Research article

Forbidden subgraphs in reduced power graphs of finite groups

  • Received: 28 January 2021 Accepted: 11 March 2021 Published: 15 March 2021
  • MSC : 05C25, 05C17

  • Let G be a finite group. The reduced power graph of G is the undirected graph whose vertex set consists of all elements of G, and two distinct vertices x and y are adjacent if either xy or yx. In this paper, we show that the reduced power graph of G is perfect and characterize all finite groups whose reduced power graphs are split graphs, cographs, chordal graphs, and threshold graphs. We also give complete classifications in the case of abelian groups, dihedral groups, and generalized quaternion groups.

    Citation: Huani Li, Ruiqin Fu, Xuanlong Ma. Forbidden subgraphs in reduced power graphs of finite groups[J]. AIMS Mathematics, 2021, 6(5): 5410-5420. doi: 10.3934/math.2021319

    Related Papers:

    [1] Shuting Chang, Yaojun Ye . Upper and lower bounds for the blow-up time of a fourth-order parabolic equation with exponential nonlinearity. Electronic Research Archive, 2024, 32(11): 6225-6234. doi: 10.3934/era.2024289
    [2] Yaning Li, Yuting Yang . The critical exponents for a semilinear fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. Electronic Research Archive, 2023, 31(5): 2555-2567. doi: 10.3934/era.2023129
    [3] Xu Liu, Jun Zhou . Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28(2): 599-625. doi: 10.3934/era.2020032
    [4] Hui Yang, Futao Ma, Wenjie Gao, Yuzhu Han . Blow-up properties of solutions to a class of p-Kirchhoff evolution equations. Electronic Research Archive, 2022, 30(7): 2663-2680. doi: 10.3934/era.2022136
    [5] Yitian Wang, Xiaoping Liu, Yuxuan Chen . Semilinear pseudo-parabolic equations on manifolds with conical singularities. Electronic Research Archive, 2021, 29(6): 3687-3720. doi: 10.3934/era.2021057
    [6] Jun Zhou . Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28(1): 67-90. doi: 10.3934/era.2020005
    [7] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li . Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28(1): 369-381. doi: 10.3934/era.2020021
    [8] Cheng Wang . Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29(5): 2915-2944. doi: 10.3934/era.2021019
    [9] Yang Cao, Qiuting Zhao . Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations. Electronic Research Archive, 2021, 29(6): 3833-3851. doi: 10.3934/era.2021064
    [10] Abdelhadi Safsaf, Suleman Alfalqi, Ahmed Bchatnia, Abderrahmane Beniani . Blow-up dynamics in nonlinear coupled wave equations with fractional damping and external source. Electronic Research Archive, 2024, 32(10): 5738-5751. doi: 10.3934/era.2024265
  • Let G be a finite group. The reduced power graph of G is the undirected graph whose vertex set consists of all elements of G, and two distinct vertices x and y are adjacent if either xy or yx. In this paper, we show that the reduced power graph of G is perfect and characterize all finite groups whose reduced power graphs are split graphs, cographs, chordal graphs, and threshold graphs. We also give complete classifications in the case of abelian groups, dihedral groups, and generalized quaternion groups.



    The convection-diffusion equation(CDE) governs the transmission of particles and energy caused by convection and diffusion. The CDE is given by

    vt+ωvz=ϑ2vz2,czd,t>0, (1.1)

    where ω denotes coefficient of viscosity and ϑ is the phase velocity respectively and both are positive. The Eq (1.1) is subject to the IC,

    v(z,0)=ϕ(z),czd (1.2)

    and the BCs,

    {v(c,t)=f0(t)v(d,t)=f1(t)t>0, (1.3)

    where, ϕ, f0 and f1 are given smooth functions.

    Numerous computational procedures have been developed in the literature for the CDE. Chawla et al. [1] developed extended one step time-integration schemes for the CDE. Daig et al. [2] discussed least-squares finite element method for the advection-diffusion equation(ADE). Mittal and Jain [3] revisited the cubic B-splines collocation procedure for the numerical treatment of the CDE. The characteristics method with cubic interpolation for ADE was presented by Tsai et al. [4]. Sari et al. [5] worked on high order finite difference schemes for solving the ADE. Taylor-Galerkin B-spline finite element method for the one-dimensional ADE was developed by Kadalbajoo and Arora [6]. Kara and Zhang [7] presented ADI method for unsteady CDE. Feng and Tian [8] found numerical solution of CDE using the alternating group explicit methods with exponential-type. Dehghan [9] used weighted finite difference techniques for the CDE. Further, Dehghan [10] developed a technique for the numerical solution of the three-dimensional advection-diffusion equation.

    A second-order space and time nodal method for CDE was conducted by Rizwan [11]. Mohebbi and Dehghan [12] presented a high order compact solution of the one-dimensional heat equation and ADE. Karahan [13,14] worked on unconditional stable explicit and implicit finite difference technique for ADE using spreadsheets. Salkuyeh [15] used finite difference approximation to solve CDE. Cao et. al [16] developed a fourth-order compact finite difference scheme for solving the CDEs. The generalized trapezoidal formula is used by Chawla and Al-Zanaidi [17] to solve CDE. Restrictive Taylor approximation has been used by Ismail et al. [18] to solve CDE. A boundary element method for anisotropic-diffusion convection-reaction equation in quadratically graded media of incompressible flow was studied by Salam et al. [19]. Azis [20] obtained standard-BEM solutions o two types of anisotropic-diffusion convection reaction equations with variable coefficients. The principle inspiration of this investigation is that the introduced scheme offers the solution as a piecewise adequately smooth continuous function enabling us to discover a numerical solution at any point in the solution domain. Moreover, it is simple to implement and has incredibly diminished computational expense. The refinement of the scheme with new approximations has elevated the accuracy of the scheme. The methodology used by von Neumann is utilized to affirm that the introduced scheme is unconditionally stable. The scheme is implemented to various test problems and the results are contrasted with the ones revealed in [25,26,27]. For further related studies, the interested reader is referred to [28,29] and references therein.

