Research article

S-function associated with fractional derivative and double Dirichlet average

  • The object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.

    Citation: Jitendra Daiya, Dinesh Kumar. S-function associated with fractional derivative and double Dirichlet average[J]. AIMS Mathematics, 2020, 5(2): 1372-1382. doi: 10.3934/math.2020094

    Related Papers:

    [1] Muhammad Imran Liaqat, Sina Etemad, Shahram Rezapour, Choonkil Park . A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients. AIMS Mathematics, 2022, 7(9): 16917-16948. doi: 10.3934/math.2022929
    [2] Musawa Yahya Almusawa, Hassan Almusawa . Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator. AIMS Mathematics, 2024, 9(11): 31898-31925. doi: 10.3934/math.20241533
    [3] Aslı Alkan, Halil Anaç . A new study on the Newell-Whitehead-Segel equation with Caputo-Fabrizio fractional derivative. AIMS Mathematics, 2024, 9(10): 27979-27997. doi: 10.3934/math.20241358
    [4] Humaira Yasmin, Aljawhara H. Almuqrin . Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies. AIMS Mathematics, 2024, 9(6): 16721-16752. doi: 10.3934/math.2024811
    [5] Mariam Sultana, Muhammad Waqar, Ali Hasan Ali, Alina Alb Lupaş, F. Ghanim, Zaid Ameen Abduljabbar . Numerical investigation of systems of fractional partial differential equations by new transform iterative technique. AIMS Mathematics, 2024, 9(10): 26649-26670. doi: 10.3934/math.20241296
    [6] Humaira Yasmin, Aljawhara H. Almuqrin . Efficient solutions for time fractional Sawada-Kotera, Ito, and Kaup-Kupershmidt equations using an analytical technique. AIMS Mathematics, 2024, 9(8): 20441-20466. doi: 10.3934/math.2024994
    [7] M. Mossa Al-Sawalha, Khalil Hadi Hakami, Mohammad Alqudah, Qasem M. Tawhari, Hussain Gissy . Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation. AIMS Mathematics, 2025, 10(3): 7099-7126. doi: 10.3934/math.2025324
    [8] Meshari Alesemi . Innovative approaches of a time-fractional system of Boussinesq equations within a Mohand transform. AIMS Mathematics, 2024, 9(10): 29269-29295. doi: 10.3934/math.20241419
    [9] Yousef Jawarneh, Humaira Yasmin, Abdul Hamid Ganie, M. Mossa Al-Sawalha, Amjid Ali . Unification of Adomian decomposition method and ZZ transformation for exploring the dynamics of fractional Kersten-Krasil'shchik coupled KdV-mKdV systems. AIMS Mathematics, 2024, 9(1): 371-390. doi: 10.3934/math.2024021
    [10] Maysaa Al Qurashi, Saima Rashid, Sobia Sultana, Fahd Jarad, Abdullah M. Alsharif . Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory. AIMS Mathematics, 2022, 7(7): 12587-12619. doi: 10.3934/math.2022697
  • The object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.


    A novel mathematical tool for characterizing non-local structures is fractional calculus (FC). Mathematical explanations of many physical problems using fractional derivatives have proved successful in recent generations when applied to situations close to reality. Many authors, including Hadamard, Riemann-Liouville, Coimbra, Grunwald-Letnikov, Riesz, Weyl, Liouville Caputo, Atangana-Baleanu, and Caputo-Fabrizio, have offered crucial definitions of fractional operators [1,2,3,4]. The underlying principle of these traditional differential equations is their reliance on integer-order derivatives, which give the order an integer numerical value indicating the number of times that a function is differentiated. As opposed to fractional partial differential equations (FPDEs), these concepts are expanded by the addition of fractional derivatives. This kind of model is necessary for the description of delayed or dependent responses, non-local interactions, and anomalous diffusion. These relationships not only have the power to explain the complex phenomena of physics, biology, finance, and engineering, but they also can predict special events. The solution of FPDEs is based on a series of special numerical methods and analytical techniques, all adapted to face the scale-free and non-integer properties of fractional derivatives. Often, the systems required for modeling real-world phenomena follow interrelated processes that can be described using systems of partial differential equations [1,2,3,4,9]. Such systems have multiple differential equations, with each one describing how the value of a specific physical quantity or any of the interacting systemic components changes with time as they progress.

    Computational models often include partial differential equations (PDEs), which are important in applications such as fluid flow, electromagnetics, population dynamics, and quantized mechanics. By studying the behavior of the components, their interactions, and their relationships with each other, one can understand how patterns and dynamics are formed and how the system stabilizes. Studying PDE systems with coupled terms is extremely complex and requires advanced mathematics, such as numerical simulations, perturbation methods, and symmetry analysis, to discern a solution [10,11,12,13,14]. The Hermite colocation method [10], the optimal homotopy asymptotic technique [11], the Adomian decomposition method [12], the homotopy perturbation transform method [13], the Pade approximation and homotopy-Pade technique [14], the invariant subspace method [20], the q-homotopy analysis transform method [21], the homotopy analysis Sumudu transform method [22], and the Sumudu transform series expansion method [23] are some of the sophisticated approaches developed for finding exact solutions to nonlinear FPDE models [24,25,26]. If perturbation methods are not used, the homotopy analysis method breaks a problem into an endless series of linear problems. This method employs the concept of homotopy from topology to derive a convergent series solution [27,28]. An approach to homotopy analysis proposed by Liao [29], and the Laplace transform [30] are combined in the Laplace homotopy perturbation method.

    A gradient of chemical molecules guides the movement of cells, a process known as chemotaxis, which is essential for cell population self-organization and developmental biology in general. In 1970, Lee Segel and Evelyn Keller presented the first mathematical model of chemotaxis. To further understand how the mould aggregation process works in the chemical-attraction-based cellular slime, they used parabolic approaches [31]. Here, we take a look at the fractional-order system of a KS model that goes like this:

    Dpφβ1(ψ,φ)a2β1(ψ,φ)ψ2+ψ(β1(ψ,φ)ϖ(β2)ψ)=0,Dpφβ2(ψ,φ)b2β2(ψ,φ)ψ2cβ1(ψ,φ)+dβ2(ψ,φ)=0,   where   0<p1, (1.1)

    having IC's:

    β1(ψ,0)=β10(ψ),β2(ψ,0)=β20(ψ). (1.2)

    The concentration of amoebae are indicated by the unknown term β1(ψ,φ), while the chemical substance of concentration is expressed by β2(ψ,φ); ψ(β1(ψ,φ)ϖ(β2)ψ); stands for the chemotactic word, indicating that the chemicals are attractive to and sensitive to the cells. The sensitivity function is denoted by ϖ(β2), and a,b,c, and d are positive constants. The parameter 0<p1 represents the order of the fractional derivative. Much recent research has focused on the KS model. For example, to solve the KS model, Atangana used a combination of methods, including a modified homotopy perturbation, the homotopy decomposition, and the Laplace transform approach [32,33,34]. Zayernouri established a fractional class of implicit Adams-Moulton and explicit Adams-Bashforth methods in [35] and so on [36,37,38].

    According to [39], the residual power series method (RPSM) was developed in 2013 by a Jordanian mathematician named Omar Abu Arqub. The RPSM is a semi-analytical approach that uses Taylor's series to integrate the residual error function. It finds convergence series solutions for differential equations. In 2013, RPSM was first used to resolve fuzzy differential equations. A new RPSM method was created by Arqub et al. [40] to quickly get power series solutions for ordinary differential equations (DEs). A new and attractive RPSM approach for fractional DEs problems was developed by Arqub et al. [41]. A novel iterative technique to estimate fractional KdV-burgers equations was presented by El-Ajou et al. [42] utilizing RPSM. A unique method was developed by Xu et al. [43] for solving Boussinesq DEs with fractional power series. Zhang et al. [44] stated that a trustworthy numerical approach was developed. More readings on RPSM may be found in [45,46,47].

