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A two-stage grasp detection method for sequential robotic grasping in stacking scenarios


  • Dexterous grasping is essential for the fine manipulation tasks of intelligent robots; however, its application in stacking scenarios remains a challenge. In this study, we aimed to propose a two-phase approach for grasp detection of sequential robotic grasping, specifically for application in stacking scenarios. In the initial phase, a rotated-YOLOv3 (R-YOLOv3) model was designed to efficiently detect the category and position of the top-layer object, facilitating the detection of stacked objects. Subsequently, a stacked scenario dataset with only the top-level objects annotated was built for training and testing the R-YOLOv3 network. In the next phase, a G-ResNet50 model was developed to enhance grasping accuracy by finding the most suitable pose for grasping the uppermost object in various stacking scenarios. Ultimately, a robot was directed to successfully execute the task of sequentially grasping the stacked objects. The proposed methodology demonstrated the average grasping prediction success rate of 96.60% as observed in the Cornell grasping dataset. The results of the 280 real-world grasping experiments, conducted in stacked scenarios, revealed that the robot achieved a maximum grasping success rate of 95.00%, with an average handling grasping success rate of 83.93%. The experimental findings demonstrated the efficacy and competitiveness of the proposed approach in successfully executing grasping tasks within complex multi-object stacked environments.

    Citation: Jing Zhang, Baoqun Yin, Yu Zhong, Qiang Wei, Jia Zhao, Hazrat Bilal. A two-stage grasp detection method for sequential robotic grasping in stacking scenarios[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 3448-3472. doi: 10.3934/mbe.2024152

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  • Dexterous grasping is essential for the fine manipulation tasks of intelligent robots; however, its application in stacking scenarios remains a challenge. In this study, we aimed to propose a two-phase approach for grasp detection of sequential robotic grasping, specifically for application in stacking scenarios. In the initial phase, a rotated-YOLOv3 (R-YOLOv3) model was designed to efficiently detect the category and position of the top-layer object, facilitating the detection of stacked objects. Subsequently, a stacked scenario dataset with only the top-level objects annotated was built for training and testing the R-YOLOv3 network. In the next phase, a G-ResNet50 model was developed to enhance grasping accuracy by finding the most suitable pose for grasping the uppermost object in various stacking scenarios. Ultimately, a robot was directed to successfully execute the task of sequentially grasping the stacked objects. The proposed methodology demonstrated the average grasping prediction success rate of 96.60% as observed in the Cornell grasping dataset. The results of the 280 real-world grasping experiments, conducted in stacked scenarios, revealed that the robot achieved a maximum grasping success rate of 95.00%, with an average handling grasping success rate of 83.93%. The experimental findings demonstrated the efficacy and competitiveness of the proposed approach in successfully executing grasping tasks within complex multi-object stacked environments.



    Fractional differential equations have gained growing attention in recent years and many monographs have appeared [1,2]. The most common definitions of fractional calculus (differentiation and integration) are for Caputo, Riemann-Liouville, and Grunwald-Letnikov derivatives [3,4,5,6,7]. Compared with these two types of definitions, the Hadamard fractional calculus, which was first introduced in 1892 by Hadamard [8], did not receive much attention. The kernel of the integrand in the definition of the fractional Hadamard derivative contains a logarithmic function with an arbitrary exponent different from the Riemann-Liouville fractional derivatives. Recently, the Hadamard derivative and Hadamard-type fractional differential equations have been useful in practical problems related to mechanics and engineering, such as fracture analysis or both planar and three-dimensional elasticities [9]. Kilbas discussed Hadamard-type fractional differential equations in different spaces [10]. Recently, Ma and Li described the properties of Hadamard calculus [11] and they also proposed the definite conditions for Hadamard-type fractional differential equations.

    The Caputo-Hadamard (C-H) fractional derivative is a kind of fractional derivative that is useful in describing abnormal diffusion processes, especially ultra-slow diffusion. Gohar et al. [12] studied the existence and uniqueness of the solution to Caputo-Hadamard fractional differential equations and the corresponding continuation theorem. Wang et al. [13] investigated the stability of the zero solution of a class of nonlinear Hadamard-type fractional differential systems by utilizing a new fractional comparison principle. Belbali et al. [14] discussed the existence, uniqueness, and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem. Aljoudi et al. [15] studied a coupled system of Caputo-Hadamard-type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. Dhaniya et al. [16] established the existence, uniqueness, and Hyers-Ulam stability of the solution to the nonlinear Langevin fractional differential equation that involves the C-H and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. Beyenea et al. [17] established sufficient conditions for the existence and uniqueness of solutions to nonlinear Caputo-Hadamard fractional differential equations involving Hadamard integrals and unbounded delays. He et al. [18] considered the Hadamard and the Caputo-Hadamard fractional derivatives and the stability of related systems without and with delay.