    The remaining part of the paper is organized in the following sequence. The numerical scheme is presented in section 2 which is based on the cubic B-spline collocation method. Section 3 deals with the Scheme's stability and convergence analysis. The comparison of our numerical results with the ones presented in [25,26,27] is also presented in this section. The study's finding is summed up in section 4.

    Derivation of the scheme

    Define Δt=TN to be the time and h=dcM the space step sizes for positive integers M and N. Let tn=nΔt,n=0,1,2,...,N, and zj=jh,j=0,1,2...M. The solution domain czd is evenly divided by knots zj into M subintervals [zj,zj+1] of uniform length, where c=z0<z1<...<zn1<zM=d. The scheme for solving (1.1) assumes approximate solution V(z,t) to the exact solution v(z,t) to be [23]

    V(z,t)=M+1j=1Dj(t)B3j(z), (2.1)

    where Dj(t) are unknowns to be calculated and B3j(z) [23] are cubic B-spline basis functions given by

    B3j(z)=16h3{(zzj)3,z[zj,zj+1],h3+3h2(zzj+1)+3h(zzj+1)23(zzj+1)3,z[zj+1,zj+2],h3+3h2(zj+3z)+3h(zj+3z)23(zj+3z)3,z[zj+2,zj+3],(zj+4z)3,z[zj+3,zj+4],0,otherwise, (2.2)

    Here, just B3j1(z),B3j(z) and B3j+1(z) are last on account of local support of the cubic B-splines so that the approximation vnj at the grid point (zj,tn) at nth time level is given as

    V(zj,tn)=Vnj=k=j+1k=j1Dnj(t)B3j(z). (2.3)

    The time dependent unknowns Dnj(t) are determined using the given initial and boundary conditions and the collocation conditions on B3j(z). Consequently, the approximations vnj and its required derivatives are found to be

    {vnj=α1Dnj1+α2Dnj+α1Dnj+1,(vnj)z=α3Dnj1+α4Dnj+α3Dnj+1, (2.4)

    where α1=16,α2=46,α3=12h,α4=0.

    The new approximation [24] for (vnj)zz is given as

    {(vn0)zz=112h2(14Dn133Dn0+28Dn114Dn2+6Dn3Dn4),(vnj)zz=112h2(Dnj2+8Dnj118Dnj+8Dnj+1+Dnj+2),j=1,2,...,M1(vnM)zz=112h2(DnM4+6DnM314DnM2+28DnM133DnM+14DnM+1), (2.5)

    The problem (2.1) subject to the weighted θ-scheme takes the form

    (vnj)t=θhn+1j+(1θ)hnj, (2.6)

    where hnj=ϑ(vnj)zzω(vnj)z and n=0,1,2,3,.... Now utilizing the formula, (vnj)t=vn+1jvnjk in (2.6) and streamlining the terms, we obtain

    vn+1j+kωθ(vn+1j)zkϑθ(vn+1j)zz=vnjkω(1θ)(vnj)z+kϑ(1θ)(vnj)zz, (2.7)

    Observe that θ=0, θ=12 and θ=1 in the system (2.7) correspond to an explicit, Crank-Nicolson and a fully implicit schemes respectively. We use the Crank-Nicolson approach so that (2.7) is evolved as

    vn+1j+12kω(vn+1j)z12kϑ(vn+1j)zz=vnj12kω(vnj)z+12kϑ(vnj)zz. (2.8)

    Subtituting (2.4) and (2.5) in (2.8) at the knot z0 returns

    (α1kωα327kϑ12h2)Dn+11+(α2+kωα42+11kϑ8h2)Dn+10+(α1+kωα327kϑ6h2)Dn+11+(7kϑ12h2)Dn+12(kϑ4h2)Dn+13+(kϑ24h2)Dn+14=(α1+kωα32+7kϑ12h2)Dn1+(α2kωα4211kϑ8h2)Dn0+(α1kωα32+7kϑ6h2)Dn1(7kϑ12h2)Dn2+(kϑ4h2)Dn3(kϑ24h2)Dn4. (2.9)

    Substituting (2.4) and (2.5) in (2.8) produces

    (kϑ24h2)Dn+1j2+(α1kωα32kϑ3h2)Dn+1j1+(α2+kωα42+3kϑ4h2)Dn+1j+(α1+kωα32kϑ3h2)Dn+1j+1(kϑ24h2)Dn+1j+2=(kϑ24h2)Dnj2+(α1+kωα32+kϑ3h2)Dnj1+(α2kωα423kϑ4h2)Dnj+(α1kωα32+kϑ3h2)Dnj+1+(kϑ24h2)Dnj+2,j=1,2,3,...,M1. (2.10)

    Subtituting (2.4) and (2.5) in (2.8) at the knot zM yields

    (kϑ24h2)Dn+1M4(kϑ4h2)Dn+1M3+(7kϑ12h2)Dn+1M2+(α1kωα327kϑ6h2)Dn+1M1+(α2+kωα42+11kϑ8h2)Dn+1M+(α1+kωα327kϑ12h2)Dn+1M+1=(kϑ24h2)DnM4+(kϑ4h2)DnM3(7kϑ12h2)DnM2+(α1+kωα32+7kϑ6h2)DnM1+(α2kωα4211kϑ8h2)DnM+(α1kωα32+7kϑ12h2)DnM+1. (2.11)

    From (2.9), (2.10) and (2.11), we acquire a system of (M+1) equations in (M+3) unknowns. To get a consistent system, two additional equations are obtained using the given boundary conditions. Consequently a system of dimension (M+3)×(M+3) is obtained which can be numerically solved using any numerical scheme based on Gaussian elimination.