    To resolve fractional-order differential equations (FODEs), the research team used two separate approaches. One approach to solving the updated equation is to project it into the space generated by the Aboodh transform. Next, the original equation may be solved by using the inverse Aboodh transform [48]. This novel methodology combines the Sumudu transform with the homotopy perturbation method. Without discretization, linearization, or perturbation, this novel approach may solve PDEs as power series expansions, irrespective of their linearity or nonlinearity. There is a significant reduction in the computations needed to find the coefficients compared to RPSM, which requires several repetitions of calculating distinct fractional derivatives throughout the solution phases. The proposed approach has the potential to provide an accurate and closed-form approximation solution.

    The Aboodh transform iterative technique (ATIM) is a significant mathematical achievement for fractional partial differential equations. Complexity and convergence issues may develop when using traditional techniques to solve partial differential equations with fractional derivatives. Keeping a steady computational economy while continually improving approximations allows our new strategy to improve accuracy continuously, avoiding these limits. Due to this discovery, we can tackle difficult problems in applied mathematics, engineering, and physics, which enhances our capacity to identify and understand complex systems governed by fractional partial differential equations [49,50,51].

    The two most basic approaches to solving fractional differential equations are the Aboodh transform iterative technique (ATIM) and the Aboodh residual power series method (ARPSM) [49,50,51,52,53], respectively. These techniques not only provide numerical solutions to PDEs that do not need discretization or linearization but also make the symbolic terms in analytical solutions instantly and visible. The primary objective of this study is to compare and contrast the performance of ARPSM and ATIM in solving the Keller-Segel (KS) model. It is worth mentioning that several linear and nonlinear fractional differential problems have been solved using these two approaches.

    Definition 2.1. [54] Let us assume that the function β1(ψ,φ) is piecewise continuous with exponential order. The Aboodh transform (AT) is defined as follows, assuming τ0 for β1(ψ,φ),

    A[β1(ψ,φ)]=Ψ(ψ,ξ)=1ξ0β1(ψ,φ)eφξdφ,  r1ξr2.

    The Aboodh inverse transform (AIT) is specifically described as follows:

    A1[Ψ(ψ,ξ)]=β1(ψ,φ)=12πiu+iuiΨ(ψ,φ)ξeφξdφ,

    where ψ=(ψ1,ψ2,,ψp)R and pN.

    Lemma 2.2. Let [55,56] β11(ψ,φ) and β12(ψ,φ) are two functions. It is assumed that they are piecewise continuous on [0,[ and exponentially ordered. Let A[β11(ψ,φ)]=Ψ1(ψ,φ),A[β12(ψ,φ)]=Ψ2(ψ,φ) and χ1,χ2 are constants. Thus, the following characteristics are true:

    (1) A[χ1β11(ψ,φ)+χ2β12(ψ,φ)]=χ1Ψ1(ψ,ξ)+χ2Ψ2(ψ,φ),

    (2) A1[χ1Ψ1(ψ,φ)+χ2Ψ2(ψ,φ)]=χ1β11(ψ,ξ)+χ2β12(ψ,φ),

    (3) A[Jpφβ1(ψ,φ)]=Ψ(ψ,ξ)ξp,

    (4) A[Dpφβ1(ψ,φ)]=ξpΨ(ψ,ξ)r1K=0β1K(ψ,0)ξKp+2,r1<pr, rN.

    Definition 2.3. [57] In terms of order p, the Caputo defines the fractional derivative of the function β1(ψ,φ) as:

    Dpφβ1(ψ,φ)=Jmpφβ1(m)(ψ,φ), r0, m1<pm,

    where ψ=(ψ1,ψ2,,ψp)Rp and m,pR,Jmpφ is the R-L integral of β1(ψ,φ).

    Definition 2.4. [58] Following is the structure of the power series notation:

    r=0r(ψ)(φφ0)rp=0(φφ0)0+1(φφ0)p+2(φφ0)2p+,

    where ψ=(ψ1,ψ2,,ψp)Rp and pN. The series concerning φ0 is referred to as a multiple fractional power series (MFPS), where the series coefficients are r(ψ)s and φ is variable.

    Lemma 2.5. Let us suppose that the exponential order function is β1(ψ,φ). In this case, the AT is defined as: A[β1(ψ,φ)]=Ψ(ψ,ξ). Hence,

    A[Drpφβ1(ψ,φ)]=ξrpΨ(ψ,ξ)r1j=0ξp(rj)2Djpφβ1(ψ,0),  0<p1, (2.1)

    where ψ=(ψ1,ψ2,,ψp)Rp and pN and Drpφ=Dpφ.Dpφ..Dpφ(rtimes).

    Proof. By induction, we are able to illustrate Eq (2.5). When r=1 is used in Eq (2.5), the following results are obtained:

    A[D2pφβ1(ψ,φ)]=ξ2pΨ(ψ,ξ)ξ2p2β1(ψ,0)ξp2Dpφβ1(ψ,0).

    Equation (2.5) is true for r=1, according to Lemma 2.2, part (4). After substituting r=2 in Eq (2.5), we get:

    A[D2prβ1(ψ,φ)]=ξ2pΨ(ψ,ξ)ξ2p2β1(ψ,0)ξp2Dpφβ1(ψ,0). (2.2)

    Equation (2.2) L.H.S. enables us to determine

    L.H.S=A[D2pφβ1(ψ,φ)]. (2.3)

    The following way is used to express Eq (2.3):

    L.H.S=A[Dpφβ1(ψ,φ)]. (2.4)

    Assume

    z(ψ,φ)=Dpφβ1(ψ,φ). (2.5)

    Thus, Eq (2.4) becomes

    L.H.S=A[Dpφz(ψ,φ)]. (2.6)

    Implementing the Caputo derivative led to a modification in Eq (2.6).

    L.H.S=A[J1pz(ψ,φ)]. (2.7)

    Equation (2.7) provides the R-L integral for AT, which allows us to deduce the following:

    L.H.S=A[z(ψ,φ)]ξ1p. (2.8)

    Equation (2.8) is changed into the following form by using the differential characteristic of the AT:

    L.H.S=ξpZ(ψ,ξ)z(ψ,0)ξ2p, (2.9)

    from Eq (2.5), we obtain:

    Z(ψ,ξ)=ξpΨ(ψ,ξ)β1(ψ,0)ξ2p,

    where A[z(ψ,φ)]=Z(ψ,ξ). Therefore, Eq (2.9) is transformed to

    L.H.S=ξ2pΨ(ψ,ξ)β1(ψ,0)ξ22pDpφβ1(ψ,0)ξ2p, (2.10)

    when r=K. Equations (2.5) and (2.10) are compatible. For r=K, let's assume that Eq (2.5) holds. Therefore, we substitute r=K into Eq (2.5):

    A[DKpφβ1(ψ,φ)]=ξKpΨ(ψ,ξ)K1j=0ξp(Kj)2DjpφDjpφβ1(ψ,0), 0<p1. (2.11)

    Next, we will show how to solve Eq (2.5) for r=K+1. Based on Eq (2.5), we may express

    A[D(K+1)pφβ1(ψ,φ)]=ξ(K+1)pΨ(ψ,ξ)Kj=0ξp((K+1)j)2Djpφβ1(ψ,0). (2.12)

    After examining the left side of Eq (2.12), we get

    L.H.S=A[DKpφ(DKpφ)], (2.13)

    let

    DKpφ=g(ψ,φ),

    by Eq (2.13), we drive

    L.H.S=A[Dpφg(ψ,φ)]. (2.14)

    Equation (2.14) is modified to provide the following result by using the R-L integral and Caputo derivative:

    L.H.S=ξpA[DKpφβ1(ψ,φ)]g(ψ,0)ξ2p. (2.15)

    Equation (2.15) is derived from Eq (2.11),

    L.H.S=ξrpΨ(ψ,ξ)r1j=0ξp(rj)2Djpφβ1(ψ,0). (2.16)

    In addition, the outcome that follows is obtained from Eq (2.16):

    L.H.S=A[Drpφβ1(ψ,0)].

    Thus, for r=K+1, the Eq (2.5) is valid. Equation (2.5) is valid for all positive integers according to the mathematical induction method.