    Due to the complex form of C-H fractional operators, one often needs to find a suitable numerical scheme to approximate it, which greatly improves the efficiency of the actual calculation process. The studies on numerical methods for nonlinear C-H fractional differential equations are still in their early stages. Gohar [19] studied finite difference methods for fractional differential equations with C-H derivatives and investigated the smoothness properties of the solution. Li et al. [20] obtained the analytical solution to a certain linear fractional partial differential equation with the C-H fractional derivative by introducing a new modified Laplace transform, and derived a numerical algorithm for such kinds of equations. Fan et al. [21] proposed three kinds of numerical formulas for approximating the C-H fractional derivatives, which are called L12 formula, L21σformula, and H2N2 formula.

    Most numerical methods for solving fractional differential equations are based on local difference schemes. Compared with the previous works, the main contribution of this paper is to extend the results in [22,23] by constructing and analyzing a nonlocal spectral collocation method for the following system of fractional pantograph delay differential equations:

    {CHDρtX1(t)=g1(t,X1(t),,XM(t),X1(qt),,XM(qt)), tI,CHDρtX2(t)=g2(t,X1(t),,XM(t),X1(qt),,XM(qt)), tI,CHDρtXM(t)=gM(t,X1(t),,XM(t),X1(qt),,XM(qt)), tI,Xi(t)=ˉXi(t), for qt i=1,2,,M,(0,t), ρ(0,1),q(0,1), (1.1)

    where gi:I×RMR are given continuous functions, I=(,e), and the C-H derivative CHDρt of order 0<ρ<1 is given by (2.2).

    The outline of this paper is as follows: In Section 2, we introduce some necessary definitions and preliminaries. In Section 3, we construct the spectral collocation scheme. In Section 4, we provide some auxiliary lemmas. The convergence analysis is discussed in Section 5. The numerical results are provided in Section 6.

    In this section, some relevant properties of the C-H fractional calculus and the logarithmic Jacobi (log J) approximation are presented.

    Definition 2.1. The C-H fractional integral with order ρ>0 is defined as [24]

    JρzX(z)=1Γ(ρ)zκρ1(z,w)X(w)dww,z>>0, (2.1)

    where κ(z,w)=log(zw).

    Definition 2.2. The C-H fractional differential operator of order 0<ρ<1 is given as [1]

    CHDρzX(z)=1Γ(1ρ)zκρ(z,w)X(w)dw. (2.2)

    Definition 2.3. Let ρ,η>1, I:=[,e], and >0. The log J functions of order p are given by [23]

    Pρ,η,p(z)=Pρ,ηp(κ2(z,)1)(η,ρ>1, >0, zI)=Γ(p+ρ+1)p!Γ(p+ρ+η+1)pk=0(pk)Γ(p+k+ρ+η+1)Γ(k+ρ+1)(κ(z,)1)k, (2.3)

    where Pρ,ηp(z) is the Jacobi polynomial and it is defined as

    Pρ,ηp(z)=Γ(p+ρ+1)Γ(p+1+ρ+η)p!pk=0(pk)Γ(p+k+ρ+η+1)Γ(k+ρ+1)(z12)k.

    We define the space of logarithmic functions of order s by

    Plogs(Ω):=span{1,κ(z,),κ(z,)2,,κ(z,)s},

    where Ω=[,+), >0. Let

    χρ,η,(z):=z1κ(z,)η(1κ(z,))ρ. (2.4)

    We denote by L2χρ,η,(I) the weighted L2 space with the following inner product and norm:

    (X,ϕ)χρ,η,=IX(z)ϕ(z)χρ,η,(z)dz,Xχρ,η,=(X,X)1/2χρ,η,. (2.5)

    One of the most important properties of the log J polynomials is that they are mutually orthogonal in L2χρ,η,(I), i.e.,