    Initial State: To begin iterative process, the initial vector D0 is required which can be obtained using the initial condition and the derivatives of initial condition as follows:

    {(v0j)z=ϕ(zj),j=0,(v0j)=ϕ(zj),j=0,1,2,...,M,(v0j)z=ϕ(zj),j=M. (2.12)

    The system (2.12) produces an (M+3)×(M+3) matrix system of the form

    HC0=b, (2.13)

    where,

    H=[α3α4α30...00α1α2α10...000α1α2α10...00...0α1α2α10...0α3α4α3],

    D0=[D01,D00,D01,...,D0M+1]T and b=[ϕ(z0),ϕ(z0),...,ϕ(zM),ϕ(zM)]T.

    The von Neumann stability technique is applied in this section to explore the stability of the given scheme. Consider the Fourier mode, Dnj=σneiβhj, where β is the mode number, h is the step size and i=1. Plug in the Fourier mode into equation (2.8) to obtain

    γ1σn+1eiβ(j4)h+γ2σn+1eiβ(j3)h+γ3σn+1eiβ(j2)h+γ4σn+1eiβ(j1)hγ1σn+1eiβ(j)h=γ1σneiβ(j4)h+γ5σneiβ(j3)h+γ6σneiβ(j2)h+γ7σneiβ(j1)h+γ1σneiβ(j)h, (3.1)

    where,

    γ1=kϑ24h2,

    γ2=α1kωα32kϑ3h2,

    γ3=α2+kωα42+3kϑ4h2,

    γ4=α1+kωα32kϑ3h2,

    γ5=α1+kωα32+kϑ3h2,

    γ6=α2kωα423kϑ4h2,

    γ7=α1kωα32+kϑ3h2.

    Dividing equation (3.1) by σneiβjh and rearranging, we obtain

    σ=γ1e2iβh+γ5eiβh+γ6+γ7eiβh+γ1e2iβhγ1e2iβh+γ2eiβh+γ3+γ4eiβhγ1e2iβh. (3.2)

    Using cos(βh)=eiβh+eiβh2 and sin(βh)=eiβheiβh2i in equation (3.2) and simplifying, we obtain

    σ=2γ1cos(2βh)+γ6+2E1cos(βh)i2F1sin(βh)2γ1cos(2βh)+γ3+2E2cos(βh)+i2F2sin(βh), (3.3)

    where, E1=α1+kϑ3h2,F1=kωα32,E2=α1kϑ3h2,F2=kωα32.

    Notice that β[π,π]. Without loss of generality, we can assume that β=0, so that Eq (3.3) reduces to

    σ=2γ1+γ6+2E12γ1+γ3+2E2,=kϑ12h2+α23kϑ4h2+2α1+2kϑ3h2kϑ12h2+α2+3kϑ4h2+2α12kϑ3h2,=2α1+α22α1+α2=1,

    which proves unconditional stability.

    In this section, we present the convergence analysis of the proposed scheme. For this purpose, we need to recall the following Theorem [21,22]:

    Theorem 1. Let v(z)C4[c,d] and c=z0<z1<...<zn1<zM=d be the partition of [c,d] and V(z) be the unique B-spline function that interpolates v. Then there exist constants λi independent of h, such that

    (vV)λih4i,i=0,1,2,3.

    First, we assume the computed B-spline approximation to (2.1) as

    V(z,t)=M1j=3Dj(t)B3j(z).

    To estimate the error, v(z,t)V(z,t) we must estimate the errors v(z,t)V(z,t) and V(z,t)V(z,t) separately. For this purpose, we rewrite the equation (2.8) as:

    v+12kω(v)z12kϑ(v)zz=r(z), (3.4)

    where, v=vn+1j,(v)z=(vn+1j)z,(v)zz=(vn+1j)zz and r(z)=vnj12kω(vnj)z+12kϑ(vnj)zz. Equation (3.4) can be written in matrix form as:

    AD=R, (3.5)

    where, R=NDn+h and

    A=[α1α2α10............0q1q2q3q4q5q60...0γ1γ2γ3γ4γ10......00......0γ1γ2γ3γ4γ10...0q6q5q4q10q2q110......0α1α2α1]

    ,

    N=[0000............0q7q8q9q4q5q60...0γ1γ5γ6γ7γ10......00......0γ1γ5γ6γ7γ10...0q6q5q4q12q8q13000......0],

    h=[f0(tn+1),0,...0,f1(tn+1)]T, Dn=[Dn1,Dn0,Dn1,...,DnM+1]T,

    q1=α1kωα327kϑ12h2,q2=α2+kωα42+11kϑ8h2,q3=α1+kωα327kϑ6h2,q4=7kϑ12h2,

    q5=kϑ4h2,q6=kϑ24h2,q7=α1+kωα32+7kϑ12h2,q8=α2kωα4211kϑ8h2,

    q9=α1kωα32+7kϑ6h2,q10=α1kωα327kϑ6h2,q11=α1+kωα327kϑ12h2,

    q12=α1+kωα32+7kϑ6h2,q13=α1kωα32+7kϑ12h2.