    Here, we find another novel way of looking to MFTS, or multiple fractional Taylor's series. The ARPSM, which will be discussed in more depth later on, will benefit from this formula.

    Lemma 2.6. Assume that β1(ψ,φ) represents the exponential order function. The expression A[β1(ψ,φ)]=Ψ(ψ,ξ) is the AT of β1(ψ,φ). The AT MFTS notation looks like this:

    Ψ(ψ,ξ)=r=0r(ψ)ξrp+2,ξ>0, (2.17)

    where, ψ=(s1,ψ2,,ψp)Rp, pN.

    Proof. Let us investigate Taylor's series' fractional order expression:

    β1(ψ,φ)=0(ψ)+1(ψ)φpΓ[p+1]++2(ψ)φ2pΓ[2p+1]+. (2.18)

    The following equality is obtained by applying the AT to Eq (2.18):

    A[β1(ψ,φ)]=A[0(ψ)]+A[1(ψ)φpΓ[p+1]]+A[1(ψ)φ2pΓ[2p+1]]+

    This is accomplished by using the AT's characteristics.

    A[β1(ψ,φ)]=0(ψ)1ξ2+1(ψ)Γ[p+1]Γ[p+1]1ξp+2+2(ψ)Γ[2p+1]Γ[2p+1]1ξ2p+2

    A distinct variant of Taylor's series in the AT is therefore obtained.

    Lemma 2.7. As stated in the new form of Taylor's series 2.17, the MFPS may be represented as A[β1(ψ,φ)]=Ψ(ψ,ξ).

    0(ψ)=limξξ2Ψ(ψ,ξ)=β1(ψ,0). (2.19)

    Proof. This can be determined from the revised version of Taylor's series:

    0(ψ)=ξ2Ψ(ψ,ξ)1(ψ)ξp2(ψ)ξ2p (2.20)

    As shown in Eq (2.20), the necessary solution may be obtained by evaluating limξ into Eq (2.19) and doing a quick computation.

    Theorem 2.8. The function A[β1(ψ,φ)]=Ψ(ψ,ξ) may be expressed in MFPS form as follows:

    Ψ(ψ,ξ)=0r(ψ)ξrp+2, ξ>0,

    where ψ=(ψ1,ψ2,,ψp)Rp and pN. Then we have

    r(ψ)=Drprβ1(ψ,0),

    where, Drpφ=Dpφ.Dpφ..Dpφ(rtimes).

    Proof. The new Taylor's series is as follows:

    1(ψ)=ξp+2Ψ(ψ,ξ)ξp0(ψ)2(ψ)ξp3(ψ)ξ2p (2.21)

    limξ, is applied to (2.21), we get

    1(ψ)=limξ(ξp+2Ψ(ψ,ξ)ξp0(ψ))limξ2(ψ)ξplimξ3(ψ)ξ2p

    After calculating the limit, we have the following equality:

    1(ψ)=limξ(ξp+2Ψ(ψ,ξ)ξp0(ψ)). (2.22)

    The result of inserting Lemma 2.5 into Eq (2.22) is as follows:

    1(ψ)=limξ(ξ2A[Dpφβ1(ψ,φ)](ξ)). (2.23)

    Furthermore, it is transformed into by using Lemma 2.6 to Eq (2.23),

    1(ψ)=Dpφβ1(ψ,0).

    Again, applying limit ξ and using the new form of Taylor's series, we obtain:

    2(ψ)=ξ2p+2Ψ(ψ,ξ)ξ2p0(ψ)ξp1(ψ)3(ψ)ξp

    We get the result from Lemma 2.6.

    2(ψ)=limξξ2(ξ2pΨ(ψ,ξ)ξ2p20(ψ)ξp21(ψ)). (2.24)

    Using Lemmas 2.5 and 2.7, we convert Eq (2.24) into

    2(ψ)=D2pφβ1(ψ,0),

    when the new Taylor's series is put through the same process, the following results are obtained:

    3(ψ)=limξξ2(A[D2pφβ1(ψ,p)](ξ)),

    Lemma 2.7 is used to derive the final equation:

    3(ψ)=D3pφβ1(ψ,0),

    in general

    r(ψ)=Drpφβ1(ψ,0).

    Consequently, proof ends here.

    The principles regulating the convergence of Taylor's series in its new form are explained and proven in the following theorem.

    Theorem 2.9. Presented in Lemma 2.6, the formula for multiple fractional Taylor's series may be represented in the following new form: A[β1(ψ,φ)]=Ψ(ψ,ξ). When |ξaA[D(K+1)pφβ1(ψ,φ)]|T, for all 0<ξs and 0<p1, the following inequality satisfies the residual RK(ψ,ξ) of the new MFTS:

    |RK(ψ,ξ)|Tξ(K=1)p+2, 0<ξs.

    Proof. Let A[Drpφβ1(ψ,φ)](ξ) is defined on 0<ξs for r=0,1,2,,K+1. Let us assume that |ξ2A[Dpsi,φ]K+1β11]|T, on 0<ξs. Determine the following relation using the new Taylor's series:

    RK(ψ,ξ)=Ψ(ψ,ξ)Kr=0r(ψ)ξrp+2. (2.25)

    Equation (2.25) is converted using Theorem 2.8,

    RK(ψ,ξ)=Ψ(ψ,ξ)Kr=0Drpφβ1(ψ,0)ξrp+2. (2.26)

    To solve Eq (2.26), multiply ξ(K+1)a+2 on both sides,

    ξ(K+1)p+2RK(ψ,ξ)=ξ2(ξ(K+1)pΨ(ψ,ξ)Kr=0ξ(K+1r)p2Drpφβ1(ψ,0)). (2.27)

    Lemma 2.5 applied to Eq (2.27) yields

    ξ(K+1)p+2RK(ψ,ξ)=ξ2A[D(K+1)pφβ1(ψ,φ)]. (2.28)

    Taking absolute of Eq (2.28), we get

    |ξ(K+1)p+2RK(ψ,ξ)|=|ξ2A[D(K+1)pφβ1(ψ,φ)]|. (2.29)

    Applying the criteria listed in Eq (2.29) yields the following result:

    Tξ(K+1)p+2RK(ψ,ξ)Tξ(K+1)p+2. (2.30)

    We use Eq (2.30) to get the necessary result,

    |RK(ψ,ξ)|Tξ(K+1)p+2.

    Thus, a new series convergence criteria is developed.

    The ARPSM rules served as the foundation for our general model solution, which we describe below.

    Step 1. The general equation may be simplified to obtain:

    Dpφβ1(ψ,φ)+ϑ(ψ)N(β1)δ(ψ,β1)=0. (3.1)

    Step 2. The two sides of Eq (3.1) are evaluated using the AT in order to get

    A[Dpφβ1(ψ,φ)+ϑ(ψ)N(β1)δ(ψ,β1)]=0, (3.2)

    transformation of Eq (3.2) by using Lemma 2.5. Thus,

    Ψ(ψ,s)=q1j=0Djφβ1(ψ,0)sqp+2ϑ(ψ)Y(s)sqp+F(ψ,s)sqp, (3.3)

    where, A[δ(ψ,β1)]=F(ψ,s),A[N(β1)]=Y(s).

    Step 3. Examine the form that the solution to Eq (3.3) takes:

    Ψ(ψ,s)=r=0r(ψ)srp+2, s>0,

    Step 4. To proceed, follow these steps:

    0(ψ)=limss2Ψ(ψ,s)=β1(ψ,0).

    The following outcome by using Theorem 2.9:

    1(ψ)=Dpφβ1(ψ,0),
    2(ψ)=D2pφβ1(ψ,0),
    w(ψ)=Dwpφβ1(ψ,0).

    Step 5. The Ψ(ψ,s) series that has been Kth truncated may be found using the formula below:

    ΨK(ψ,s)=Kr=0r(ψ)srp+2, s>0,
    ΨK(ψ,s)=0(ψ)s2+1(ψ)sp+2++w(ψ)swp+2+Kr=w+1r(ψ)srp+2.