    (Pρ,η,m(z),Pρ,η,j(z))χρ,η,=0,jm,Pρ,η,j(z)χρ,η,=ˆθρ,ηj=Γ(j+ρ+1)Γ(j+η+1)(2j+ρ+η+1)j!Γ(j+ρ+η+1). (2.6)

    We define the following first-order differential operator:

    D1logϕ(z)=ddκ(z,)ϕ(z)=zϕ(z), (2.7)

    and an induction leads to

    Dklogϕ(z)=kD1logD1logD1logϕ(z). (2.8)

    We also define the non-uniformly weighted log J Sobolev space as

    Bi,ρ,η(I):={ϕ:DjlogϕL2χρ+j,η+j,(I),0ji},iN,

    with

    (ψ,ϕ)Bi,ρ,η=ik=0(Dklogψ,Dklogϕ)χρ+k,η+k,,ϕBi,ρ,η=(ϕ,ϕ)1/2Bi,ρ,η,|ϕ|Bi,ρ,η=Dilogϕχρ+i,η+i,.

    For the usual shifted-weighted Jacobi Sobolev space, we define

    Biρ,η(Λ):={ϕ:jzϕL2χρ+j,η+j(Λ),0ji},iN,

    where χρ,η=(z+1)ρzη with zΛ=[0,1] is the classical Jacobi weight function.

    Assume that x0<x1<<xM1<xM in I are the roots of Pρ,η,M+1(x). Let z(x)=logx. Then zj:=z(xj)=logxj,0jM, are zeros of Pρ,ηM+1(x), and {χi}Mi=0 are the corresponding weights.

    The log J-Gauss quadrature enjoys the exactness

    IX(z)χρ,η,(z)dz=Mi=0X(zi)χi,X(z)Plog2M+1. (2.9)

    Hence,

    Mk=0Pρ,η,q(zk)Pρ,η,j(zk)χk=ˆθρ,ηqδq,j, 0q+j2M+1. (2.10)

    For any X(ez)C(I), the log J-Gauss interpolation operator Iρ,η,z,M:C(I)PlogM is determined uniquely by

    Iρ,η,z,MX(zq)=X(zq),0qM. (2.11)

    From the above condition, we have Iρ,η,z,MX=X for all XPlogM. On the other hand, since Iρ,η,z,MXPlogM, we can write

    Iρ,η,z,MX(x)=Mi=0ˆXρ,η,iPρ,η,i(x),ˆXρ,η,i=1ˆθρ,ηiMj=0X(xj)Pρ,η,i(xj)χj, XPlogM(I). (2.12)

    The L(I) space is the set of all measurable functions that are essentially bounded. That is, functions g that are bounded almost everywhere on a set of finite measures. The essential supermom norm is used to define the norm of this space and is given as

    g=ess supxI|g(x)|.

    Definition 2.4. Let A(z)=(aij(z))m×n be an (m×n) matrix function with zI. We consider the non-negative real-valued function

    |A(z)|=mi=1nj=1|aij(z)|, (2.13)

    and the norms

    Aχρ,η,:=(I|A(z)|2χρ,η,dz)1/2,A:=esssupzI|A(z)|. (2.14)

    Proposition 2.1. It holds for any ψ(ex)Bmρ,η(Λ), m1 and M+1mq0

    Dqlog(ψIρ,η,Mψ)χρ+q,η+q,c(1+Mm)!M!Mq(1+m)/2mx{ψ(ex)}χρ+m,η+m, (2.15)

    and it takes the form

    Dqlog(ψIρ,η,Mψ)χρ+q,η+q,cMqmmx{ψ(ex)}χρ+m,η+m, c1,for fixedmandM1. (2.16)

    In the case of q=0,1, we can write

    ψIρ,η,Mψχρ,η,cMmmx{ψ(ex)}χρ+m,η+m, (2.17)
    x(ψIρ,η,Mψ)˜χρ,η,cM1mmx{ψ(ex)}χρ+m,η+m, (2.18)

    where ˜χρ,η,=x(1log(x))ρ+1(log(x))η+1.

    Lemma 2.1. [23] For any ρ,η(1,12) and for all ψ(x)B1,ρ,η(I), ψ(ξ)=0 for some ξI, it holds

    ψ2xψ1/2˜χρ,η,ψ1/2χρ,η,. (2.19)

    Proposition 2.2. [23] For ρ,η(1,12],

    ψIρ,η,MψcM1/2mmxψ(ex)χρ+m,η+m,ψ(ex)Bmρ,η(Λ), m1. (2.20)

    Lemma 2.2. [23]

    Iρ,η,M:=maxxIMj=0|hρ,η,j(x)|={O(logM),1<ρ,η12,O(Mμ+12),μ=max(ρ,η), otherwise, (2.21)

    where {hρ,η,j(x)}Mj=0 are the logarithmic Lagrange interpolation functions that are related to Pρ,η,M+1(x).

    To begin with, we rewrite the differential equation (1.1) in the following equivalent compact integral form:

    Z(t)=Z+1Γ(ρ)t(κ(t,s))ρ1Q(s,Z(s),Z(qs))dss, t(,e], (3.1)

    where

    Z(t)=[X1(t),X2(t),,XM(t)]T,Z=[ˉX1(),ˉX2(),,ˉXM()]T,Z(qt)={[X1(qt),X2(qt),,XM(qt)]T,ifqt>,[ˉX1(qt),ˉX2(qt),,ˉXM(qt)]T,ifqt,Q(t)=[g1,g2,,gM]T.

    In the following, we will make some useful transformations, which in turn are the basis for the numerical solution scheme and its numerical analysis. In order to convert the integral interval (,t) to I, we consider

    κ(s,)=κ(t,)κ(r,),

    or

    s=s(t,r)=(r)κ(t,).

    Hence, the system(3.1) becomes

    Z(t)=Z+(κ(t,))ρΓ(ρ)I(1κ(r,))ρ1G(s(t,r),Z(s(t,r)),Z(qs(t,r)))drr. (3.2)

    The non-polynomial spectral collocation scheme for (3.2) is to find Xm,N(t)PlogN(I), m=1,2,,M such that

    ZN(t)=Z+1Γ(ρ)I0,0,t,N(κ(t,))ρIr1(1κ(r,))ρ1Iρ1,0,r,NQ(s(t,r),ZN(s(t,r)),ZN(qs(t,r)))dr, (3.3)

    where

    ZN(t)=[X1,N,X2,N,,XM,N]T,

    and Iρ,η,z,N the log J-Gauss interpolation operator in the z-direction. For simplicity, we will consider the trial functions as

    Xm,N(t)=Ni=0Xm,iP0,0,i(t),m=1,,M. (3.4)

    Also, we can use the following approximation:

    I0,0,t,NIρ1,0,r,N(κ(t,))ρgm(s(t,r),ZN(s(t,r)),ZN(qs(t,r)))=Ni=0Nj=0vm,i,jP0,0,i(t)Pρ1,0,j(r),m=1,,M. (3.5)

    A straightforward calculation by using (3.5) and (2.6) gives

    1Γ(ρ)I0,0,t,N[(κ(t,))ρIr1(1κ(r,))ρ1Iρ1,0,r,Ngm(s(t,r),ZN(s(t,r)),ZN(qs(t,r)))dr]=1Γ(ρ)Ni=0Nj=0vm,i,jP0,0,i(t)Ir1(1κ(r,))ρ1Pρ1,0,j(r)dr=1Γ(ρ+1)Ni=0vm,i,0P0,0,i(t),m=1,,M. (3.6)

    Let {χρ,η,p,xρ,η,p}Np=0 be the weights and the nodes of Gauss-type logarithmic Jacobi interpolation. A direct application of (3.5) and (2.12) yields

    vm,i,0=ρ(2i+1)Np=0Nq=0(κ(t0,0,p,))ρP0,0,i(t0,0,p)×gm(s(t0,0,p,rρ1,0,q),ZN(s(t0,0,p,rρ1,0,q)),ZN(qs(t0,0,p,rρ1,0,q)))χ0,0,pχρ1,0,q. (3.7)

    Hence, we deduce that

    Ni=0Xm,iP0,0,i(t)=XP0,0,0(t)+1Γ(ρ+1)Ni=0vm,i,0P0,0,i(t). (3.8)

    We compared the coefficients of (3.8) to get

    Xm,0=Z+vm,0,0Γ(ρ+1),Xm,i=vm,i,0Γ(ρ+1),1iN,m=1,,M, (3.9)

    where Z is the vector of initial values defined in (3.1).

    Here, we derive the rate of convergence of the scheme (3.3) in the L2χ0,0,-norm. Accordingly, we introduce some lemmas.

    Let rρ,η,i be the log J-Gauss nodes in I, and sρ,η,i=s(x,rρ,η,i). The mapped log J-Gauss interpolation operator x˜Iρ,η,s,N:C(,x)PlogN(,x) is defined by

    x˜Iρ,η,s,Nu(sρ,η,i)=u(sρ,η,i),0iN. (4.1)

    Hence,

    x˜Iρ,η,s,Nu(sρ,η,i)=u(sρ,η,i)=u(s(x,rρ,η,i))=Iρ,η,r,Nu(s(x,rρ,η,i)), (4.2)

    and

    x˜Iρ,η,s,Nu(s)=Iρ,η,r,Nu(s(x,r))|κ(r,)=κ(s,)κ(x,). (4.3)

    Moreover, the following results can be easily derived:

    xs1(κ(x,s))ρ1x˜Iρ1,0,s,NX(s)ds=(κ(x,))ρIr1(1κ(r,))ρ1Iρ1,0,r,NX(s(x,r))dr=(κ(x,))ρNj=0X(s(x,rρ1,0,j))χρ1,0,j=(κ(x,))ρNj=0X(sρ1,0,j)χρ1,0,j. (4.4)

    Similarly,

    xs1(κ(x,s))ρ1(x˜Iρ1,0,s,NX(s))2ds=(κ(x,))ρNj=0X2(sρ1,0,j)χρ1,0,j. (4.5)

    Then, for any 1sN+1, we have

    xs1(κ(x,s))ρ1|(Ix˜Iρ1,0,s,N)X(s)|2ds=(κ(x,))ρIr1(1κ(r,))ρ1|(IIρ1,0,r,N)X(s(x,r))|2drcN2m(κ(x,))ρIr1(1κ(r,))ρ+m1(κ(r,))m|Dmlog,rX(s(x,r))|2dr=cN2mxs1(κ(x,s))ρ+m1(κ(s,))m|Dmlog,sX(s)|2ds, (4.6)

    where I is the identity operator.

    Lemma 4.1. The following estimate holds for the error function eN(x)=Z(x)ZN(x):

    eNχ0,0,3j=1Ξjχ0,0,, (4.7)

    where

    Ξ1=Z(x)I0,0,x,NZ(x),Ξ2=I0,0,x,NxR(x,s)(Ix˜Iρ1,0,s,N)Q(s,Z(s),Z(qs))ds,Ξ3=I0,0,x,NxR(x,s)x˜Iρ1,0,s,N(Q(s,Z(s),Z(qs))Q(s,ZN(s),ZN(qs)))ds,

    and R(x,s)=(Rij) with Rij=s1(κ(x,s))ρ1Γ(ρ)δij, i, j=1,,M.

    Proof.

    eNχ0,0,ZI0,0,x,NZχ0,0,+I0,0,x,NZZNχ0,0,. (4.8)

    It is clear from (3.1) that

    I0,0,x,NZ(x)=Z+1Γ(ρ)I0,0,x,Nxs1(κ(x,s))ρ1Q(s,Z(s),Z(qs))ds, (4.9)

    and

    ZN(x)=Z+1Γ(ρ)I0,0,x,Nxs1(κ(x,s))ρ1x˜Iρ1,0,s,NQ(s,ZN(s),ZN(qs))ds. (4.10)

    Subtracting (4.9) from (4.10) yields

    I0,0,x,NZ(x)ZN(x)=1Γ(ρ)I0,0,x,Nxs1(κ(x,s))ρ1(Q(s,Z(s),Z(qs))x˜Iρ1,0,s,NQ(s,ZN(s),ZN(qs)))ds, (4.11)

    which has the form:

    I0,0,x,NZ(x)ZN(x)=1Γ(ρ)I0,0,x,Nxs1(κ(x,s))ρ1(Ix˜Iρ1,0,s,N)Q(s,Z(s),Z(qs))ds+1Γ(ρ)I0,0,x,Nxs1(κ(x,s))ρ1x˜Iρ1,0,s,N(Q(s,Z(s),Z(qs))Q(s,ZN(s),ZN(qs)))ds. (4.12)

    Theorem 5.1. Let Z(x) be the solutions of the systems (3.1) and (3.3), respectively. Then we have the following estimate:

    ZZNχ0,0,cNm(DmlogZ2χm,m,+DmlogQ(x,Z(x),Z(qx))2χρ+m1,m,), (5.1)

    where 1mN+1 and N1.

    Proof. Using Proposition 2.1, we get

    Ξ1χ0,0,=ZI0,0,x,NZχ0,0,cNmDmlogZ2χm,m,cNmmxZ(ex)χm,m. (5.2)

    Using the log J-Gauss integration formula, gives

    Ξ2χ0,0,=I0,0,x,NxR(x,s)(Ix˜Iρ1,0,s,N)Q(s,Z(s),Z(qs))dsχ0,0,=Mk=1I0,0,x,NxRkk(x,s)(Ix˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))dsχ0,0,=[Iχ0,0,(Mk=1I0,0,x,NxRkk(x,s)(Ix˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))ds)dx]1/2=[Nj=0χ0,0,j(Mk=1x0,0,jRkk(x0,0,j,s)(Ix0,0,j˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))ds)2]1/2[Nj=0χ0,0,jMk=1(x0,0,jRkk(x0,0,j,s)(Ix0,0,j˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))ds)2Mk=1(1)2]1/2.

    Using Cauchy-Schwarz inequality leads to the following estimate:

    Ξ2χ0,0,C[Nj=0Mk=1χ0,0,jx0,0,jRkk(x0,0,j,s) dsx0,0,jRkk(x0,0,j,s)|(Ix0,0,j˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))|2ds]1/2C[Nj=0Mk=1χ0,0,j(κ(x0,0,j,))ρx0,0,js1(κ(x0,0,j,s))ρ1|(Ix0,0,j˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))|2ds]1/2C(Nj=0χ0,0,j(κ(x0,0,j,))ρ)1/2(Mk=1x0,0,js1(κ(x0,0,j,s))ρ1|(Ix0,0,j˜Iρ1,0,s,N)gk(s,Z(s),Z(qs))|2ds)1/2C(Nj=0χ0,0,j(κ(x0,0,j,))ρ)1/2(x0,0,js1(κ(x0,0,j,s))ρ1|(Ix0,0,j˜Iρ1,0,s,N)Q(s,Z(s),Z(qs))|2ds)1/2cNm[Nj=0χ0,0,j(κ(x0,0,j,))ρx0,0,js1(κ(x0,0,j,s))ρ+m1(κ(s,))m|Dmlog,sQ(s,Z(s),Z(qs))|2ds]1/2cNmDmlogQ(,Z(),Z(q))2χρ+m1,m,. (5.3)

    An estimate for the term E3χ0,0, can be obtained by using the log J-Gauss integration formula, to give

    Ξ32χ0,0,=R(x,s)I0,0,x,Nxx˜Iρ1,0,s,N(Q(s,Z(s),Z(qs))Q(s,ZN(s),ZN(qs)))ds2χ0,0,=1Γ2(ρ)Iχ0,0,×(Mk=1I0,0,x,Nxs1(κ(x,s))ρ1x˜Iρ1,0,s,N(gk(s,Z(s),Z(qs))gk(s,ZN(s),ZN(qs)))ds)2dx=1Γ2(ρ)Nj=0χ0,0,j×(x0,0,js1(κ(x,s))ρ1Mk=1x0,0,j˜Iρ1,0,s,N(gk(s,Z(s),Z(qs))gk(s,ZN(s),ZN(qs)))ds)2.

    Using the Cauchy-Schwarz inequality, we get

    Ξ32χ0,0,1Γ2(ρ)Nj=0χ0,0,jx0,0,js1(κ(x,s))ρ1ds×x0,0,js1(κ(x,s))ρ1(|Mk=1x0,0,j˜Iρ1,0,s,N(gk(s,Z(s),Z(qs))gk(s,ZN(s),ZN(qs)))|)2ds1Γ2(ρ)Nj=0χ0,0,j(logx)ρx0,0,js1(κ(x,s))ρ1×(Mk=1|x0,0,j˜Iρ1,0,s,N(gk(s,Z(s),Z(qs))gk(s,ZN(s),ZN(qs)))|)2ds, (5.4)

    and using the logarithmic Jacobi-Gauss quadrature formula (4.4), we obtain

    Ξ3χ0,0,1Γ(ρ+1)[Nj=0ρχ0,0,j(κ(x0,0,j,))2ρ×Nq=0χρ1,0,q(Mk=1|gk(s(x0,0,j,rρ1,0,q),Z(s(x0,0,j,rρ1,0,q)),Z(qs(x0,0,j,rρ1,0,q)))gk(s(x0,0,j,rρ1,0,q),ZN(s(x0,0,j,rρ1,0,q)),ZN(qs(x0,0,j,rρ1,0,q)))|)2]1/2. (5.5)

    Using the Lipschitz condition, we obtain

    Ξ3χ0,0,LΓ(ρ+1)×[Nj=0ρχ0,0,j(κ(x0,0,j,))2ρNq=0(Mi=1χρ1,0,q|Xi(s(x0,0,j,rρ1,0,q))XN,i(s(x0,0,j,rρ1,0,q))|)2]1/2, (5.6)

    using (4.5), we get

    Ξ3χ0,0,LΓ(ρ+1)×[Nj=0ρχ0,0,j(κ(x0,0,j,))ρx0,0,js1(κ(x0,0,j,s))ρ1(Mi=1|x0,0,j˜Iρ1,0,s,N(Xi(s)XN,i(s))|)2ds]1/2.E3χ0,0,LΓ(ρ+1)(Nj=0ρχ0,0,j(κ(x0,0,j,))ρ)1/2×max0jN(x0,0,js1(κ(x0,0,j,s))ρ1(Mi=1|x0,0,j˜Iρ1,0,s,N(Xi(s)Xi,N(s))|)2ds)1/2. (5.7)

    For any x0,0,jI. Let f(ρ)=(κ(x0,0,j,))ρ. We note that f(ρ) is a convex function of ρ. Hence, by Jensen's inequality for all ρ(0,1),

    f(ρ)=(1ρ)f(0)+ρf(1).

    The above inequality yields

    ρNj=0χ0,0,j(κ(x0,0,j,))ρρNj=0χ0,0,j[1ρ+ρ(κ(x0,0,j,))]ρ[1ρ+ρIs1(logxa)dx]ρ(1ρ2)12. (5.8)

    Hence, by using the above inequality, the triangle inequality, (4.6) and (4.5), we deduce that

    Ξ3χ0,0,L2Γ(ρ+1)max0jN(x0,0,js1(κ(x0,0,j,s))ρ1(Mi=1|x0,0,j˜Iρ1,0,s,N(Xi(s)XN,i(s))|)2ds)1/2L2 Γ(ρ+1)×max0jN[(x0,0,js1(κ(x0,0,j,s))ρ1(Mi=1|x0,0,j˜Iρ1,0,s,NXi(s)Xi(s)|)2ds)1/2+(x0,0,js1(κ(x0,0,j,s))ρ1(Mi=1|Xi(s)XN,i(s)|)2ds)1/2]cNmmax0jN(xj(κ(s,))m(Mi=1|Dmlog,sXi(s)|)2ds)1/2+L2 Γ(ρ+1)×max0jN(x0,0,js1(κ(x0,0,j,s))ρ1(Mi=1|Xi(s)XN,i(s)|)2ds)1/2cNmDmlogZ2χm,m,+L2Γ(ρ+1)eN2χm,m,. (5.9)

    Hence, a combination of (5.2), (5.3), (5.9) and the Lipschitz constant L<Γ(ρ+1) leads to the desired result.

    In order to illustrate the significance of our key findings, we provide two numerical examples in this section.

    Example 6.1. We consider the following initial value problem:

    CH1DρtX(t)=g(x),X(1)=0, t(1,e), ρ(0,1]. (6.1)

    Table 1 shows a comparison of the maximum absolute errors that are obtained from the method that we have presented and those given in [22] and [21]. The numerical results depict that, by using the method proposed in this paper, higher accuracy is achieved.

    Table 1.  A comparison between the maximum absolute errors of presented method and methods in [22] and [21] with ρ=0.5 for Example 6.1.
    N Error Error [22] N[21] Error [21]
    9 5.3874×109 9.1283×106 20 1.2500×103
    10 2.7917×109 2.2831×106 40 2.8647×104
    11 1.5313×109 7.1562×107 80 6.6144×105
    12 8.8111×1010 2.6679×107 160 1.5345×105

     | Show Table
    DownLoad: CSV

    To investigate numerically the stability of the spectral collocation scheme, we consider the initial value problem (6.1) and the following problems, whose right-hand side, the initial value, and the order of the differential operator suffer perturbations.

    CH1DρtY(t)=g(x)+εg,Y(1)=0,t(1,e),ρ=0.5. (6.2)
    CH1Dρ+ερtY(t)=g(x),Y(1)=0,t(1,e),ρ=0.5,ερ(0.5,0.5). (6.3)
    CH1DρtY(t)=g(x),Y(1)=εY0,t(1,e),ρ=0.5. (6.4)

    The maximum absolute errors |XNYN|, where XN is the numerical solution of problem (6.1) and YN is the numerical solution of the perturbed problems (6.2), (6.3), and (6.4), are displayed in Tables 2, 3, and 4, respectively. We observe that XNYN=O(εg), XNYN=O(ερ), and XNYN=O(εy0), respectively, independently of N.

    Table 2.  Maximum of the absolute errors, |XNYN|, where XN is the numerical solution of problem (6.1) and YN is the numerical solution of the perturbed problem (6.2) with several values of εg.
    N εg=0.1 εg=0.01 εg=0.001
    5 1.130×101 1.131×102 1.131×103
    10 1.127×101 1.127×102 1.127×103
    15 1.128×101 1.128×102 1.128×103

     | Show Table
    DownLoad: CSV
    Table 3.  Maximum of the absolute errors, |XNYN|, where XN is the numerical solution of problem (6.1) and YN is the numerical solution of the perturbed problem (6.3) with several values of ερ.
    N ερ=0.1 ερ=0.01 ερ=0.001
    5 1.192×101 1.249×102 1.255×103
    10 1.192×101 1.249×102 1.255×103
    15 1.192×101 1.249×102 1.255×103

     | Show Table
    DownLoad: CSV
    Table 4.  Maximum of the absolute errors, |XNYN|, where XN is the numerical solution of problem (6.1) and YN is the numerical solution of the perturbed problem (6.4) with several values of εY0.
    N εY0=0.1 εY0=0.01 εY0=0.001
    5 1.000×101 1.000×102 1.000×103
    10 1.000×101 1.000×102 1.000×103
    15 1.000×101 1.000×102 1.000×103

     | Show Table
    DownLoad: CSV

    Example 6.2. We consider the following coupled system:

    CH1DρtX1(t)=X22(qt)+g1(t),ρ(0,1),CH1DρtX2(t)=X21(qt)+g2(t),ρ(0,1). (6.5)

    For this problem, the exact solution is given as

    X1(t)=(logt)5+2(logt)3,
    X2(t)=(logt)4+2(logt)3.

    We employ the proposed method to solve this problem with various N and ρ values. In Table 5, we list the errors for different values of N and ρ. The numerical results show the convergence of the scheme, which confirms our error analysis.

    Table 5.  The errors with the fractional orders ρ=0.2,0.4,0.6,0.8, and q=3/4 for Example 6.2.
    The errors for X1
    N ρ=0.2 ρ=0.4 ρ=0.6 ρ=0.8
    5 6.8×108 4.6×107 1.3×106 2.1×106
    10 6.7×1010 6.0×109 2.4×108 5.0×108
    15 3.9×1011 4.2×1010 1.9×109 4.9×109
    20 5.0×1012 6.0×1011 3.1×1010 8.8×1010
    25 9.9×1013 1.3×1011 7.5×1011 2.3×1010
    30 2.6×1013 7.2×1013 2.3×1011 7.6×1011
    The errors for X2
    N ρ=0.2 ρ=0.4 ρ=0.6 ρ=0.8
    5 5.7×108 4.3×107 1.3×106 2.1×106
    10 5.3×1010 5.3×109 2.2×108 4.9×108
    15 3.0×1011 3.6×1010 1.8×109 4.7×109
    20 3.8×1012 5.2×1011 2.9×1010 8.4×1010
    25 7.6×1013 1.1×1011 7.0×1011 2.2×1010
    30 2.0×1013 3.21×1012 2.1×1011 7.2×1011

     | Show Table
    DownLoad: CSV

    We provided a collocation spectral scheme for nonlinear systems of fractional pantograph delay differential equations. We constructed a mapped Jacobi spectral collocation scheme, described its effective implementation, and derived its convergence analysis. In addition, we provided a numerical example to support our theoretical analysis. The numerical results demonstrate the accuracy and effectiveness of the proposed scheme. We also conclude that the described technique produces very accurate results, even when employing a small number of base functions. Preserving some important mathematical properties and physical structures, such as existence, positivity preservation, the maximum principle, long-time behavior, and singular solutions, may be considered in future work [25,26].

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

    This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23095).

    The authors assert that they do not have any known competing financial interests or personal relationships that could have influenced the work reported in this paper.



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