    If we replace v by V in (3.4), then the resulting equation in matrix form becomes

    AD=R. (3.6)

    Subtracting (3.6) from (3.5), we obtain

    A(DD)=(RR). (3.7)

    Now using (3.4), we have

    |r(zj)r(zj)|=|(v(zj)v(zj))+kω2(vz(zj)vz(zj))kϑz(vzz(zj)vzz(zj))||(v(zj)v(zj))|+|kω2(vz(zj)vz(zj))|+|kϑ2(vzz(zj)vzz(zj))|. (3.8)

    From (3.8) and Theorem (1), we have

    RRλ0h4+kω2λ1h3+kϑ2λ2h2=(λ0h2+kω2λ1h+kϑ2λ2)h2=M1h2, (3.9)

    where M1=λ0h2+kω2λ1h+kϑ2λ2. It is obvious that the matrix A is diagonally dominant and thus nonsingular, so that

    (DD)=A1(RR). (3.10)

    Now using (3.9), we obtain

    DDA1RRA1(M1h2). (3.11)

    Let aj,i denote the entries of A and ηj,0jM+2 is the summation of jth row of the matrix A, then we have

    η0=M+2i=0a0,i=2α1+α2,
    η1=M+2i=0a1,i=2α1+α2+kωα42,
    ηj=M+2i=0aj,i=2α1+α2+kωα42,2jM
    ηM+1=M+2i=0aM+1,i=2α1+α2+kωα42,
    ηM+2=M+2i=0aM+2,i=2α1+α2.

    From the theory of matrices we have,

    M+2j=0a1k,jηj=1,k=0,1,...,M+2, (3.12)

    where a1k,j are the elements of A1. Therefore

    A1=M+2j=0|a1k,j|1minηk=1ξl1|ξl|,0k,lM+2. (3.13)

    Substituting (3.13) into (3.11) we see that

    DDM1h2|ξl|=M2h2, (3.14)

    where M2=M1|ξl| is some finite constant.

    Theorem 2. The cubic B-splines {B1,B0,....BM+1} defined in relation (2.2) satisfy the inequality

    M1j=3|B3j(z)|53,0z1.

    Proof. Consider,

    |M1j=3B3j(z)|M1j=3|B3j(z)|=|B3j1(z)|+|B3j(z)|+|B3j+1(z)|=16+46+16=1.

    Now for z[zj+1,zj+2], we have

    |B3j2(z)|46

    |B3j1(z)|16,

    |B3j(z)|46,

    |B3j+1(z)|16.

    Then, we have

    M1j=3|B3j(z)|=|B3j2(z)|+|B3j1(z)|+|B3j(z)|+|B3j+1(z)|53

    as required.

    Now, consider

    V(z)V(z)=M1j=3(DjDj)B3j(z). (3.15)

    Using (3.14) and Theorem 2, we obtain

    V(z)V(z)=M1j=3(DjDj)B3j(z)|M1j=3B3j(s)|(DjDj)53M2h2. (3.16)

    Theorem 3. Let v(z) be the exact solution and let V(z) be the cubic collocation approximation to v(z) then the provided scheme has second order convergence in space and

    v(z)V(z)μh2,whereμ=λ0h2+53M2h2.

    Proof. From Theorem 1, we have

    v(z)V(z)λ0h4. (3.17)

    From (3.16) and (3.17), we obtain

    v(z)V(z)v(z)V(z)+V(z)V(z)λ0h4+53M2h2=μh2. (3.18)

    where μ=λ0h2+53M2.

    In this section, some numerical calculations are performed to test the accuracy of the offered scheme. In all examples, we use the following error norms

    L=maxj|Vnum(zj,t)vexact(zj,t)|. (3.19)
    L2=hj|Vnum(zj,t)vexact(zj,t)|2. (3.20)

    The numerical order of convergence p is obtained by using the following formula:

    p=Log(L(n)/L(2n))Log(L(2n)/L(n)), (3.21)

    where L(n) and L(2n) are the errors at number of partition n and 2n respectively.

    Example 1. Consider the CDE,

    vt+0.1vz=0.012vz2,0z1,t>0, (3.22)

    with IC,

    v(z,0)=exp(5z)sin(πz) (3.23)

    and the BCs,

    v(0,t)=0,v(1,t)=0. (3.24)

    The analytic solution of the given problem is v(z,t)=exp(5z(0.25+0.01π2)t)sin(πz). To acquire the numerical results, the offered scheme is applied to Example 1. The absolute errors are compared with those obtained in [25] at various time stages in Table 1. In Table 2, absolute errors and error norms are presented at time stages t=5,10,100. Figure 1 displays the comparison that exists between exact and numerical solutions at various time stages. Figure 2 depicts the 2D and 3D error profiles at t=1. A 3D comparison between the exact and numerical solutions is presented to exhibit the exactness of the scheme in Figure 3.