    Step 6. Remember that, in order to derive the following, you must take into consideration both the Kth-truncated Aboodh residual function and the Aboodh residual function (ARF) from (3.3) separately:

    ARes(ψ,s)=Ψ(ψ,s)q1j=0Djφβ1(ψ,0)sjp+2+ϑ(ψ)Y(s)sjpF(ψ,s)sjp,

    and

    AResK(ψ,s)=ΨK(ψ,s)q1j=0Djφβ1(ψ,0)sjp+2+ϑ(ψ)Y(s)sjpF(ψ,s)sjp. (3.4)

    Step 7. Put ΨK(ψ,s) into Eq (3.4) rather than use its expansion form,

    AResK(ψ,s)=(0(ψ)s2+1(ψ)sp+2++w(ψ)swp+2+Kr=w+1r(ψ)srp+2)q1j=0Djφβ1(ψ,0)sjp+2+ϑ(ψ)Y(s)sjpF(ψ,s)sjp. (3.5)

    Step 8. Equation (3.5) may be solved by multiplying both sides by sKp+2,

    sKp+2AResK(ψ,s)=sKp+2(0(ψ)s2+1(ψ)sp+2++w(ψ)swp+2+Kr=w+1r(ψ)srp+2q1j=0Djφβ1(ψ,0)sjp+2+ϑ(ψ)Y(s)sjpF(ψ,s)sjp). (3.6)

    Step 9. After taking lims, we calculate the solution to Eq (3.6), which is:

    limssKp+2AResK(ψ,s)=limssKp+2(0(ψ)s2+1(ψ)sp+2++w(ψ)swp+2+Kr=w+1r(ψ)srp+2q1j=0Djφβ1(ψ,0)sjp+2+ϑ(ψ)Y(s)sjpF(ψ,s)sjp).

    Step 10. Solving the above equation will provide the value of K(ψ),

    lims(sKp+2AResK(ψ,s))=0,

    where K=w+1,w+2,.

    Step 11. Using a K-truncated series of Ψ(ψ,s), replace the values of K(ψ) to get the K-approximate solution of Eq (3.3).

    Step 12. Solve ΨK(ψ,s) using the AIT to get the K-approximate solution β1K(ψ,φ).

    Consider the following PDE of space and time fractional order:

    Dpφβ1(ψ,φ)=Φ(β1(ψ,φ),Dφψβ1(ψ,φ),D2φψβ1(ψ,φ),D3φψβ1(ψ,φ)), 0<p,φ1, (3.7)

    having the IC's

    β1(k)(ψ,0)=hk, k=0,1,2,,m1, (3.8)

    the function β1(ψ,φ) is unknown, while Φ(β1(ψ,φ),Dφψβ1(ψ,φ),D2φψβ1(ψ,φ),D3φψβ1(ψ,φ)) may be a nonlinear operator or linear of β1(ψ,φ),Dφψβ1(ψ,φ),D2φψβ1(ψ,φ) and D3φψβ1(ψ,φ). Applying the AT to both sides of Eq (3.7) yields the following equation; for convenience, we will denote β1(ψ,φ) using the symbol β1,

    A[β1(ψ,φ)]=1sp(m1k=0β1(k)(ψ,0)s2p+k+A[Φ(β1(ψ,φ),Dφψβ1(ψ,φ),D2φψβ1(ψ,φ),D3φψβ1(ψ,φ))]), (3.9)

    as a result of using the AIT to solve this problem,

    β1(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k+A[Φ(β1(ψ,φ),Dφψβ1(ψ,φ),D2φψβ1(ψ,φ),D3φψβ1(ψ,φ))])]. (3.10)

    The solution obtained by using the iterative Aboodh transform method is represented as an infinite series,

    β1(ψ,φ)=i=0β1i. (3.11)

    Since Φ(β1,Dφψβ1,D2φψβ1,D3φψβ1) is either a nonlinear or linear operator, which can be decomposed as follows:

    Φ(β1,Dφψβ1,D2φψβ1,D3φψβ1)=Φ(β10,Dφψβ10,D2φψβ10,D3φψβ10)+i=0(Φ(ik=0(β1k,Dφψβ1k,D2φψβ1k,D3φψβ1k))Φ(i1k=1(β1k,Dφψβ1k,D2φψβ1k,D3φψβ1k))). (3.12)

    Equation (3.10) is changed to the following equation by substituting the values of (3.12) and (3.11).

    i=0β1i(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k+A[Φ(β10,Dφψβ10,D2φψβ10,D3φψβ10)])]+A1[1sp(A[i=0(Φik=0(β1k,Dφψβ1k,D2φψβ1k,D3φψβ1k))])]A1[1sp(A[(Φi1k=1(β1k,Dφψβ1k,D2φψβ1k,D3φψβ1k))])], (3.13)
    β10(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k)],β11(ψ,φ)=A1[1sp(A[Φ(β10,Dφψβ10,D2φψβ10,D3φψβ10)])],β1m+1(ψ,φ)=A1[1sp(A[i=0(Φik=0(β1k,Dφψβ1k,D2φψβ1k,D3φψβ1k))])]A1[1sp(A[(Φi1k=1(β1k,Dφψβ1k,D2φψβ1k,D3φψβ1k))])], m=1,2,. (3.14)

    The m-term of Eq (3.7) may be analytically approximated using the following expression:

    β1(ψ,φ)=m1i=0β1i. (3.15)

    Examine the time-fractional KS model with sensitivity term ϖ(β2)=1, as shown in [23]. Then, ψ(β1(ψ,φ)ϖ(β2)ψ)=0,

    Dpφβ1(ψ,φ)a2β1(ψ,φ)ψ2=0,Dpφβ2(ψ,φ)b2β2(ψ,φ)ψ2cβ1(ψ,φ)+dβ2(ψ,φ)=0,   where   0<p1, (4.1)

    having IC's:

    β1(ψ,0)=l1eψ2,β2(ψ,0)=l2eψ2. (4.2)

    Using Eq (4.2), AT is applied to Eq (4.1) in order to get

    β1(ψ,s)l1eψ2s2asp[2β1(ψ,φ)ψ2]=0,β2(ψ,s)l2eψ2s2bsp[2β2(ψ,φ)ψ2]csp[β1(ψ,φ)]+dsp[β2(ψ,φ)]=0. (4.3)

    The kth-truncated term series are

    β1(ψ,s)=l1eψ2s2+kr=1fr(ψ,s)srp+1,β2(ψ,s)=l2eψ2s2+kr=1jr(ψ,s)srp+1,  r=1,2,3,4 (4.4)

    Aboodh residual functions (ARFs) are

    AφRes(ψ,s)=β1(ψ,s)l1eψ2s2asp[2β1(ψ,φ)ψ2]=0,AφRes(ψ,s)=β2(ψ,s)l2eψ2s2bsp[2β2(ψ,φ)ψ2]csp[β1(ψ,φ)]+dsp[β2(ψ,φ)]=0, (4.5)

    and the kth-LRFs as:

    AφResk(ψ,s)=β1k(ψ,s)l1eψ2s2asp[2β1k(ψ,φ)ψ2]=0,AφResk(ψ,s)=β2k(ψ,s)l2eψ2s2bsp[2β2k(ψ,φ)ψ2]csp[β1k(ψ,φ)]+dsp[β2k(ψ,φ)]=0, (4.6)

    To determine fr(ψ,s) and jr(ψ,s), for r=1,2,3,.... Then, we iteratively solve lims(srp+1) by multiplying the resulting equation by srp+1, substituting the rth-Aboodh residual function Eq (4.6) for the rth-truncated series Eq (4.4). AφResβ1,r(ψ,s))=0 and AφResβ2,r(ψ,s))=0, and r=1,2,3,. Putting a=0.5,b=3,c=1 and d=0.8 and taking the values of l1=160 and l2=120, we find the first few terms as:

    f1(ψ,s)=eψ2(320β22160),j1(ψ,s)=eψ2(1440ψ2656), (4.7)
    f2(ψ,s)=eψ2(640ψ41920ψ2+480),j2(ψ,s)=eψ2(17280ψ451904ψ2+12941), (4.8)

    and so on.

    Putting fr(ψ,s), for r=1,2,3,, in Eq (4.4), we get

    β1(ψ,s)=eψ2(320ψ2160)sp+1+eψ2(640ψ41920ψ2+480)s2p+1+160eψ2s2+,β2(ψ,s)=eψ2(1440ψ2656)sp+1+eψ2(17280ψ451904ψ2+12941)s2p+1+120eψ2s2+. (4.9)

    The AIT may be used to get

    β1(ψ,φ)=eψ2φp(320ψ2160)Γ(p+1)+eψ2φ2p(640ψ41920ψ2+480)Γ(2p+1)+160eψ2+,β2(ψ,s)=eψ2φp(1440ψ2656)Γ(p+1)+eψ2φ2p(17280ψ451904ψ2+12941)Γ(2p+1)+120eψ2+. (4.10)
    Dpφβ1(ψ,φ)=a2β1(ψ,φ)ψ2,Dpφβ2(ψ,φ)=b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ),   where   0<p1, (4.11)

    having IC's:

    β1(ψ,0)=l1eψ2,β2(ψ,0)=l2eψ2, (4.12)

    By using the AT on each side of Eq (4.11), we are able to get the following result:

    A[Dpφβ1(ψ,φ)]=1sp(m1k=0β1(k)(ψ,0)s2p+k+A[a2β1(ψ,φ)ψ2]),A[Dpφβ2(ψ,φ)]=1sp(m1k=0β2(k)(ψ,0)s2p+k+A[b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ)]), (4.13)

    using the AIT on each side of 4.13, we get the following result:

    β1(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k+A[a2β1(ψ,φ)ψ2])],β2(ψ,φ)=A1[1sp(m1k=0β2(k)(ψ,0)s2p+k+A[b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ)])]. (4.14)

    The equation that is produced as a consequence of applying the AT in an iterative manner is as follows:

    β10(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k)]=A1[β1(ψ,0)s2]=l1eψ2,
    β20(ψ,φ)=A1[1sp(m1k=0β2(k)(ψ,0)s2p+k)]=A1[β2(ψ,0)s2]=l2eψ2.

    We replaced the RL integral in Eq (4.11) to get the equivalent variant.

    β1(ψ,φ)=l1eψ2A[a2β1(ψ,φ)ψ2],β2(ψ,φ)=l2eψ2A[b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ)]. (4.15)

    Putting a=0.5,b=3,c=1, and d=0.8 and taking the values of l1=160 and l2=120, the following terms are then acquired by using the ATIM procedure:

    β10(ψ,φ)=160eψ2,β20(ψ,φ)=120eψ2,β11(ψ,φ)=eψ2(320ψ2160)φpΓ(p+1),β21(ψ,φ)=eψ2(1440ψ2656)φpΓ(p+1),β12(ψ,φ)=eψ2(640ψ41920ψ2+478)φ2pΓ(2p+1),β22(ψ,φ)=eψ2zeta2p((364.8832ψ2)Γ(p+1)+p(17280ψ451072ψ2+12576)Γ(p))Γ(p+1)Γ(2p+1). (4.16)

    The following is the final ATIM solution:

    β1(ψ,φ)=β10(ψ,φ)+β11(ψ,φ)+β12(ψ,φ)+,β2(ψ,φ)=β20(ψ,φ)+β21(ψ,φ)+β22(ψ,φ)+. (4.17)
    β1(ψ,φ)=160eψ2+eψ2(320ψ2160)φpΓ(p+1)+eψ2(640ψ41920ψ2+478)φ2pΓ(2p+1)+,β2(ψ,φ)=120eψ2+eψ2(1440ψ2656)φpΓ(p+1)+eψ2zeta2p((364.8832ψ2)Γ(p+1)+p(17280ψ451072ψ2+12576)Γ(p))Γ(p+1)Γ(2p+1)+. (4.18)

    Examine the KS model of fractional order as stated in [23] with sensitivity term ϖ(β2)=β2(ψ,φ). Then, the function ψ(β1(ψ,φ)ϖ(β2)ψ)=β1(ψ,φ)2β2(ψ,φ)ψ2+β1(ψ,φ)ψβ2(ψ,φ)ψ,

    Dpφβ1(ψ,φ)a2β1(ψ,φ)ψ2+β1(ψ,φ)2β2(ψ,φ)ψ2+β1(ψ,φ)ψβ2(ψ,φ)ψ=0,Dpφβ2(ψ,φ)b2β2(ψ,φ)ψ2cβ1(ψ,φ)+dβ2(ψ,φ)=0,   where   0<p1, (5.1)

    having IC's:

    β1(ψ,0)=l1eψ2,β2(ψ,0)=l2eψ2. (5.2)

    AT is applied to Eq (5.1), the following results are obtained using Eq (5.2):

    β1(ψ,s)l1eψ2s2asp[2β1(ψ,φ)ψ2]+1spAφ[A1φβ1(ψ,φ)×A1φ2β2(ψ,φ)ψ2]+1spAφ[A1φβ1(ψ,φ)ψ×A1φβ2(ψ,φ)ψ]=0,β2(ψ,s)l2eψ2s2bsp[2β2(ψ,φ)ψ2]csp[β1(ψ,φ)]+dsp[β2(ψ,φ)]=0. (5.3)

    The kth truncated term series are

    β1(ψ,s)=l1eψ2s2+kr=1fr(ψ,s)srp+1,β2(ψ,s)=l2eψ2s2+kr=1jr(ψ,s)srp+1,  r=1,2,3,4. (5.4)

    Aboodh residual functions (ARFs) are

    AφRes(ψ,s)=β1(ψ,s)l1eψ2s2asp[2β1(ψ,φ)ψ2]+1spAφ[A1φβ1(ψ,φ)×A1φ2β2(ψ,φ)ψ2]+1spAφ[A1φβ1(ψ,φ)ψ×A1φβ2(ψ,φ)ψ]=0,AφRes(ψ,s)=β2(ψ,s)l2eψ2s2bsp[2β2(ψ,φ)ψ2]csp[β1(ψ,φ)]+dsp[β2(ψ,φ)]=0, (5.5)

    and the kth-LRFs as:

    AφResk(ψ,s)=β1k(ψ,s)+l1eψ2s2asp[2β1k(ψ,φ)ψ2]+1spAφ[A1φβ1k(ψ,φ)×A1φ2β2k(ψ,φ)ψ2]+1spAφ[A1φβ1k(ψ,φ)ψ×A1φβ2k(ψ,φ)ψ]=0,AφResk(ψ,s)=β2k(ψ,s)l2eψ2s2bsp[2β2k(ψ,φ)ψ2]csp[β1k(ψ,φ)]+dsp[β2k(ψ,φ)]=0. (5.6)

    To determine fr(ψ,s) and jr(ψ,s), for r=1,2,3,.... Then, we iteratively solve lims(srp+1) by multiplying the resulting equation by srp+1, substituting the rth-Aboodh residual function Eq (5.6) for the rth-truncated series Eq (5.4). AφResβ1,r(ψ,s))=0 and AφResβ2,r(ψ,s))=0, and r=1,2,3,. Putting a=0.5,b=3,c=1, and d=0.8 and taking the values of l1=160 and l2=120, we find the first few terms as:

    f1(ψ,s)=eψ2(320ψ2160)38400e2ψ2,j1(ψ,s)=eψ2(1440ψ2656), (5.7)
    f2(ψ,s)=e3ψ2(18432000ψ2+9216000)+e2ψ2(7859201612800ψ2)+eψ2(640ψ41920ψ2+480),j2(ψ,s)=eψ2(17280ψ451904ψ2+12941)38400e2ψ2, (5.8)

    and so on.

    Equation (5.4) is used to obtain fr(ψ,s) for r=1,2,3,,

    β1(ψ,s)=160eψ2s2+eψ2(320ψ2160)38400e2ψ2sp+1+e3ψ2(18432000ψ2+9216000)+e2ψ2(7859201612800ψ2)+eψ2(640ψ41920ψ2+480)s2p+1+,β2(ψ,s)=120eψ2s2+eψ2(1440ψ2656)sp+1+eψ2(17280ψ451904ψ2+12941)38400e2ψ2s2p+1+. (5.9)

    AIT is applied to get

    β1(ψ,φ)=160eψ2+eψ2φp(320ψ2160)38400e2ψ2Γ(p+1)+e3ψ2φ2p(18432000ψ2+9216000)+e2ψ2(7859201612800ψ2)+eψ2(640ψ41920ψ2+480)Γ(2p+1)+,β2(ψ,φ)=120eψ2s2+eψ2φp(1440ψ2656)Γ(p+1)+eψ2φ2p(17280ψ451904ψ2+12941)38400e2ψ2Γ(2p+1)+. (5.10)
    Dpφβ1(ψ,φ)=a2β1(ψ,φ)ψ2β1(ψ,φ)2β2(ψ,φ)ψβ1(ψ,φ)ψβ2(ψ,φ)ψ,Dpφβ2(ψ,φ)=b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ),   where   0<p1, (5.11)

    having IC's:

    β1(ψ,0)=l1eψ2,β2(ψ,0)=l2eψ2, (5.12)

    when applying the AT to both sides of Eq (5.11), we get the following result:

    A[Dpφβ1(ψ,φ)]=1sp(m1k=0β1(k)(ψ,0)s2p+k+A[a2β1(ψ,φ)ψ2β1(ψ,φ)2β2(ψ,φ)ψβ1(ψ,φ)ψβ2(ψ,φ)ψ]),A[Dpφβ2(ψ,φ)]=1sp(m1k=0β2(k)(ψ,0)s2p+k+A[b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ)]), (5.13)

    applying the AIT to both sides of Eq (5.13) yields the following result:

    β1(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k+A[a2β1(ψ,φ)ψ2β1(ψ,φ)2β2(ψ,φ)ψβ1(ψ,φ)ψβ2(ψ,φ)ψ])],β2(ψ,φ)=A1[1sp(m1k=0β2(k)(ψ,0)s2p+k+A[b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ)])]. (5.14)

    This equation is obtained by using the AT's iterative procedure:

    β10(ψ,φ)=A1[1sp(m1k=0β1(k)(ψ,0)s2p+k)]=A1[β1(ψ,0)s2]=l1eψ2,
    β20(ψ,φ)=A1[1sp(m1k=0β2(k)(ψ,0)s2p+k)]=A1[β2(ψ,0)s2]=l2eψ2.

    The RL integral is applied to Eq (5.1) to yield the equivalent form.

    β1(ψ,φ)=l1eψ2A[a2β1(ψ,φ)ψ2β1(ψ,φ)2β2(ψ,φ)ψβ1(ψ,φ)ψβ2(ψ,φ)ψ],β2(ψ,φ)=l2eψ2A[b2β2(ψ,φ)ψ2+cβ1(ψ,φ)dβ2(ψ,φ)]. (5.15)

    Putting a=0.5,b=3,c=1, and d=0.8 and taking the values of l1=160 and l2=120, the following terms are then acquired by using the ATIM procedure: These terms are obtained using the ATIM process,

    β10(ψ,φ)=160eψ2,β20(ψ,φ)=120eψ2,β11(ψ,φ)=e2β22(eβ22(320β22160)38400)φpΓ(p+1),β21(ψ,φ)=ex2(1440x2656)tpΓ(p+1),β12(ψ,φ)=(225 46pe2ψ2φp(pΓ(p)(4pΓ(p+12)(eψ2(240ψ410.66ψ2+eψ2(ψ40.044ψ2+0.727)+174.66)φ2pΓ(2p+1)+0.0416Γ(p+1)Γ(3p+1))+φpΓ(p+1)Γ(3p+1)(3.10179ψ2+(0.0012ψ4+35.4528ψ2+17.7236)sinh(ψ2)+(0.00123ψ435.4453ψ217.7254)cosh(ψ2)1.5115))0.0738Γ(p+1)Γ(2p+1)Γ(3p+1)))/(pΓ(p)Γ(p+12)Γ(p+1)2Γ(3p+1)),β22(ψ,φ)=(e2ψ2φ2p(eψ2((524.81152ψ2)Γ(p+1)+p(17280ψ450752ψ2+12416)Γ(p))38400pΓ(p)))/(Γ(p+1)Γ(2p+1)). (5.16)

    The following is the ATIM procedure's ultimate solution:

    β1(ψ,φ)=β10(ψ,φ)+β11(ψ,φ)+β12(ψ,φ)+,β2(ψ,φ)=β20(ψ,φ)+β21(ψ,φ)+β22(ψ,φ)+. (5.17)
    β1(ψ,φ)=160eψ2+(e2β22(eβ22(320β22160)38400)φp)/(Γ(p+1))(225 46pe2ψ2φp×(pΓ(p)(4pΓ(p+12)(eψ2(240ψ410.66ψ2+eψ2(ψ40.044ψ2+0.727)+174.66)φ2pΓ(2p+1)+0.0416Γ(p+1)Γ(3p+1))+φpΓ(p+1)Γ(3p+1)(3.10179ψ2+(0.0012ψ4+35.4528ψ2+17.7236)sinh(ψ2)+(0.00123ψ435.4453ψ217.7254)cosh(ψ2)1.5115))0.0738Γ(p+1)Γ(2p+1)Γ(3p+1)))/(pΓ(p)Γ(p+12)Γ(p+1)2Γ(3p+1))+, (5.18)
    β2(ψ,φ)=120eψ2+(ex2(1440x2656)tp)/(Γ(p+1))+(e2ψ2φ2p(eψ2((524.81152ψ2)Γ(p+1)+p(17280ψ450752ψ2+12416)Γ(p))38400pΓ(p)))/(Γ(p+1)Γ(2p+1))+. (5.19)

    In Problem 1, we embark on a comprehensive exploration of the solutions β1(ψ,φ) and β2(ψ,φ) through both graphical and numerical analyses employing two distinct methodologies: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). Beginning with β1(ψ,φ), Figure 1 offers an insightful depiction of the approximate solution obtained via ARPSM for a specific value of p=1. Building upon this foundation, Figure 2 extends the analysis, providing both 3D and 2D representations to elucidate the influence of varying p on the solution when φ=0.1. Similarly, Figures 3 and 4 delve into the corresponding analyses for β2(ψ,φ). These visualizations offer a nuanced understanding of how changes in the parameter p affect the behavior of the solutions across different dimensions. In conjunction with the graphical exploration, Tables 1 and 2 complement our investigation by presenting detailed fractional order analyses for ARPSM applied to β1(ψ,φ) and β2(ψ,φ), respectively. These tables provide valuable insights into the fractional characteristics of the solutions and contribute to a comprehensive understanding of their properties.

    Figure 1.  Approximate solution of β1(ψ,φ) via ARPSM for p=1..
    Figure 2.  β1(ψ,φ), (a) shows three-dimensional analysis of different values of p; (b) shows two-dimensional analysis of different values of p at φ=0.1 via ARPSM.
    Figure 3.  Approximate solution of β2(ψ,φ) via ARPSM for p=1.
    Figure 4.  β2(ψ,φ): (a) shows three-dimensional analysis of different values of p; (b) shows two-dimensional analysis of different values of p at φ=0.1 via ARPSM.
    Table 1.  Analysis of various values of fractional order of ARPSM of Problem 1 of β1(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.0048028 0.0022338 0.00098417 0.00043033
    0.4 0.0082096 0.0049227 0.00273349 0.00144405
    0.6 0.0112581 0.0078699 0.00506050 0.00305916
    0.8 0.0140966 0.0110044 0.00787829 0.00527565
    1.0 0.0167894 0.0142882 0.0111338 0.00809353

     | Show Table
    DownLoad: CSV
    Table 2.  Analysis of various values of fractional order of ARPSM of Problem 1 of β2(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.121739 0.054477 0.022375 0.0086336
    0.4 0.211321 0.124300 0.066837 0.0334867
    0.6 0.291879 0.201620 0.127169 0.0745724
    0.8 0.367092 0.284277 0.200911 0.1318910
    1.0 0.438569 0.371156 0.286582 0.2054420

     | Show Table
    DownLoad: CSV

    Shifting focus to the ATIM method, Figures 5 and 7 display the approximate solutions of β1(ψ,φ) and β2(ψ,φ) for p=1, respectively. Figures 6 and 8 further extend the analysis, offering insights into the impact of varying p at φ=0.1. The fractional order sensitivity is examined through Tables 3 and 4 for β1(ψ,φ) and β2(ψ,φ) under ATIM.

    Figure 5.  Approximate solution of β1(ψ,φ) via ATIM for p=1..
    Figure 6.  β1(ψ,φ): (a) shows three-dimensional analysis of different values of p; (b) shows two dimensional analysis of different values of p at φ=0.1 via ATIM.
    Figure 7.  Approximate solution of β2(ψ,φ) via ATIM for p=1..
    Figure 8.  β2(ψ,φ): (a) shows three-dimensional analysis of different values of p; (b) shows two-dimensional analysis of different values of p at φ=0.1 via ATIM.
    Table 3.  Analysis of various values of fractional order of ATIM of Problem 1 of β1(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.0048028 0.0022338 0.00098416 0.00043033
    0.4 0.0082095 0.0049226 0.00273345 0.00144403
    0.6 0.0112579 0.0078698 0.00506043 0.00305912
    0.8 0.0140964 0.0110042 0.00787818 0.00527558
    1.0 0.0167892 0.0142880 0.01113370 0.00809342

     | Show Table
    DownLoad: CSV
    Table 4.  Analysis of various values of fractional order of ATIM of Problem 1 of β2(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.121748 0.054481 0.022376 0.0086342
    0.4 0.211336 0.124308 0.066842 0.0334890
    0.6 0.291899 0.201634 0.127178 0.0745776
    0.8 0.367118 0.284297 0.200924 0.1319000
    1.0 0.438600 0.371182 0.286602 0.2054560

     | Show Table
    DownLoad: CSV

    In order to facilitate a comprehensive comparison, Tables 5 and 6 juxtapose the results obtained from both ARPSM and ATIM for β1(ψ,φ) and β2(ψ,φ) in Problem 1. These tables provide a nuanced understanding of the performance of each method, shedding light on their strengths and limitations in solving the given fractional-order equations. The two most basic approaches to solving fractional differential equations are the ATIM and the ARPSM, as stated in [52,53] and [49,50,51], respectively. These techniques provide numerical solutions to PDEs that do not need discretization or linearization, making the symbolic terms in analytical solutions instantly visible. The primary objective of this study is to compare and contrast the performance of ARPSM and ATIM in solving the Keller-Segel (KS) model. It is worth mentioning that several linear and nonlinear fractional differential problems have been solved using these two approaches.

    Table 5.  Problem 1: comparison of both methods for β1(ψ,φ).
    p=1.00 p=0.80 p=0.60 p=0.40
    φ ARPSM ATIM ARPSM ATIM ARPSM ATIM ARPSM ATIM
    0.2 0.00043033 0.00043033 0.00098417 0.00098416 0.0022338 0.0022338 0.0048028 0.0048028
    0.4 0.00144405 0.00144403 0.00273349 0.00273345 0.0049227 0.0049226 0.0082096 0.0082095
    0.6 0.00305916 0.00305912 0.00506050 0.00506043 0.0078699 0.0078698 0.0112581 0.0112579
    0.8 0.00527565 0.00527558 0.00787829 0.00787818 0.0110044 0.0110042 0.0140966 0.0140964
    1.0 0.00809353 0.00809342 0.01113380 0.01113370 0.0142882 0.0142880 0.0167894 0.0167892

     | Show Table
    DownLoad: CSV
    Table 6.  Problem 1: comparison of both methods for β2(ψ,φ).
    p=1.00 p=0.80 p=0.60 p=0.40
    φ ARPSM ATIM ARPSM ATIM ARPSM ATIM ARPSM ATIM
    0.2 0.0086336 0.0086342 0.022375 0.022376 0.054477 0.054481 0.121739 0.121748
    0.4 0.0334867 0.0334890 0.066837 0.066842 0.124300 0.124308 0.211321 0.211336
    0.6 0.0745724 0.0745776 0.127169 0.127178 0.201620 0.201634 0.291879 0.291899
    0.8 0.1318910 0.1319000 0.200911 0.200924 0.284277 0.284297 0.367092 0.367118
    1.0 0.2054420 0.2054560 0.286582 0.286602 0.371156 0.371182 0.438569 0.438600

     | Show Table
    DownLoad: CSV

    In Problem 2, the analysis of solutions β1(ψ,φ) and β2(ψ,φ) is carried out using the ARPSM and the ATIM. For β1(ψ,φ), Figure 9 illustrates the approximate solution via ARPSM for p=1. Subsequently, Figure 10 presents 3D and 2D analyses, demonstrating the influence of varying p on the solution at φ=0.1. Analogously, Figures 11 and 12 provide the corresponding analyses for β2(ψ,φ). Complementing the graphical exploration, Table 7 details the fractional order analysis for ARPSM of β1(ψ,φ), and Table 8 does the same for β2(ψ,φ).

    Figure 9.  Approximate solution of β1(ψ,φ) via ARPSM for p=1..
    Figure 10.  2D analysis of different values of p at φ=0.1.
    Figure 11.  Approximate solution of β2(ψ,φ) via ARPSM for p=1..
    Figure 12.  2D analysis of different values of p at φ=0.1.
    Table 7.  Analysis of various values of fractional order of ARPSM of Problem 2 of β1(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.0048027 0.0022338 0.00098415 0.00043032
    0.4 0.0082094 0.0049226 0.00273344 0.00144403
    0.6 0.0112578 0.0078698 0.00506040 0.00305910
    0.8 0.0140963 0.0110042 0.00787813 0.00527555
    1.0 0.0167891 0.0142880 0.01113360 0.00809337

     | Show Table
    DownLoad: CSV
    Table 8.  Analysis of various values of fractional order of ARPSM of Problem 2 of β2(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.121739 0.054477 0.022375 0.0086336
    0.4 0.211321 0.124300 0.066837 0.0334867
    0.6 0.291879 0.201620 0.127169 0.0745724
    0.8 0.367092 0.284277 0.200911 0.1318910
    1.0 0.438569 0.371156 0.286582 0.2054420

     | Show Table
    DownLoad: CSV

    Shifting focus to the ATIM method, Figures 13 and 15 display the approximate solutions of β1(ψ,φ) and β2(ψ,φ) for p=1, respectively. Figures 14 and 16 further extend the analysis, offering insights into the impact of varying p at φ=0.1. The fractional order sensitivity is examined through Tables 9 and 10 for β1(ψ,φ) and β2(ψ,φ) under ATIM.

    Figure 13.  Approximate solution of β1(ψ,φ) via ATIM for p=1..
    Figure 14.  2D analysis of different values of p at φ=0.1.
    Figure 15.  Approximate solution of β2(ψ,φ) via ATIM for p = 1.
    Figure 16.  2D analysis of different values of p at φ=0.1.
    Table 9.  Analysis of various values of fractional order of ATIM of Problem 2 of β1(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.0047363 0.00220436 0.00097225 0.00042586
    0.4 0.0080934 0.00485484 0.00269728 0.00142614
    0.6 0.0110971 0.00775928 0.00499109 0.00301877
    0.8 0.0138937 0.01084780 0.00776810 0.00520373
    1.0 0.0165466 0.01408330 0.01097610 0.00798096

     | Show Table
    DownLoad: CSV
    Table 10.  Analysis of various values of fractional order of ATIM of Problem 2 of β2(ψ,φ).
    φ p=0.40 p=0.60 p=0.80 p=1.00
    0.2 0.121748 0.054481 0.0223765 0.0086342
    0.4 0.211336 0.124308 0.0668425 0.0334890
    0.6 0.291899 0.201634 0.1271780 0.0745776
    0.8 0.367118 0.284297 0.2009240 0.1319000
    1.0 0.438600 0.371182 0.2866020 0.2054560

     | Show Table
    DownLoad: CSV

    To facilitate a comprehensive comparison, Tables 11 and 12 juxtapose the results obtained from both ARPSM and ATIM for β1(ψ,φ) and β2(ψ,φ) in Problem 2. These tables provide a nuanced understanding of the performance of each method, shedding light on their strengths and limitations in solving the given fractional-order equations in the context of Problem 2.

    Table 11.  Problem 2 comparison of both methods for β1(ψ,φ).
    p=1.00 p=0.80 p=0.60 p=0.40
    φ ARPSM ATIM ARPSM ATIM ARPSM ATIM ARPSM ATIM
    0.2 0.0004303 0.0004258 0.0009841 0.0009722 0.0022338 0.0022043 0.004802 0.0047363
    0.4 0.0014440 0.0014261 0.0027334 0.0026972 0.0049226 0.0048548 0.008209 0.0082096
    0.6 0.0030591 0.0030187 0.0050604 0.0049910 0.0078698 0.0077592 0.011257 0.0110971
    0.8 0.0052755 0.0052037 0.0078781 0.0077681 0.0110042 0.0108478 0.014096 0.0138937
    1.0 0.0080933 0.0079809 0.0111336 0.0109761 0.0142880 0.0140833 0.016789 0.0165466

     | Show Table
    DownLoad: CSV
    Table 12.  Problem 2 comparison of both methods for β2(ψ,φ).
    p=1.00 p=0.80 p=0.60 p=0.40
    φ ARPSM ATIM ARPSM ATIM ARPSM ATIM ARPSM ATIM
    0.2 0.0086336 0.0086342 0.022375 0.022376 0.054477 0.054481 0.121739 0.121748
    0.4 0.0334867 0.0334890 0.066837 0.066842 0.124300 0.124308 0.211321 0.211336
    0.6 0.0745724 0.0745776 0.127169 0.127178 0.201620 0.201634 0.291879 0.291899
    0.8 0.1318910 0.1319000 0.200911 0.200924 0.284277 0.284297 0.367092 0.367118
    1.0 0.2054420 0.2054560 0.286582 0.286602 0.371156 0.371182 0.438569 0.438600

     | Show Table
    DownLoad: CSV

    In summary, this research has focused on improving numerical methods designed for solving the fractional Keller-Segel (KS) model, which is a crucial framework for studying chemotaxis phenomena. By utilizing the Caputo operator framework, we have employed two distinct methodologies: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). These approaches have enabled us to obtain accurate solutions to the fractional KS equation, contributing to a better understanding of chemotactic behavior in biological systems. Through a comparative analysis of ARPSM and ATIM, we have revealed their individual strengths and applications in addressing complex fractional models. This work not only advances numerical techniques tailored for fractional differential equations but also improves our understanding of chemotaxis dynamics through precise modeling.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The researcher would like to thank the Deanship of Scientific Research, Shaqra University, for funding the publication of this project.

    The authors declare that they have no competing interests.



    [1] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303. doi: 10.1088/1751-8113/40/24/003
    [2] B. C. Carlson, Lauricella's hypergeometric function FD, J. Math. Anal. Appl., 7 (1963), 452-470. doi: 10.1016/0022-247X(63)90067-2
    [3] B. C. Carlson, A connection between elementary and higher transcendental functions, SIAM J. Appl. Math., 17 (1969), 116-148. doi: 10.1137/0117013
    [4] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.
    [5] B. C. Carlson, B-splines, hypergeometric functions and Dirichlet average, J. Approx. Theory, 67 (1991), 311-325. doi: 10.1016/0021-9045(91)90006-V
    [6] W. Zu Castell, Dirichlet splines as fractional integrals of B-splines, Rocky Mountain J. Math., 32 (2002), 545-559. doi: 10.1216/rmjm/1030539686
    [7] J. Daiya, Representing double Dirichlet average in term of K-Mittag-Leffler function associated with fractional derivative, Journal of Chemical, Biological and Physical Sciences, Section C, 6 (2016), 1034-1045.
    [8] J. Daiya, J. Ram, Dirichlet averages of generalized Hurwitz-Lerch zeta function, Asian J. Math. Comput. Res., 7 (2015), 54-67.
    [9] R. Diaz, E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat., 15 (2007), 179-192.
    [10] A. Erdélyi, W. Magnus, F. Oberhettinger, et al. Tables of Integral Transforms, Vol. II, McGrawHill, New York-Toronto-London, 1954.
    [11] S. C. Gupta and B. M. Agrawal, Double Dirichlet average and fractional derivative, Ganita Sandesh, 5 (1991), 47-52.
    [12] R. K. Gupta, B. S. Shaktawat, D. Kumar, Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function, J. Raj. Acad. Phy. Sci., 15 (2016), 117-126.
    [13] R. K. Gupta, B. S. Shaktawat, D. Kumar, A study of Saigo-Maeda fractional calculus operators associated with the multiparameter K-Mittag-Leffler function, Asian J. Math. Comput. Res., 12 (2016), 243-251.
    [14] R. K. Gupta, B. S. Shaktawat, D. Kumar, Generalized fractional differintegral operators of the K-series, Honam Math. J., 39 (2017), 61-71. doi: 10.5831/HMJ.2017.39.1.61
    [15] V. Kiryakova, A brief story about the operators of the generalized fractional calculus, Fract. Calc. Appl. Anal., 11 (2008), 203-220.
    [16] D. Kumar, On certain fractional calculus operators involving generalized Mittag-Leffler function, Sahand Communication in Mathematical Analysis, 3 (2016), 33-45.
    [17] D. Kumar, R. K. Gupta, D. S. Rawat, et al. Hypergeometric fractional integrals of multiparameter K-Mittag-Leffler function, Nonlinear Science Letters A: Mathematics, Physics and Mechanics, 9 (2018), 17-26.
    [18] D. Kumar, S. D. Purohit, Fractional differintegral operators of the generalized Mittag-Leffler type function, Malaya J. Mat., 2 (2014), 419-425.
    [19] D. Kumar, S. Kumar, Fractional integrals and derivatives of the generalized Mittag-Leffler type function, Internat. Sch. Res. Not., 2014 (2014), 1-5. doi: 10.1093/imrn/rns215
    [20] Shy-Der Lin, H. M. Srivastava, Some miscellaneous properties and applications of certain operators of fractional calculus, Taiwanese J. Math., 14 (2010), 2469-2495. doi: 10.11650/twjm/1500406085
    [21] P. Massopust, B. Forster, Multivariate complex B-splines and Dirichlet averages, J. Approx. Theory, 162 (2010), 252-269. doi: 10.1016/j.jat.2009.05.002
    [22] E. Neuman, Stolarsky means of several variables, J. Inequal. Pure Appl. Math., 6 (2005), 1-10.
    [23] E. Neuman, P. J. Van Fleet, Moments of Dirichlet splines and their applications to hypergeometric functions, J. Comput. Appl. Math., 53 (1994), 225-241. doi: 10.1016/0377-0427(94)90047-7
    [24] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications ("Nauka i Tekhnika", Minsk, 1987), Gordon and Breach Science Publishers, UK, 1993.
    [25] R. K. Saxena, J. Daiya, Integral transforms of the S-function, Le Mathematiche, 70 (2015), 147-159.
    [26] R. K. Saxena, J. Daiya, A. Singh, Integral transforms of the k-Mittag-Leffler function Eγk,α,β(z), Le Mathematiche, 69 (2014), 7-16.
    [27] R. K. Saxena, T. K. Pogány, J. Ram, et al. Dirichlet averages of generalized multi-index MittagLeffler functions, Armen. J. Math., 3 (2010), 174-187.
    [28] T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072.
  • Reader Comments
  • © 2020 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(4154) PDF downloads(342) Cited by(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog