Research article

Drift coefficient inversion problem of Kolmogorov-type equation

  • Kolmogorov-type equations often appear in stochastic analysis and have important applications in financial derivatives pricing, stochastic control and other fields. In this paper, we consider an inverse problem of reconstructing drift coefficient in a Kolmogorov-type equation. Being different from other works, the unknown drift coefficient is related to both temporal and spatial variables, which makes theoretical analysis rather difficult. Until now, documents dealt with evolutional inverse drift problems are quite few. Inspired by the Rothe's idea, we introduce a new time semi-discrete scheme to find the optimal solution at each time layer. Then we construct an approximate solution of the unknown drift coefficient and strictly analyze its convergence. After establishing the necessary conditions for the limit minimizer, we prove the uniqueness and stability of the global optimal solution.

    Citation: Liu Yang, Lijun Yin, Zuicha Deng. Drift coefficient inversion problem of Kolmogorov-type equation[J]. AIMS Mathematics, 2021, 6(4): 3432-3454. doi: 10.3934/math.2021205

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  • Kolmogorov-type equations often appear in stochastic analysis and have important applications in financial derivatives pricing, stochastic control and other fields. In this paper, we consider an inverse problem of reconstructing drift coefficient in a Kolmogorov-type equation. Being different from other works, the unknown drift coefficient is related to both temporal and spatial variables, which makes theoretical analysis rather difficult. Until now, documents dealt with evolutional inverse drift problems are quite few. Inspired by the Rothe's idea, we introduce a new time semi-discrete scheme to find the optimal solution at each time layer. Then we construct an approximate solution of the unknown drift coefficient and strictly analyze its convergence. After establishing the necessary conditions for the limit minimizer, we prove the uniqueness and stability of the global optimal solution.


    The Korteweg–de Vries–Burgers (KdV-Burgers) model, which comes up in several practical situations, such as the turbulence of undular bores in shallow water [1], the transport of liquids carrying gas bubbles [2], the waves that go through an elastic pipe that is filled with a viscous liquid [3], and weakly nonlinear plasma waves that have specific dissipative properties [4], has received a lot of interest over the last few decades. It may also be utilized as a nonlinear model in ferroelectricity theory, turbulence, circuit theory, and other fields [5,6]. The typical version of the KdV-Burger's equation is

    Ut+σ3Ux3+ζ2Ux2+ηUUx=0,t>0. (1.1)

    The KdV [7] and Burgers models [8] are commonly believed to be combined in Eq (1.1). Johnson [9] discovered that a specific limit of the matter led to the proposed model, where U(x,t) is proportional to the radial disturbance of the pipe wall, and x and t are the characterizing and temporal variables, in a field of wave propagation in fluid-filled elastic pipes. The model (1.1) was correct in the far-field of a near-field solution that was originally linear (small amplitude). Nonlinearity (UUx), dispersion (3Ux3), and dissipation (2Ux2) all exist in this equation, which is the basic version of a wave model. Fractional differential equations (FDEs) are extensions of differential equations (DEs) having integer order. FDEs have ample applications in different domains of sciences [10,11,12,13,14]. Due to the wide applications of FDEs, several operators have been defined in the literature [15,16]. The recent operators that are frequently used for studying DEs are the Caputo-Fabrizio (CF) [17] and Atangana-Baleanu (AB) [18] operators. These operators are dependent on the exponential and Mittag-Leffler kernels, respectively. The literature has several applications for the CF and the AB operators. For instance, HIV-1 infection has been investigated via the CF operator in [19]. Ahmad et al. studied the fractional-order Ambartsumian equation through the CF operator [20]. The Φ 4-model has been investigated using the CF and AB operators by Rahman et al. [21]. More applications can be found in the literature [22,23,24]. The fractional non-linear KdV-Burger's equation is taken into consideration in the form:

    αUtα+σ3Ux3+ζ2Ux2+ηUUx=0,0<α1,t>0, (1.2)

    with

    U(x,0)=R(x),

    where U is a function of x and t, x represents the space variable, t represents the time variable and η is a positive constant. We analyze Eq (1.2) in two ways: writing it first in the Atangana-Baleanu-Caputo (ABC) sense and then in the CF sense.

    One of the most significant areas of research for FDEs is the quest for accurate and numerical solutions to FDEs. To date, many strategies for obtaining numerical and precise solutions of FDEs have been developed. A number of FDEs have been examined using these approaches. For example, Ahmad et al. [25] used the Laplace transform to find series of third order dispersive fractional PDEs. A generalized differential transform approach has been developed to solve fractional order PDEs by Odibat and Momani [26]. The homotopy perturbation technique has been proposed to solve the KdV-Burger's fractional PDE by Wang [27]. The Laplace transform was also observed to have a number of advantages, including its convergence to an exact solution of a problem after a certain iteration and that it does not allow any perturbation or discretization. Here, we utilize the double Laplace (DL) transform to compute a series solution of the considered equation.

    The rest of article is organized as follows: Section 2 contains some basic definitions and a remark. The existence and uniqueness of the IVPs are presented in Section 3. In Section 4, the proposed techniques are presented. Section 5 consists of the application and comparison between results and diagrams of the proposed method. Finally, the conclusion is presented in Section 6.

    In this part, we provide some definitions, remarks, and lemmas about fractional calculus. Additionally, we provide a definition of the DL transformation and decomposition technique.

    Definition 1. [18] Let UH1(c,d), c>d and β(0,1]. Then the ABC operator is expressed as

    ABCDβU(t)=B(β)(1β)tcU(s)Eβ(β(ts)β(1β))ds,

    where B(β) is a normalizing factor with the conditions B(0)=B(1)=1, and Eβ is the Mittag-Leffler function

    Eβ(x)=k=0xkΓ(βk+1),0<β<1.

    Definition 2. [17] Let UH1(c,d), d>c and β(0,1]. Then the CF operator is written as

    CFDβU(t)=E(β)1βtcU(t)exp(β(ts)1β)ds,

    where E(β) is the normalizing factor such that E(0)=E(1)=1. When U(t) H1(c,d) then the above equation can be written for u ϵL1(,d) and any β ϵ(0,1] as

    CFDβU(t)=βE(β)1βt(U(t)U(s))exp(β(ts)1β)ds.

    Remark 1. For the above definitions, n=[β]+1, [β] is the greatest integer not greater than β, and "Γ" is the well-known gamma function that can be calculated as

    Γ(β)=0essβ1ds.

    Definition 3. Suppose that U is a function for x,t>0. The DL transformation of U is expressed as [28]

    LxLt[U]=0epx0estUdtdx,

    where p and s are complex numbers.

    Definition 4. Application of the DL transform on the ABC operator is as follows:

    LxLt{ABCDθxU}=B(θ)(1θ)(pθ+θ(1θ))[pθˉU(p,s)n1k=0pθ1kLt{kU(0,t)xk}],

    and

    LxLt{ABCDϑtU}=B(ϑ)(1ϑ)(sϑ+ϑ(1ϑ))[sϑˉU(p,s)m1k=0sϑ1kLx{kU(x,0)tk}],

    where, n=[θ]+1,m=[ϑ]+1.

    Definition 5. Application of the DL transformation on the CF operator is as follows:

    LxLt{CFDθ+nxU}=E(θ)p+(1θ)p[pn+1ˉU(p,s)ni=0pniLt{iU(0,t)xi}],

    and

    LxLt{CFDϑ+mtU}=E(ϑ)s+(1ϑ)s[sm+1ˉU(p,s)mi=0smiLx{iU(x,0)ti}],

    where, n=[θ]+1,m=[ϑ]+1.

    From the interpretation provided above, it is clear that

    LxLtU(x)v(t)=ˉU(p)ˉv(s)=LxU(x)Ltv(t).

    A complex double-integral formulation is used to represent the inverse DL transform L1xL1t{ˉU}=U:

    L1xL1t{ˉU}=12πia+iaiestb+ibiepxˉU(p,s)dpds,

    where, ˉU(p,s), is an analytic function for all p and s that are described in the region Re (p)a and Re (s)b, where a,bR to be chosen appropriately.

    The existence and uniqueness of the IVPs are studied in this part employing α-type ϝ-contraction. For this purpose, assume that (Z,d) is a complete metric space, and is the collection of strictly increasing functions ϝ:R+R having the following required characteristics:

    limnϝ(cn)= if and only if, for each {cn}, limn(cn)=0;

    ● there exists υ(0,1) such that limc0+cυϝ(c)=0.

    Definition 6. [29] Let Q:ZZ be self mapping with α:Z×Z[0,). If

    α(X,W)1α(QX,QW)1,

    for all X,WZ, then Q is referred to as α-admissible.

    Definition 7. [30] Suppose that Q:ZZ, α:Z×Z{}[0,), and there exists ω>0 such that

    ω+α(X,W)ϝ(d(QX,QW))ϝ(d(X,W))

    for each X,WZ with d(QX,QW)>0. Then, Q is called an α-type ϝ-contraction.

    Theorem 1. [30] Let (Z,d) be a complete metric space and Q:ZZ be an α-type ϝ-contraction such that

    1) there exists XZ such that α(X,QX)1;

    2) if there exist {Xn}Z with α(Xn,Xn+1)1 and XnX, then α(Xn,X)1 for all nN;

    3) ϝ is continuous.

    Then, Q has a fixed point XZ. Also for XZ, the sequence {QnX}nN is convergent to X

    Let Z=C([0,1]2,R), where C is the space of all continuous functions X:[0,1]×[0,1]R, and d(X(x,t),W(x,t))=supx,t[0,1]{|X(x,t)W(x,t)|}. Then we can write the IVP (1.1) in the CF fractional derivative sense as

    CFDαtX(x,t)=F(x,t,X(x,t)),0<α1, (3.1)

    with initial condition

    X(x,0)=g(x),

    where F(x,t,X(x,t))=σUxxxζUxxηUUx.

    The following theorem demonstrates the existence of a solution of the problem (3.1).

    Theorem 2. There exists G:R2R such that

    1) |F(x,t,X)F(x,t,W)|2γM(γ)2eb|X(x,t)W(x,t)| for (x,t)[0,1]2 and X,WR;

    2) there exists X1Z such that G(X1,QX1)0, where Q:ZZ is defined by

    QX=X+CF0IγF(x,t,X(x,t));

    3) for X,WZ,G(X,W)0 implies that G(QX,QW)0;

    4) {Xn}Z,limnXn=X, where XZ and G(Xn,Xn+1)0 implies that G(Xn,X)0, for all nN.

    Then, there exists at least one fixed point of Q that is the solution of the given model (3.1).

    Proof. To prove that Q has a fixed point, we consider

    |QXQW||QXQW+1|=|CFI[F(x,tX)F(x,tW)]||CFI[F(x,tX)F(x,tW)]+1|(2(1γ)(2γ)M(γ)|F(x,tX)F(x,tW)|+2γ(2γ)M(γ)τ0|F(x,tX)F(x,tW)|dτ)×(2(1γ)(2γ)M(γ)|F(x,tX)F(x,tW)|+2γ(2γ)M(γ)τ0|F(x,tX)F(x,tW)|dτ+1))(2(1γ)(2γ)M(γ).(2γM(γ))2eb|XW|+2γ(2γ)M(γ).2γM(γ)2τ0eb|XW|dτ)×(2(1γ)(2γ)M(γ).(2γM(γ))2eb|XW|+2γ(2γ)M(γ).2γM(γ)2τ0eb|XW|dτ+1)(ebsupx,t[0,1]|X(x,t)W(x,t)|)(ebsupx,t[0,1]|X(x,t)W(x,t)|+1)=(ebd(X,W))(ebd(X,W)+1)=eb[eb(d(X,W))2+d(X,W)]eb[(d(X,W))2+d(X,W)].

    Thus, for X,WZ with G(X,W)0, we obtain

    (d(QX,QW))2+d(QX,QW)
    eb[(d(X,W))2+d(X,W)].

    Taking the ln on both sides, we have

    b+ln[d(QX,QW))2+d(QX,QW]ln[d(X,W))2+d(X,W].

    If F:[0,)R is defined by F(u)=ln[u2+u],u>0, then Fδ.

    Now, define α:Z×Z{}[0,) as

    α(X,W)={1,ifG(X(x,t),W(x,t))0forallx,t[0,1],,otherwise.

    Then,

    b+α(X,W)F(d(QX,QW))F(d(X,W)),

    for X,WZ with d(QX,QW)>0. Now, by G3,

    α(X,W)1G(X,W)0G(QX,QW)0α(QX,QW)1,

    for all X,WZ. From G2, there exists XZ such that α(X,QX)1. Therefore by G4 and Theorem 1, there exists XZ such that X=QX. Hence, X is the solution of the problem (3.1)

    Similarly, we can write the IVP (1.1) in the ABC sense as

    ABCDαtX(x,t)=F(x,t,X(x,t)),0<α1, (3.2)

    with initial condition

    X(x,0)=g(x),

    where F(x,t,X(x,t))=σUxxxζUxxηUUx.

    The following theorem shows the existence of a solution of the problem (3.2).

    Theorem 3. There exists G:R2R such that

    1) |F(x,t,X)F(x,t,W)|ΓγM(γ)(1γ)Γγ+1eb2|X(x,t)W(x,t)| for (x,t)[0,1]2 and X,WR;

    2) there exists X1Z such that G(Y1,QX1)0, where Q:ZZ is defined by

    QX=X+ABC0IγF(x,t,X(x,t));

    3) for X,WZ,G(X,W)0 implies that G(QX,QW)0;

    4) {Xn}Z,limnXn=X, where XZ and G(Xn,Xn+1)0 implies that G(Xn,X)0, for all nN.

    Then, there exists at least one fixed point of Q which is the solution of the problem (3.2).

    Proof.

    |QXQW|2=|AB0Iγ[F(x,t,X(x,t))F(x,t,W(x,t))]|2|{1γM(γ)[F(x,t,X)F(x,t,W)]+γM(γ)0Iγ[F(x,t,X(x,t))F(x,t,W(x,t))]}|2{1γM(γ)|F(x,t,X)F(x,t,W)|+γM(γ)0Iγ|F(x,t,X(x,t))F(x,t,W(x,t))|}2{1γM(γ).M(γ)Γγ(1γ)Γγ+1eb2|XW|2+γM(γ)M(γ)Γγ(1γ)Γγ+10Iγ1.eb2|XW|2}2={M(γ)Γγ(1γ)Γγ+1eb2|XW|2}2{1γM(γ)+γM(γ)γΓγ}2{M(γ)Γγ(1γ)Γγ+1eb2supx,t[0,1]|X(x,t)W(x,t)|2}2{1γM(γ)+γM(γ)γΓγ}2={M(γ)Γγ(1γ)Γγ+1eb2d(X,W)}2{1γM(γ)+γM(γ)Γγ}2=ebd(X,W).

    Consequently,

    d(QX,QW)ebd(X,W).

    Applying "ln'' on both sides, we have

    ln(d(QX,QW))ln(ebd(X,W)),

    and

    b+ln(d(QX,QW)ln(d(X,W)).

    Let ϝ:[0,)R be defined by ϝ(λ)=lnλ, where λ>0. Then it is easy to show that ϝ.

    Now, define α:Z×Z{}[0,) by

    α(X,W)={1,ifG(X(x,t),W(x,t))0forallx,t[0,1],,otherwise.

    Thus, b+α(X,W)ϝ(d(QX,QW))ϝ(d(X,W)) for X,WZ with d(QX,QW)0. Therefore, Q is an α-type ϝ-contraction. From (G3), we have

    α(X,W)1G(X,W)0G(QX,QW)α(QX,QW)1,

    for all x,t[0,1]. Thus, Q is α-admissible. From (G2), there exists XZ with α(X,QX)1. From (G4) and Theorem[29], there exists XZ such that QX. Hence, X is the solution of the IVP (3.2).

    Here, we briefly introduce the suggested technique MDLDM, which combines the DL transformation and Adomian decomposition method (ADM) for obtaining a series solution of non-linear ordinary DEs and PDEs. It is a very effective approach for obtaining the approximate values of dynamical problems such as KdV-Burgers, Sine-Gordon and KdV type equations. Here, we first introduce the technique and then the application of the given method on Eq (1.1).

    Take the following form:

    LU+RU+NU=f(x,t),tR, (4.1)

    where L is linear, R and N are operators containing linear and non-linear terms, respectively, and f(x,t) is some external function. An, the well-known Adomian polynomials [31] of the functions U0,U1,U2,, can be described as

    An=1n!dndλn[nk=0λkUkUkx]λ=0. (4.2)

    Here, first take Eq (1.1) in the form of ABC sense as:

     ABCDαtU+σUxxx+ζUxx+ηUUx=0, (4.3)

    with initial conditions

    U(x,0)=R(x).

    By applying the DL transform method on both sides of Eq (4.3), we obtain

    LxLtABCDαtU=1[σLxLtUxxx+ζLxLtUxx+ηLxLtUUx].

    Using initial condition and after some calculation, we have

    LxLtU=1sLxR(x)(1α+αsα)LxLt[σUxxx+ζUxx+ηUUx].

    Consider the series form

    U=n=0Un. (4.4)

    The non-linear term UUx can be calculated using "ADM"; we obtain

    UUx=n=0An,
    LxLtn=0Un=1sLxR(x)(1α+αsα)[LxLt(σn=0Unxxx+ζn=0Unxx+ηn=0An)]. (4.5)

    Applying the double Laplace inverse transform and Equating terms of both sides,

    U0=R(x),U1=L1xL1t(1α+αsα)LxLt(σU0xxx+ζU0xx+ηA0),U2=L1xL1t(1α+αsα)LxLt(σU1xxx+ζU1xx+ηA1),U3=L1xL1t(1α+αsα)LxLt(σU2xxx+ζU2xx+ηA2),.

    The final solution can be calculated as follows:

    U=n=0Un. (4.6)

    Now, we take Eq (1.1) in CF fractional derivative sense as:

     CFDαtU+σUxxx+ζUxx+ηUUx=0, (4.7)

    with initial conditions

    U(x,0)=R(x).

    Using (3.1), we obtain the following series solutions:

    U0=R(x).U1=L1xL1t((1s)α+ss)LxLt(σU0xxx+ζU0xx+ηA0),U2=L1xL1t((1s)α+ss)LxLt(σU1xxx+ζU1xx+ηA1),U3=L1xL1t((1s)α+ss)LxLt(σU2xxx+ζU2xx+ηA2),.

    The final form can be expressed as

    U=n=0Un. (4.8)

    Here, we consider a numerical example in the form of time fractional derivatives in ABC and CF operators.

    Example 1. Consider Eq (1.1) in the following form:

    αUtα+σ3Ux3+ζ2Ux2+ηUUx=0,0<α1,t>0, (5.1)

    with initial condition

    U0=a03ζ2tanh2(ζx10σ)25ησ+6ζ2tanh(ζx10σ)25ησ.

    The exact solution of Eq (5.1) is:

    U=a0+6ζ225ησtanh(ζ10σ(x+(3ζ225a0ησ)t25σ))3ζ225ησtanh2(ζ10σ(x+(3ζ225a0ησ)t25σ)).

    Here, we will discuss two cases.

    Case Ⅰ:

    Consider Eq (5.1) in ABC form as follows:

     ABCDαtU+σUxxx+ζUxx+ηUUx=0. (5.2)

    The approximate solutions of Eq (5.2) by using the techniques discussed in Section 4 are obtained in the series up to O(3) and given by

    U0=a03ζ2tanh2(ζx10σ)25ησ+6ζ2tanh(ζx10σ)25ησ,U1=3ζ3(tα+Γ(α)αΓ(α))(3ζ225a0ησ)(tanh(ζx10σ)1)sech2(ζx10σ)3125ησ3Γ(α),U2=3ζ4781250ησ5Γ(α)Γ(1+2α)((t2αα+2(α1)2Γ(2α))Γ(1+α)2tα(α1)Γ(1+2α))(3ζ225a0ησ)2(tanh(ζx10σ)1)(3tanh(ζx10σ)+1)sech2(ζx10σ),
    U3=3ζ5(3ζ225a0ησ)2sech6(xζ10σ)(tanh(ζx10σ)1)781250000σ7η(Γ(α))2Γ(1+α)Γ(1+2α)Γ(1+3α)[24(1+α)2(ζ)2(Γ(α))2(tαα(1+α)Γ(1+α))Γ(1+2α)Γ(1+3α)(cosh(xζ10σ)sinh(xζ10σ))(cosh(xζ10σ)+3sinh(xζ10σ))24t2αζ2Γ(1+α)Γ(1+2α)(tααΓ(1+2α)(1+α)Γ(1+3α))(2+cosh(xζ5σ)sinh(xζ5σ))+Γ(α)Γ(1+α)(2tα(1+α)2Γ(1+3α)(αΓ(2α)(123ζ2+225a0ησ+4(9ζ2+25a0ησ)cosh(xζ5σ)+5(3ζ225a0ησ)cosh(2xζ5σ)30(ζ2+5a0ησ)sinh(xζ5σ)+9ζ2sinh(2xζ5σ)75a0ησsinh(2xζ5σ))+Γ(1+2α)(75ζ2+225a0ησ+4(3ζ2+25a0ησ)cosh(xζ5σ)+5(3ζ225a0ησ)cosh(2xζ5σ)6(ζ2+25a0ησ)sinh(xζ5σ)+9ζ2sinh(2xζ5σ)75a0ησsinh(2xζ5σ)))+Γ(1+α)(t3αα2Γ(1+2α)(123ζ2+225a0ησ+4(9ζ2+25a0ησ)cosh(xζ5σ)+5(3ζ225a0ησ)cosh(2xζ5σ)30(ζ2+5a0ησ)sinh(xζ5σ)+9ζ2sinh(2xζ5σ)75a0ησsinh(2xζ5σ))(1+α)Γ(1+3α)(3t2αα(91ζ2+225a0ησ+20(ζ2+5a0ησ)cosh(xζ5σ)+5(3ζ225a0ησ)cosh(2xζ5σ)14ζ2sinh(xζ5σ)150a0ησsinh(xζ5σ)+9ζ2sinh(2xζ5σ)75a0ησsinh(2xζ5σ))+2(1+α)2Γ(2α)(123ζ2+225a0ησ+4(9ζ2+25a0ησ)cosh(xζ5σ)+5(3ζ225a0ησ)cosh(2xζ5σ)30(ζ2+5a0ησ)sinh(xζ5σ)+9ζ2sinh(2xζ5σ)75a0ησsinh(2xζ5σ)))))].

    The simplified form of U3 is:

    U3=3ζ5(3ζ225a0ησ)2sech4(xζ10σ)(tanh(ζx10σ)1)1562500000σ7η[43(1+expxζ5σ)2(expxζ5σ(6(α1)3(3(1+3expxζ5σ25exp2xζ5σ+exp3xζ5σ+4exp4xζ5σ)ζ225a0(1+expxζ5σ)2(17expxζ5σ+4exp2xζ5σ)ησ)+t2αα(9(α1)Γ(2α)((3+17expxζ5σ91exp2xζ5σ+3exp3xζ5σ+12exp4xζ5σ)ζ225a0(1+expxζ5σ)2(17expxζ5σ+4exp2xζ5σ)ησ)2tααΓ(3α)(3(1+11expxζ5σ41exp2xζ5σ+exp3xζ5σ+4exp4xζ5σ)ζ225a0(1+expxζ5σ)2(17expxζ5σ+4exp2xζ5σ)ησ)))48(2expxζ5σ1)t2αζ2(3(α1)Γ(3α)+tαΓ(1+2α))(Γ(α))2Γ(3α))24tα(α1)Γ(α1)(9ζ2+175a0ησ+5(3ζ225a0ησ)cosh[xζ5σ]18ζ2sech2[xζ5σ]+3(3ζ225a0ησ)sinh[xζ5σ]12ζ2tanh[xζ5σ])].

    Therefore,

    U=U0+U1+U2+U3..... (5.3)

    Case Ⅱ:

    Consider Eq (1.2) in the CF sense as

     CFDαtU+σUxxx+ζUxx+ηUUx=0. (5.4)

    The approximate solutions of Eq (5.4) obtained by using the approaches discussed in Section 4 up to O(3) are given by

    U0==a0(3ζ2)tanh2(ζx10σ)25ησ+(6ζ2)tanh(ζx10σ)25ησ,U1=3ζ3(1+(t1)α)(3ζ225a0ησ)(tanh(ζx10σ)1)sech2(ζx10σ)3125ησ3,U2=3ζ4(α2(t24t+2)+4α(t1)+2)(3ζ225a0ησ)2(tanh(ζx10σ)1)(3tanh(ζx10σ)+1)sech2(ζx10σ)1562500ησ5,
    U3=ζ5(3ζ225a0ησ)2sech6(xζ10σ)1562500000ησ7[4(3(618(1+t)α3(612t+t2)α2+(618t+3t2+t3)α3)ζ225a0(6+18(1+t)α+9(24t+t2)α2+(6+18t9t2+t3)α3)ησ)cosh(xζ5σ)5(6+18(1+t)α+9(24t+t2)α2+(6+18t9t2+t3)α3)(3ζ225a0ησ)cosh(2xζ5σ)3(3(50+150(1+t)α+(150300t+59t2)α2+(50+150t59t2+3t3)α3)ζ2+75a0(6+18(1+t)α+9(24t+t2)α2+(6+18t9t2+t3)α3)ησ+2(3(26(1+t)α+(6+12t+t2)α2+(26tt2+t2)α3)ζ225a0(6+18(1+t)α+9(24t+t2)α2+(6+18t9t2+t3)α3)ησ)sinh(xζ5σ)+(6+18(1+t)α+9(24t+t2)α2+(6+18t9t2+t3)α3)(3ζ225a0ησ)sinh(2xζ5σ))](1+tanh(xζ10σ)).

    The simplified form is

    U3=ζ5(3ζ225a0ησ)2sech6(xζ10σ)(tanh(xζ10σ)1)1562500000ησ7[9(50+α(150t(α1)259t2(α1)α+3t3α250(3+(α3)α)))ζ2225a0(6+18(t1)α+9(2+(t4)t)α2+(6+(t6)(t3)t)α3)ησ+12(618(t1)α3(6+(t12)t)α2+(6+(t3)t(t+6))α3)ζ2cosh[xζ5σ]+(6+18(t1)α+9(2+(t4)t)α2+(6+(t6)(t3)t)α3)(3ζ225a0ησ)(5cosh[2xζ5σ]3sinh[2xζ5σ])18(2+α(66t+(6+t(t+12))α+(2+(t3)t(t+2))α2))ζ2sinh[xζ5σ]+50a0(6+18(1+t)α+9(24t+t2)α2+(6+18t9t2+t3)α3)ση(2cosh[xζ5σ]+3sinh[xζ5σ])]

    The final result in the series form up to O(3) is given by

    U=U0+U1+U2+U3..... (5.5)

    The errors between the exact and approximate solutions in the ABC and CF senses are shown in the Tables 1 and 2, respectively.

    Table 1.  The comparison of approximated ABC and exact solutions for α=1 at various points for Example 1.
    (x, t) U Exact ExactU (x, t) U Exact ExactU
    (4, 0.01) 3.5967 3.5967 4.437×1011 (2, 0.01) 3.42484 3.42484 1.488×109
    (1, 0.01) 2.50239 2.50239 7.202×109 (0, 0.01) 0.02862 0.02862 1.666×108
    (1, 0.01) 1.13462 1.13462 4.376×109 (2, 0.01) 1.19852 1.19852 2.930×1010
    (4, 0.01) 1.2000 1.2000 1.184×1013 (4, 0.1) 3.5959 3.5959 4.635×107
    (2, 0.1) 3.3840 3.3840 1.515×105 (1, 0.1) 2.2941 2.2942 7.761×105
    (0, 0.1) 0.2693 0.2695 1.725×104 (1, 0.1) 1.1556 1.1556 4.302×105
    (2, 0.1) 1.1990 1.1990 2.715×106 (4, 0.1) 1.2000 1.2000 1.085×109
    (4, 0.05) 3.5963 3.5963 2.827×108 (2, 0.05) 3.4077 3.4077 9.378×107
    (1, 0.05) 2.4131 2.4131 4.656×106 (0, 0.05) 0.1395 0.1395 1.06×105
    (1, 0.05) 1.1448 1.1448 2.716×106 (2, 0.05) 1.1987 1.1987 1.769×107
    (4, 0.05) 1.2000 1.2000 7.099×1011 (4, 0.2) 3.5948 3.5948 7.795×106
    (2, 0.2) 3.3282 3.3279 2.4641×104 (1, 0.2) 2.0304 2.0317 1.3403×103
    (0, 0.2) 0.4958 0.4986 2.8191×103 (1, 0.2) 1.1720 1.1713 6.7111×104
    (2, 0.2) 1.1994 1.1994 4.007×105 (4, 0.2) 1.2000 1.2000 1.5911×108

     | Show Table
    DownLoad: CSV
    Table 2.  The comparison of approximated the CF and exact solution for α=1 at various points for Example 1.
    (x, t) U Exact ExactU (x, t) U Exact ExactU
    (4, 0.01) 3.5967 3.5967 4.437×1011 (2, 0.01) 3.4248 3.4248 1.488×109
    (1, 0.01) 2.5023 2.5023 7.202×109 (0, 0.01) 0.02862 0.02862 1.666×108
    (1, 0.01) 1.13462 1.13462 4.376×109 (2, 0.01) 1.19852 1.19852 2.930×1010
    (4, 0.01) 1.2000 1.2000 1.184×1013 (4, 0.1) 3.5959 3.5959 4.635×107
    (2, 0.1) 3.3840 3.3840 1.515×105 (1, 0.1) 2.2941 2.2942 7.761×105
    (0, 0.1) 0.2693 0.2695 1.725×104 (1, 0.1) 1.1556 1.1556 4.302×105
    (2, 0.1) 1.1990 1.1990 2.715×106 (4, 0.1) 1.2000 1.2000 1.085×109
    (4, 0.05) 3.5963 3.5963 2.827×108 (2, 0.05) 3.4077 3.4077 9.378×107
    (1, 0.05) 2.4131 2.4131 4.656×106 (0, 0.05) 0.1395 0.1395 1.06×105
    (1, 0.05) 1.1448 1.1448 2.716×106 (2, 0.05) 1.1987 1.1987 1.769×107
    (4, 0.05) 1.2000 1.2000 7.099×1011 (4, 0.2) 3.5948 3.5948 7.795×106
    (2, 0.2) 3.3282 3.3279 2.4641×104 (1, 0.2) 2.0304 2.0317 1.3403×103
    (0, 0.2) 0.4958 0.4986 2.8191×103 (1, 0.2) 1.1720 1.1713 6.7111×104
    (2, 0.2) 1.1994 1.1994 4.007×105 (4, 0.2) 1.2000 1.2000 1.5911×108

     | Show Table
    DownLoad: CSV

    Take the parameteric values as η=1, a0=0 and ζ and σ=0.1 for numerical computations. Tables 1 and 2 show the errors between the exact and approximated values in the ABC and CF senses, while the exact and computational values are also shown in these tables. Figure 1 depicts the comparison of approximated solutions in ABC and CF forms for different values of α at t=1. The surfaces in Figure 2 show the exact and approximated solutions of the ABC and the CF forms, respectively, for Eq (1.2) at α=1. Figures 3 and 4 present the approximate solutions when α=0.95 and α=0.90, respectively. These graphical representations show the behaviour of ABC and CF operators respectively for the proposed problem. Figures 5 and 6 depict the behaviours of surfaces for various values of α at different time levels. The first representation depicts the ABC form, while the second shows the CF form. It is straightforward to deduce that, as the fractional parameter α decreases, the wave response bifurcates into a wave but only for small values of x, and we observe that the amplitude of the wave grows over time t.

    Figure 1.  Comparison plots of approximated solutions of ABC and CF, respectively, for various values α.
    Figure 2.  Comparison plots of exact solutions, approximated solutions of ABC and CF for α=1 respectively.
    Figure 3.  Comparison plots of approximated solutions of ABC and CF, respectively, for α=0.95.
    Figure 4.  Comparison plots of approximated solutions of ABC and CF, respectively, for α=0.9.
    Figure 5.  Comparison plots of approximated solutions of ABC and CF, respectively, for α=0.5.
    Figure 6.  Comparison plots of approximated solutions of ABC and CF, respectively, for α=0.25.

    In this manuscript, the time fractional KdV-Burger's model with initial conditions under two non-local operators with exponential kernel and Mittag-Leffler kernel has been investigated. The existence of the solution for both operators has been demonstrated through fixed point results of α-type ϝ contraction. The MDLDM was utilized to compute a series solutions that tends to the exact values for the special case α=1. As a result, we found highly accurate computed solutions to the fractional KdV-Burger's equation. The solutions show how consistently accurate the technique is and how broadly applicable it is to fractional nonlinear evolution problems. Quick convergence is seen when solutions are simulated numerically. Furthermore, neither linearization nor perturbation are needed. Therefore, it offers more authentic series solutions that typically converge quickly in actual physical problems.

    The authors declare that they have no conflict of interest.



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