    Table 1.  Absolute errors are when k=0.001 and h=0.005 for Example 1.
    t z=0.1 z=0.3 z=0.5 z=0.7 z=0.9
    PM CNM[25] PM CNM[25] PM CNM[25] PM CNM[25] PM CNM[25]
    0.2 4.42 ×1010 1.88 ×105 4.14 ×109 3.61 ×105 1.50 ×108 1.39 ×105 3.54 ×108 2.05 ×104 4.44 ×108 9.67 ×104
    0.4 8.55 ×1010 3.23 ×105 7.73 ×109 6.76 ×105 2.80 ×108 2.61 ×105 6.60 ×108 3.84 ×104 8.12 ×108 1.64 ×103
    0.6 1.23 ×109 4.15 ×105 1.08 ×108 9.45 ×105 3.92 ×108 3.66 ×105 9.24 ×108 5.37 ×104 1.12 ×107 2.10 ×103
    0.8 1.57 ×109 4.81 ×105 1.34 ×108 1.17 ×104 4.87 ×108 4.56 ×105 1.15 ×107 6.64 ×104 1.37 ×107 2.42 ×103
    1.0 1.86 ×109 5.26 ×105 1.57 ×108 1.35 ×104 5.68 ×108 5.31 ×105 1.34 ×107 7.68 ×104 1.57 ×107 2.63 ×103
    10 9.67 ×1010 7.17 ×106 7.10 ×109 2.87 ×105 2.46 ×108 2.31 ×105 5.58 ×108 1.11 ×104 5.96 ×108 2.86 ×104
    20 6.10 ×1011 2.57 ×107 4.41 ×1010 1.13 ×106 1.51 ×109 1.41 ×106 3.36 ×109 2.10 ×106 3.55 ×109 7.60 ×106

     | Show Table
    DownLoad: CSV
    Table 2.  Absolute errors when M=40 and k=0.01 for Example 1.
    z t=5 t=10 t=100
    0.1 8.17 ×107 3.13 ×107 8.20 ×1020
    0.3 6.55 ×106 2.38 ×106 5.87 ×1019
    0.5 2.42 ×105 8.46 ×106 1.99 ×1018
    0.7 5.79 ×105 1.96 ×105 4.39 ×1018
    0.9 6.57 ×105 2.15 ×105 1.15 ×1017

     | Show Table
    DownLoad: CSV
    Figure 1.  The approximate (stars, circles, triangles) and exact (solid lines) solutions at various time stages when M=200,k=0.001 for Example 1.
    Figure 2.  2D and 3D error profiles when t=1,M=100,k=0.01 for Example 1.
    Figure 3.  The exact and approximate solutions when t=1,M=100,k=0.01 for Example 1.

    The approximate solution when t=1,k=0.01 and M=20 for Example 1 is given by

    V(z,1)={1.73472×1018+2.21674z+11.0117z2+26.9257z3,z[0,120]0.000773044+2.26312z+10.0841z2+33.1101z3,z[120,110]0.00726607+2.45791z+8.13619z2+39.6031z3,z[110,320]804.1213145.31z+4185.85z21843.97z3,z[1720,910]1242.214605.6z+5808.4z22444.91z3,z[910,1920]1863.6565.99z+7871.97z23168.97z3,z[1920,1].

    Example 2. Consider the CDE,

    vt+0.22vz=0.52vz2,0z1,t>0, (3.25)

    with IC,

    v(z,0)=exp(0.22z)sin(πz) (3.26)

    and the BCs,

    v(0,t)=0,v(1,t)=0. (3.27)

    The analytic solution is v(z,t)=exp(0.22z(0.0242+0.5π2)t)sin(πz). The numerical results are obtained by utilizing the proposed scheme. In Table 3, the comparative analysis of absolute errors is given with that of [25]. Absolute errors and errors norms at time levels t=5,10,100 are computed in Table 4. Figure 4 shows a very close comparison between the exact and numerical solutions at various stages of time. Figure 5 plots 2D and 3D absolute errors at t=1. In Figure 6, a tremendous 3D contrast between the exact and approximate solutions is depicted.

    Table 3.  Absolute errors when k=0.002 and h=0.002 for Example 2.
    z t=0.8 t=1.0
    Present scheme CNM[25] Present scheme CNM[25]
    0.1 1.94 ×107 2.12 ×107 9.01 ×108 1.79 ×107
    0.3 5.32 ×107 5.85 ×107 2.47 ×107 4.92 ×107
    0.5 6.87 ×107 7.62 ×107 3.19 ×107 6.40 ×107
    0.7 5.81 ×107 6.49 ×107 2.69 ×107 5.45 ×107
    0.9 2.32 ×107 2.61 ×107 1.07 ×107 2.19 ×107

     | Show Table
    DownLoad: CSV
    Table 4.  Absolute errors when M=20 and k=0.01 for Example 2.
    z t=5 t=10 t=100
    0.1 2.75 ×1014 1.41 ×1023 1.36 ×10216
    0.3 7.52 ×1014 3.04 ×1025 1.32 ×1023
    0.5 9.71 ×1014 1.81 ×1024 3.31 ×1024
    0.7 8.21 ×1014 6.98 ×1022 4.24 ×1022
    0.9 3.28 ×1014 2.17 ×1019 2.17 ×1019

     | Show Table
    DownLoad: CSV
    Figure 4.  The approsimate (stars, circles, triangles) and exact (solid lines) solutions at various time stages when M=500,k=0.002 for Example 2.
    Figure 5.  2D and 3D error profiles when t=1,M=100,k=0.01 for Example 2.
    Figure 6.  The exact and approximate solutions when t=1,M=100,k=0.01 for Example 2.

    The numerical solution when t=1,k=0.01 and M=20 for Example 2 is given by

    V(z,1)={2.71051×1020+0.0220426z+0.0044697z20.0330726z3,z[0,120]6.20887×107+0.0220053z+0.0052147z20.0380397z3,z[120,110]7.22875×107+0.0220457z+0.0048116z20.0366959z3,z[110,320]0.0164802+0.0968639z0.116698z2+0.0363097z3,z[1720,910]0.0199915+0.108568z0.129703z2+0.0411262z3,z[910,1920]0.0196901+0.107616z0.128701z2+0.0407746z3,z[1920,1].

    Example 3. Consider the CDE,

    vt+0.1vz=0.22vz2,0z1,t>0, (3.28)

    with IC,

    v(z,0)=exp(0.25z)sin(πz) (3.29)

    and the BCs,

    v(0,t)=0,v(1,t)=0. (3.30)

    The analytic solution is v(z,t)=exp(0.25z(0.0125+0.2π2)t)sin(πz). By utilizing the proposed scheme the numerical results are acquired. An excellent comparison between absolute errors computed by our scheme and the scheme of [25] is discussed in Table 5. In Table 6, absolute errors and error norms are computed at different time stages. A close comparison between the exact and numerical solutions at different time stages is depicted in Figure 7. Figure 8 plots 2D and 3D absolute errors at t=1. Figure 9 deals with the 3D comparison that occurs between the exact and numerical solutions. The numerical solution when t=1,k=0.01 and M=20 for Example 3 is given by

    V(z,1)={4.33681×1019+0.430962z+0.107784z20.708327z3,z[0,120]2.81208×106+0.430793z+0.111159z20.730824z3,z[120,110]2.06532×106+0.430816z+0.110935z20.730077z3,z[110,320]0.337164+1.95012z2.33266z2+0.719619z3,z[1720,910]0.398612+2.15495z2.56025z2+0.803911z3,z[910,1920]0.451717+2.32264z2.73678z2+0.865849z3,z[1920,1].
    Table 5.  Absolute errors when k=0.005 at h=0.01 for Example 3.
    t z=0.1 z=0.3 z=0.5 z=0.7 z=0.9
    PM CNM[25] PM CNM[25] PM CNM[25] PM CNM[25] PM CNM[25]
    0.2 6.95 ×107 3.72 ×106 1.91 ×106 1.02 ×105 2.49 ×106 1.32 ×105 2.12 ×106 1.12 ×105 8.50 ×107 4.48 ×106
    0.4 9.35 ×107 2.80 ×106 2.57 ×106 7.68 ×106 3.34 ×106 9.93 ×106 2.84 ×106 8.41 ×106 1.14 ×106 3.36 ×106
    0.6 9.43 ×107 1.57 ×106 2.59 ×106 4.29 ×106 3.37 ×106 5.55 ×106 2.87 ×106 4.69 ×106 1.15 ×106 1.88 ×106
    0.8 8.45 ×107 7.77 ×107 2.33 ×106 2.13 ×106 3.02 ×106 2.75 ×106 2.57 ×106 2.33 ×106 1.03 ×106 9.29 ×107
    1.0 7.10 ×107 3.61 ×107 1.95 ×106 9.88 ×107 2.54 ×106 1.28 ×106 2.16 ×106 1.08 ×106 8.67 ×107 4.31 ×107

     | Show Table
    DownLoad: CSV
    Table 6.  Absolute errors when M=20 and k=0.01 for Example 3.
    z t=5 t=10 t=100
    0.1 5.21 ×109 5.07 ×1013 3.18 ×1022
    0.3 1.44 ×108 1.40 ×1012 4.69 ×1087
    0.5 1.87 ×108 1.82 ×1012 8.27 ×1024
    0.7 1.59 ×108 1.55 ×1012 8.47 ×1022
    0.9 6.39 ×109 6.21 ×1013 2.17 ×1019

     | Show Table
    DownLoad: CSV
    Figure 7.  The approximate (stars, circles, triangles) and exact (solid lines) solutions at various time stages when M=100,k=0.005 for Example 3.
    Figure 8.  2D and 3D error profiles when t=1,M=100,k=0.01 for Example 3.
    Figure 9.  The exact and approximate solutions when t=1,M=100,k=0.01 for Example 3.

    Example 4. Consider the CDE,

    vt+0.8vz=0.12vz2,0z1,t>0, (3.31)

    with IC,

    v(z,0)=exp((z2)280) (3.32)

    and the BCs,

    v(0,t)=2020+texp((20.8t)20.4(20+t)), (3.33)
    v(1,t)=2020+texp((10.8t)20.4(20+t)). (3.34)

    The analytic solution of the given problem is v(z,t)=2020+texp((z20.8t)20.4(20+t)). In Table 7, a comparison between absolute errors calculated by our scheme and the scheme of [26] is presented. Absolute errors and errors norms at time levels t=5,10,100 are presented in Table 8. Figure 10 illustrates the behavior of numerical solutions at various time stages. Figure 11 depicts the 2D and 3D graphs of absolute errors. In Figure 12, an excellent 3D contrast between the exact and numerical solutions shows the enormous accuracy of the scheme.

    Table 7.  Absolute errors and convergence orders when t=1.0 and k=0.001 for Example 4.
    h Present scheme CuTBS[26]
    L2 L p L2 L
    1/4 7.10 ×107 1.20 ×106 1.20 ×104 1.09 ×104
    1/8 5.33 ×108 8.50 ×108 2.21360 2.98 ×105 2.35 ×105
    1/16 5.44 ×109 8.49 ×109 2.02852 7.56 ×106 5.76 ×106
    1/32 2.34 ×109 3.61 ×109 2.01005 1.91 ×106 1.43 ×106
    1/64 2.41 ×109 3.30 ×109 2.01012 4.77 ×107 3.55 ×107
    1/128 2.13 ×109 3.29 ×109 2.03705 1.17 ×107 8.65 ×108

     | Show Table
    DownLoad: CSV
    Table 8.  Absolute errors when M=20 and k=0.01 for Example 4.
    z t=5 t=10 t=100
    0.1 1.29 ×108 3.75 ×1010 4.14 ×1025
    0.3 4.20 ×108 1.43 ×109 3.10 ×1025
    0.5 7.29 ×108 2.87 ×109 1.94 ×1025
    0.7 9.66 ×108 4.30 ×109 6.62 ×1024
    0.9 7.31 ×108 3.51 ×109 1.26 ×1060

     | Show Table
    DownLoad: CSV
    Figure 10.  The approximate (stars, circles, triangles) and exact (solid lines) solutions at various time stages when M=100,k=0.001 for Example 4.
    Figure 11.  2D and 3D error profiles when t=1,M=100,k=0.01 for Example 4.
    Figure 12.  The exact and approximate solutions when t=1,M=100,k=0.01 for Example 4.

    The approximate solution when t=1,k=0.01 and M=20 for Example 4 is given by

    V(z,1)={0.383764+0.255843z+0.0396075z20.0119477z3,z[0,120]0.383764+0.255836z+0.039733z20.0127849z3,z[120,110]0.383765+0.255811z+0.0399901z20.0136416z3,z[110,320]0.385493+0.247569z+0.0547152z20.0241989z3,z[1720,910]0.385817+0.24649z+0.0559143z20.024643z3,z[910,1920]0.386119+0.245535z+0.0569187z20.0249954z3,z[1920,1].

    Example 5. Consider the CDE,

    vt+0.1vz=0.022vz2,0z1,t>0, (3.35)

    with IC,

    v(z,0)=exp(1.17712434446770z) (3.36)

    and the BCs,

    v(0,t)=exp(0.09t), (3.37)
    v(1,t)=exp(1.177124344467700.09t). (3.38)

    The analytic solution of the given problem is v(z,t)=exp(1.17712434446770z0.09t). The numerical outcomes are acquired by using the introduced scheme. The absolute errors and errors norms are computed in Tables 9 by applying the presented scheme on Example 5 and are compared with those obtained in [27]. In Table 10, absolute errors and error norms are presented at various time stages. Figure 13 plots the behavior of exact and numerical solutions at various time stages. Figure 14 depicts the 2D and 3D absolute error profiles at t=1. In Figure 15, a 3D contrast between the exact and numerical solutions is presented and all the graphs are in good agreement.

    Table 9.  Absolute errors and error norms when h=0.01 and k=0.001 for Example 5.
    z t=1 t=2
    CuBQI[27] Present method CuBQI[27] Present method
    0.1 2.1506 ×106 2.5219 ×1011 2.8217 ×106 3.3391 ×1011
    0.5 7.0601 ×106 7.6365 ×1011 1.2276 ×105 1.3300 ×1010
    0.9 7.6594 ×106 7.8200 ×1011 1.1643 ×105 1.1782 ×1010
    L2 6.4790 ×107 6.9230 ×1011 1.0719 ×106 1.1447 ×1010
    L 9.1107 ×106 9.7331 ×1011 1.5204 ×105 1.6236 ×1010

     | Show Table
    DownLoad: CSV
    Table 10.  Absolute errors when M=40 and k=0.01 for Example 5.
    z t=5 t=10 t=100
    0.1 4.46 ×109 3.42 ×109 1.12 ×1012
    0.3 1.41 ×108 1.14 ×108 3.81 ×1012
    0.5 2.34 ×108 1.99 ×108 6.73 ×1012
    0.7 2.89 ×108 2.50 ×108 8.51 ×1012
    0.9 1.94 ×108 1.65 ×108 5.59 ×1012

     | Show Table
    DownLoad: CSV
    Figure 13.  The approximate (stars, circles, triangles) and exact (solid lines) solutions at various time stages when M=100,k=0.001 for Example 5.
    Figure 14.  2D and 3D error profiles when t=1,M=100,k=0.01 for Example 5.
    Figure 15.  The exact and approximate solutions when t=1,M=100,k=0.01 for Example 5.

    The approximate solution when t=1,k=0.01 and M=20 for Example 5 is given by

    V(z,1)={0.913931+1.07581z+0.632996z2+0.255851z3,z[0,120]0.913929+1.07593z+0.630674z2+0.271329z3,z[120,110]0.913913+1.07642z+0.625737z2+0.287787z3,z[110,320]0.815184+1.50737z0.052669z2+0.695802z3,z[1720,910]0.78445+1.60981z0.166497z2+0.73796z3,z[910,1920]0.746014+1.73119z0.294263z2+0.78279z3,z[1920,1].

    This research uses a new approximation for second-order derivatives in the cubic B-spline collocation method to obtain the numerical solution of the CDE. The smooth piecewise cubic B-splines have been used to approximate derivatives in space whereas a usual finite difference has been used to discretize the time derivative. Special consideration is paid to the stability and convergence analysis of the scheme to ensure the errors do not amplify. The approximate solutions and error norms are contrasted with those reported previously in the literature. From this analysis, we can conclude that the estimated solutions are in perfect accord with the actual solutions. The scheme can be applied to a wide range of problems in science and engineering.

    The authors are thankful to the worthy reviewers and editors for the useful and valuable suggestions for the improvement of this paper which led to a better presentation.

    The authors have no conflict of interest.



    [1] G. Aalipour, S. Akbari, P. J. Cameron, R. Nikandish, F. Shaveisi, On the structure of the power graph and the enhanced power graph of a group, Electron. J. Combin., 24 (2017), 3–16.
    [2] J. Abawajy, A. Kelarev, M. Chowdhury, Power graphs: A survey, Electron. J. Graph Theory Appl., 1 (2013), 125–147.
    [3] T. Anitha, R. Rajkumar, On the power graph and the reduced power graph of a finite group, Commun. Algebra, 47 (2019), 3329–3339. doi: 10.1080/00927872.2018.1555842
    [4] T. Anitha, R. Rajkumar, Characterization of groups with planar, toroidal or projective planar (proper) reduced power graphs, J. Algebra Appl., 19 (2020), 2050099. doi: 10.1142/S0219498820500991
    [5] B. Bollobás, Mordern graph theory, New York: Springer, 1998.
    [6] D. Bubboloni, M. A. Iranmanesh, S. M. Shaker, Quotient graphs for power graphs, Rend. Semin. Mat. Univ. Padova, 138 (2017), 61–89. doi: 10.4171/RSMUP/138-3
    [7] P. J. Cameron, The power graph of a finite group, II, J. Group Theory, 13 (2010), 779–783.
    [8] P. J. Cameron, S. Ghosh, The power graph of a finite group, Discrete Math., 311 (2011), 1220–1222. doi: 10.1016/j.disc.2010.02.011
    [9] P. J. Cameron, P. Manna, R. Mehatari, Forbidden subgraphs of power graphs, Preprint, 2020. Available from: arXiv: 2010.05198v2.
    [10] I. Chakrabarty, S. Ghosh, M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum, 78 (2009), 410–426. doi: 10.1007/s00233-008-9132-y
    [11] M. Deaconescu, Classification of finite groups with all elements of prime order, Proc. Am. Math. Soc., 106 (1989), 625–629. doi: 10.1090/S0002-9939-1989-0969518-2
    [12] A. Doostabadi, A. Erfanian, D. G. M. Farrokhi, On power graphs of finite groups with forbidden induced subgraphs, Indagat. Math. (NS), 25 (2014), 525–533. doi: 10.1016/j.indag.2014.01.003
    [13] M. Feng, X. Ma, K. Wang, The structure and metric dimension of the power graph of a finite group, Eur. J. Combin., 43 (2015), 82–97. doi: 10.1016/j.ejc.2014.08.019
    [14] S. Foldes, P. L. Hammer, Split graphs, In: Proceedings of the 8th South-Eastern Conference on Combinatorics, Graph Theory and Computing, (1977), 311–315.
    [15] D. Gorenstein, Finite groups, New York: Chelsea Publishing Co., 1980.
    [16] G. Higman, Finite groups in which every element has prime power order, J. London Math. Soc., s1-32 (1957), 335–342. doi: 10.1112/jlms/s1-32.3.335
    [17] D. L. Johnson, Topics in the theory of group presentations, London Math. Soc. Lecture Note Ser., Cambridge University Press, 1980.
    [18] A. V. Kelarev, Ring constructions and applications, World Scientific, 2002.
    [19] A. V. Kelarev, Graph algebras and automata, New York: Marcel Dekker, 2003.
    [20] A. V. Kelarev, Labelled Cayley graphs and minimal automata, Australas. J. Combin., 30 (2004), 95–101.
    [21] A. V. Kelarev, S. J. Quinn, A combinatorial property and power graphs of groups, Contrib. General Algebra, 12 (2000), 229–235.
    [22] A. V. Kelarev, J. Ryan, J. Yearwood, Cayley graphs as classifiers for data mining: The influence of asymmetries, Discrete Math., 309 (2009), 5360–5369. doi: 10.1016/j.disc.2008.11.030
    [23] X. Ma, Perfect codes in proper reduced power graphs of finite groups, Commun. Algebra, 48 (2020), 3881–3890. doi: 10.1080/00927872.2020.1749845
    [24] X. Ma, G. L. Walls, K. Wang, Power graphs of (non) orientable genus two, Commun. Algebra, 47 (2019), 276–288. doi: 10.1080/00927872.2018.1476522
    [25] A. R. Moghaddamfar, S. Rahbariyan, W. J. Shi, Certain properties of the power graph associated with a finite group, J. Algebra Appl., 13 (2014), 1450040. doi: 10.1142/S0219498814500406
    [26] R. Rajkumar, T. Anitha, Reduced power graph of a group, Electron. Notes Discrete Math., 63 (2017), 69–76. doi: 10.1016/j.endm.2017.10.063
    [27] R. Rajkumar, T. Anitha, Some results on the reduced power graph of a group, Southeast Asian Bull. Math., 2018. Available from: arXiv: 1804.00728v2.
    [28] D. B. West, Introduction to graph theory, 2 Eds., Englewood Cliffs, NJ: Prentice Hall, 2001.
  • Reader Comments
  • © 2021 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(2648) PDF downloads(140) Cited by(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog