000418 -0.000422 -0.000422 -0.000422
    (0.4,0.05) -0.000813 -0.000829 -0.000837 -0.000837 -0.000837
    (0.6,0.05) -0.001201 -0.001224 -0.001235 -0.001235 -0.001235

     | Show Table
    DownLoad: CSV

    The main goal of this work as to develop a fractional-order Kersten-Krasil' shchik coupled KdV-mKdV nonlinear system approximate analytical solution. Using the NDM, we were able to achieve this goal. The numerical solutions can be achieved in two steps. The targeted problems were first simplified using the NT; then, the decomposition approach was employed to get the solutions. The method's fundamental benefit is that it gives the user an analytical approximation, and in many cases an exact solution, in a quickly converging sequence with elegantly computed terms. The suggested method has been applied to obtain the solution of the given two problems. The method's small computational size in comparison to the computational size required by other numerical methods, as well as its rapid convergence, demonstrate that it is reliable and a significant improvement on existing methods in terms of solving the generalized Kersten-Krasil' shchik coupled KdV-mKdV equation. The results we obtained have been illustrated with the help of plots and tables which confirm the validity of the proposed method. Furthermore, the proposed method is simple, straightforward, and requires minimal computational time; it may also be extended to solve other fractional-order partial differential equations.

    This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation.

    The authors declare that they have no competing interests.



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