Processing math: 56%
Research article Special Issues

Torque analysis and slot regions assignment of a DC-excited flux-modulated machine with two stator windings

  • Received: 07 July 2017 Accepted: 23 October 2017 Published: 23 November 2017
  • This paper presents a DC-excited flux-modulated (DCEFM) machine. It has one rotor with pr poles and two sets of windings on one stator. One of the windings has pf pole-pairs and is excited with DC current. The second winding has pa pole-pairs and is excited with AC sinusoidal current. The relationship between pr, pf and pa is pr = pf + pa. The purpose of this paper is to analyze the torque of the machine in order to optimize the slot regions assigned to the two stator windings for generating the highest torque at the admissible copper losses. Firstly, the torque equation of the machine is derived. The methods described in this paper are also feasible for other machine types such as variable flux reluctance machine, field excited flux-switching machine. By using the finite element analysis (FEA) the analytical results are verified. The results show that the magneto motive forces (MMFs) of the two windings need to be approximately equal to obtain optimal torque.

    Citation: Jing Ou, Yingzhen Liu, Martin Doppelbauer. Torque analysis and slot regions assignment of a DC-excited flux-modulated machine with two stator windings[J]. AIMS Electronics and Electrical Engineering, 2017, 1(1): 4-17. doi: 10.3934/ElectrEng.2017.1.4

    Related Papers:

    [1] Zihan Yue, Wei Jiang, Boying Wu, Biao Zhang . A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation. AIMS Mathematics, 2024, 9(3): 7040-7062. doi: 10.3934/math.2024343
    [2] Farman Ali Shah, Kamran, Zareen A Khan, Fatima Azmi, Nabil Mlaiki . A hybrid collocation method for the approximation of 2D time fractional diffusion-wave equation. AIMS Mathematics, 2024, 9(10): 27122-27149. doi: 10.3934/math.20241319
    [3] Xiangmei Li, Kamran, Absar Ul Haq, Xiujun Zhang . Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature. AIMS Mathematics, 2020, 5(5): 5287-5308. doi: 10.3934/math.2020339
    [4] Chao Wang, Fajie Wang, Yanpeng Gong . Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method. AIMS Mathematics, 2021, 6(11): 12599-12618. doi: 10.3934/math.2021726
    [5] Fouad Mohammad Salama, Nur Nadiah Abd Hamid, Norhashidah Hj. Mohd Ali, Umair Ali . An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions. AIMS Mathematics, 2022, 7(2): 2370-2392. doi: 10.3934/math.2022134
    [6] Xiangyun Qiu, Xingxing Yue . Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method. AIMS Mathematics, 2024, 9(10): 27150-27166. doi: 10.3934/math.20241320
    [7] Bin Fan . Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives. AIMS Mathematics, 2024, 9(3): 7293-7320. doi: 10.3934/math.2024354
    [8] Bin He . Developing a leap-frog meshless methods with radial basis functions for modeling of electromagnetic concentrator. AIMS Mathematics, 2022, 7(9): 17133-17149. doi: 10.3934/math.2022943
    [9] Hijaz Ahmad, Muhammad Nawaz Khan, Imtiaz Ahmad, Mohamed Omri, Maged F. Alotaibi . A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models. AIMS Mathematics, 2023, 8(8): 19677-19698. doi: 10.3934/math.20231003
    [10] Sayed Saifullah, Amir Ali, Zareen A. Khan . Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel. AIMS Mathematics, 2022, 7(4): 5275-5290. doi: 10.3934/math.2022293
  • This paper presents a DC-excited flux-modulated (DCEFM) machine. It has one rotor with pr poles and two sets of windings on one stator. One of the windings has pf pole-pairs and is excited with DC current. The second winding has pa pole-pairs and is excited with AC sinusoidal current. The relationship between pr, pf and pa is pr = pf + pa. The purpose of this paper is to analyze the torque of the machine in order to optimize the slot regions assigned to the two stator windings for generating the highest torque at the admissible copper losses. Firstly, the torque equation of the machine is derived. The methods described in this paper are also feasible for other machine types such as variable flux reluctance machine, field excited flux-switching machine. By using the finite element analysis (FEA) the analytical results are verified. The results show that the magneto motive forces (MMFs) of the two windings need to be approximately equal to obtain optimal torque.


    Recently, partial differential equations (PDEs) with fractional derivatives have gained significant attention of the research community in applied sciences and engineering. Such equations are encountered in various applications (continuum mechanics, gas dynamics, hydrodynamics, heat and mass transfer, wave theory, acoustics, multiphase flows, chemical engineering, etc.). Numerous phenomenon in Chemistry, Physics, Biology, Finance, Economics and other relevant fields can be modelled using PDEs of fractional order [1,2,3,4,5]. In literature a significant theoretical work on the explicit solution of fractional order differential equations can be found [6,7] and there references. Since the explicit solutions can be obtained for special cases and most of the time the exact/analytical solutions are cumbersome for differential equations of non-integer order, therefore an alternative way is to find the solutions numerically. Various computational methods have been developed for approximation of differential equations of fractional order. The authors in [8], for example, have analyzed the implicit finite difference method and proved its unconditional convergence and stability. In [9] approximate solution of fractional diffusion equation is obtained via compact finite difference scheme. Liu et al.[10] studied the sub-diffusion equation having non-linear source term using analytical and numerical techniques. In [11] a numerical scheme for the solution of turbulent Riesz type diffusion equation is presented. The authors in [12] have solved diffusion-wave equations of fractional order using a compact finite difference method which is based on its equivalent integro-differential form. Garg et al.[13] utilized the matrix method for approximation of space-time wave-diffusion equation of non-integer order. In [14] the authors solved multi-term wave-diffusion equation of fractional order via Galerkin spectral method and a high order difference scheme. Two finite difference methods for approximating wave-diffusion equations are proposed in [15]. Bhrawy et al. [16] utilized Jacobi operational matrix based spectral tau algorithm for numerical solution of diffusion-wave equation of non-integer order. In [4] the authors proposed numerical schemes for approximating the multi-term wave-diffusion equation. The Legendre wavelets scheme for diffusion wave equations is proposed in [17]. The authors [18] presented a numerical scheme which is based on alternating direction implicit method and compact difference method for 2-D wave-diffusion equations. Similarly a compact difference scheme [19] is utilized for approximation of 1-D and 2-D diffusion-wave equations. Yang et al. [20] proposed a fractional multi-step method for the approximation of wave-diffusion equation of non-integer order. A spectral collocation method and its convergence analysis are presented in [21] for fractional wave-diffusion equation.

    Since all these methods are mesh dependent and in modern problems these methods have been facing difficulties due to complicated geometries. Meshfree methods, as an alternative numerical method have attracted the researchers. Some meshless methods have been devoloped such as element-free Galerkin method(EFG)[22], reproducing kernel particle method (RKPM)[23], singular boundary method [24], the boundary particle method [25], Local radial point interpolation method (MLRPI) [26] and so on.

    Numerous meshless methods have been developed for the approximation of fractional PDEs. Dehghan et al. [27] analyzed a meshless scheme for approximation of diffusion-wave equation of non-integer order and proved its stability and convergence. In [28] the authors presented an implicit meshless scheme for approximation of anomalous sub-diffusion equation. Diffusion equations of fractional orders are apprximated via RBF based implicit meshless method in [29]. Hosseini et al. [26] developed a local radial point interpolation meshless method based on the Galerkin weak form for numerical solution of wave-diffusion equation of non integer order. In [30] the authors approximated distributed order diffusion-wave equation of fractional order using meshless method. The authors in [31] proposed a meshless point collocation method for approximation of 2D multi-term wave-diffusion equations. In [32] the authors proposed a local meshless method for time fractional diffusion-wave equation. Kansa method [33] is utilized for numerical solution of fractional diffusion equations. Zhuang et al. [34] proposed an implicit MLS meshless method for time fractional advection diffusion equation. The numerical solution of 2D wave-diffusion equation is studied in [35] using implicit MLS meshless method. The mentioned methods are meshfree time stepping methods and these methods faces stability restriction in time, and in these methods for convergence a very small step size is required. To overcome the issue of time instability some transformations may be used.

    In literature some valuable work is available on resolving the problem of time instability. The researchers have coupled the Laplace transform with other well known numerical methods. For example the Laplace transform with Kansa method [33,36], finite element method [37,38], finite difference method [39], RBF method on unit sphere [40] and the references therein. In the present work we have coupled the Laplace transform with local meshless method for approximating the solution of the multi-term diffusion wave equation of fractional order.

    In our numerical scheme we transform the multi-term time fractional wave-diffusion equation to a time independent problem with Laplace transformation. The reduced problem is then approximated using local meshless method in Laplace space. Finally the solution of the original problem is obtained using contour integration. We apply the proposed method to multi-term fractional wave-diffusion equation of the form [14]

    Pα,α1,α2,...,αm(Dτ)U(χ,τ)=KLU(χ,τ)+f(χ,τ),forχΩ,KRτ>0, (2.1)

    where

    Pα,α1,α2,...,αm(Dτ)=Dατ+mj=1djDαjτ,

    1<αm<...<α1<α<2, and dj0,j=1,2,...,m,mN are constants. Dαjτ is a Caputo derivative of order αj defined by

    Dαjτf(τ)=1Γ(nαj)τaf(n)(ν)dν(τν)αj+1n,forn1<αj<n,nN, (2.2)

    also for n=2, we have

    Dαjτf(τ)=1Γ(nαj)τa2f(ν)ν2dν(τν)αj1,forαj(1,2). (2.3)

    The initial conditions for the above Eq (2.1) are

    U(χ,0)=U0(χ),U(χ,0)τ=U1(χ). (2.4)

    and the boundary conditions are

    BU(χ,τ)=q(χ,τ),χΩ, (2.5)

    where L is the governing linear differential operator, and B is the boundary differential operator. By applying the Laplace transformation to Eq (2.1), we get

    ˆU(χ,s)=W(s;L)ˆg(χ,s), (2.6)

    where

    W(s;L)=(sαI+sα1I+...+sαmIKL)1,

    and

    ˆg(χ,s)=sα1U0(χ)+sα2U1(χ)+sα11U0(χ)+sα12U1(χ)+...+sαm1U0(χ)+sαm2U1(χ)+ˆf(χ,s).

    Similarly applying the Laplace transform to (2.5), we get

    BˆU(χ,s)=ˆq(χ,s), (2.7)

    Hence, the system of time-independent equations is obtained as

    ˆU(χ,s)=W(s;L)ˆg(χ,s), (2.8)
    BˆU(χ,s)=ˆq(χ,s), (2.9)

    In our method first we represent the solution U(χ,τ) of the original problem (2.1) as a contour integral

    U(χ,τ)=12πiΓesτˆU(χ,s)ds, (2.10)

    where, for Resω with ω appropriately large, and Γ is an initially appropriately chosen line Γ0 perpendicular to the real axis in the complex plane, with Ims±. The integral (2.10) is just the inverse transform of ˆU(χ,τ), with the condition that ˆU(χ,τ) must be analytic to the right of Γ0. To make sure the contour of integration remains in the domain of analyticity of ˆU(χ,τ), we select Γ as a deformed contour in the set ΣΥϕ={s0:|args|<ϕ}{0}, which behaves as a pair of asymptotes in the left half plane, with Res when Ims±, which force esτ to decay towards both ends of Γ. In our work we have used two types of contours, the first contour is the hyperbolic contour Γ1 due to [38] with parametric representation

    s(ξ)=Υ+(1sin(ηιξ)),ξR,(Γ1) (2.11)

    where,

    >0,0<η<ϕπ2,andΥ>0. (2.12)

    By writing s=x+ιy, we observe that Γ1 is the left branch of the hyperbola

    (xΥsinη)2(ycosη)2=1, (2.13)

    the asymptotes for (2.13) are y=±(xΥ)cotη, and x-intercept at s=Υ+(1sinη). The condition (2.12) confirms that Γ1 lies in the sector ΣΥϕ=Υ+ΣϕΣϕ, and grows into the left half plane. From (2.11) and (2.10), we have the following integral

    U(χ,τ)=12πies(ξ)τˆU(χ,s(ξ))ˊs(ξ)dξ. (2.14)

    Finally to approximate Eq (2.14), the trapezoidal rule with step k is used as

    Uk(χ,τ)=k2πiMj=MesjτˆU(χ,sj)ˊsj,forξj=jk,sj=s(ξj),sj=s(ξj). (2.15)

    The second contour employed in this work is the Talbot's contour [41], though ignored by many researchers, yet it is one of the best method for numerical inverting the Laplace transform [42]. The authors in [43] have optimized the Talbot's contour for approximating the solution of parabolic PDEs. Other works on Talbot's method can be found in [44,45] and there references. In our work we have employed the improved Talbot's method [46] for numerical inversion of Laplace transform. The Talbot's contour has parametric representation of the form

    s(ξ)=Mτθ(ξ),θ(ξ)=σ+μξcot(γξ)+νιξ,πξπ,(Γ2) (2.16)

    where the parameters σ,μ,ν, and γ are to be specified by the user. From (2.16) and (2.10) we have

    U(χ,τ)=12πiππes(ξ)τˆU(χ,s(ξ))ˊs(ξ)dξ. (2.17)

    We use M-panel mid-point rule with uniform spacing k=2πM, to approximate the integral (2.17) as

    Uk(χ,τ)=1MiMj=1esjτˆU(χ,sj)ˊsj,forξj=π+(j12)k,sj=s(ξj),sj=s(ξj). (2.18)

    To obtain the solution Uk(χ,τ), first we must solve system of 2M+1 equations given in (2.8) and (2.9) for quadrature points sj,|j|M. For this purpose the local meshless method is used to discretize operators L,B.

    Given a set of points {χi}Ni=1inRd,whered1 the approximate function for ˆU(χ) using local meshless method has the form,

    ˆU(χi)=χjΩiλijϕ(χiχij), (2.19)

    where λi={λij}nj=1 is the expansion coefficients vector, ϕ(r) is a kernel function, r=χiχj is the distance between the centers χi and χj. Ω, and Ωi are global domain and local domains respectively. The sub-domain Ωi contains the center χi, and around it, its n neighboring centers. Thus we obtain n×n linear systems

    (ˆU(χi1)ˆU(χi2)...ˆU(χin))=(ϕ(χi1χi1)ϕ(χi1χi2)...ϕ(χi1χin)ϕ(χi2χi1)ϕ(χi2χi2)...ϕ(χi2χin)............ϕ(χinχi1)ϕ(χinχi2)...ϕ(χinχin))(λi1λi2...λin),i=1,2,...,N, (2.20)

    which can be written as,

    ˆUi=Φiλi,1iN, (2.21)

    the matrix Φi contains elements in the form bikj=ϕ(χikχij),whereχik,χijΩi, the unknowns λi={λij:j=1,...,n} are obtained by solving each of the N systems in (2.21). For the differential operator L we have the form,

    LˆU(χi)=χjΩiλijLϕ(χiχij), (2.22)

    the above Eq (2.22) can be expressed as a dot product

    LˆU(χi)=λiνi, (2.23)

    where νi is a n-row vector and λi is a n-column vector, entries of the n-column vector νi are given as

    νi=Lϕ(χiχij),χijΩi, (2.24)

    eliminating the co efficient λi from (2.21), and (2.23) we have the following expression

    LˆU(χi)=νi(Φi)1ˆUi=ϖiˆUi (2.25)

    where,

    ϖi=νi(Φi)1, (2.26)

    thus at each node χi the approximation of the operator L via local meshless method is given as

    LˆUDˆU, (2.27)

    In (2.27) D is a sparse differentiation matrix obtained via local meshless method as an approximation to L. The matrix D has order N×N, it has n non-zero entries, and Nn zero entries, where N is number of centers in global domain, and n is the number of centers in local domain. The boundary operator B can be discretized in similar way.

    In order to solve the multi-term time fractional diffusion wave equation using our proposed method, the local meshless method and Laplace transformation is used. In our numerical scheme first the Laplace transform is applied to time dependent equation which eliminates the time variable, and this process causes no error. Then the local meshless method is utilized for approximating time independent equation. The error estimate for local meshless method is of order O(η1ϵh), 0<η<1, ϵ is the shape parameter and h is the fill distance. In the process of approximating the integrals (2.14) and (2.17) convergence is achieved at different rates depending on the paths Γ1, and Γ2. In approximating the integrals (2.14) and (2.17) the convergence order rely upon on the step k of the quadrature rule and the time domain [t0,T] for Γ1. The proof for the order of quadrature error for the path Γ1 is given in the next theorem.

    Theorem 3.1 ([38], Theorem 2.1). Let U(χ,τ) be the solution of (2.1) with ˆf(χ,τ) analytic in ΣΥϕ. Let ΓΩrΣΥϕ, and define b>0 by coshb=1θτ1sin(η), where τ1=t0T, 0<τ0<T, 0<θ<1.0 and let =θ¯rMbT. Then for Eq (2.15), with k=bM¯rlog2, we have |U(χ,τ)Uk(χ,τ)|CQeΥτ1l(ρrM)eμM(U0+ˆf(χ,τ)ΣΥϕ), for μ=¯r(1θ)b, ρr=θ¯rτ1sin(ηr1)b, ¯r=2πr1, r1>0, τ0τT, C=Cη,r1,β, and l(x)=max(1,log(1x)). Hence the error estimate for the proposed scheme is

    errorest(Γ1)=|U(χ,τ)Uk(χ,τ)|=O(l(ρrM)eμM).

    The authors in [46] derived the optimal values of the parameters for the Talbot's contour (Γ2) defined in (2.16) as given below

    σ=0.6122,μ=0.5017,ν=0.2645,andγ=0.6407,

    with corresponding error estimate as

    errorest(Γ2)=|U(χ,τ)Uk(χ,τ)|=O(e1.358M).

    To investigate the stability of the systems (2.8) and (2.9), we represent the system in discrete form as

    MˆU=b, (4.1)

    the matrix MN×N is sparse matrix obtained using local meshless method. For the system (4.1) the constant of stability is defined as

    C=supˆU0ˆUMˆU, (4.2)

    for any discrete norm . defined on RN the constant C is finite. From (4.2) we may write

    M1ˆUMˆUC, (4.3)

    Similarly for the pseudoinverse M of M, we can write

    M=supH0MHH. (4.4)

    Thus we have

    MsupH=MˆH0MMˆUMˆU=supˆU0ˆUMˆU=C. (4.5)

    We can see that Eqs (4.3) and (4.5) confirms the bounds for the stability constant C. Calculating the pseudoinverse for approximating the system (4.1) numerically be quite expansive computationally, but it confirms the stability. The MATLAB's function condest can be used to estimate M1 in case of square systems, thus we have

    C=condest(M)M (4.6)

    This work well with less number of computations for our sparse differentiation matrix M. Figure 1(a) shows the bounds for the constant C of our system (2.8) and (2.9) for Problem 1 using the Talbot's contour Γ2. Selecting N=80, M=80, n=15, and α=1.8,α1=1.7,α2=1.6,c=0.6 at τ=1, we have 1.00C4.5501. It is observed that the upper and lower bounds for the stability constant are very small numbers, which guarantees that the proposed local meshless scheme is stable.

    Figure 1.  In (a) the plot shows the constant of stability of our proposed method for the matrix \bmM corresponding to Problem 1, using the Talbot's contour Γ2. In (b) the Talbot's contour is shown.

    The numerical examples are given to validate our proposed Laplace transform based local meshless scheme. In our computations we have considered different 1D and 2D linear multi term wave-diffusion equations. In our numerical examples we have utilized the multi-quadrics(MQ) kernel function ϕ(r,ε) = (1+(εr)2)12. We have used the uncertainty principal due to [47] for optimization of the shape parameter. The accuracy of the method is measured using L error defined by

    L=U(χ,τ)Uk(χ,τ)=max1jN(|U(χ,τ)Uk(χ,τ)|)

    is used. Here Uk and U are the numerical and exact solutions respectively.

    In the first test problem we consider the following linear fractional equation

    DατU(χ,τ)+Dα1τU(χ,τ)+Dα2τU(χ,τ)D2χU(χ,τ)=f(χ,τ), (5.1)

    where

    f(χ,τ)=(6τ3αΓ(4α)+6τ3α1Γ(4α1)+6τ3α2Γ(4α2))1cosh(χc)(2τ3cosech(cχ)2cosh(cχ)3τ3cosh(cχ)).

    The exact solution of the problem is

    U(χ,τ)=τ3cosh(χc),cR.

    The boundary and initial conditions are

    U(10,τ)=τ3cosh(10c),U(10,τ)=τ3cosh(10c), (5.2)

    and

    U0(χ)=U1(χ)=0. (5.3)

    The points along the hyperbolic contour Γ1 are calculated using the statement ξ=M:k:M, and along Talbot's contour Γ2 using the relation ξj=π+(j12)k,wherej=1:M,andk=2πM. The parameters used in our computations for the contour Γ1 are θ=0.10,η=0.15410,τ1=t0T,r1=0.13870,ˉr=2r1π,Υ=2.0 The results obtained for the parameters α,α1,α2, and c along the contour Γ1 are displayed in Table 1, and along Γ2 are displayed in Table 2. The exact and numerical spacetime solutions for the given problem is depicted in Figure 2(a) and in Figure 2(b) respectively. The absolute error and error estimate are displayed in Figure 3(a). Figure 3(b) shows error functions for various values of αj. The results confirms that our numerical scheme is accurate, stable and can solve multi-term time fractional wave-diffusion equations with less computation time.

    Table 1.  Numerical solution in the domain [0,1] and τ=1 obtained using hyperbolic contour Γ1.
    α=1.5, α1=1.4, α2=1.3, c=0.5
    N n M L c κ errorest(Γ1) C.TIME(sec)
    80 20 35 7.77×105 10.0 1.13×10+12 3.14×101 0.343148
    55 7.61×105 10.0 1.13×10+12 3.64×102 1.222487
    75 7.61×105 10.0 1.13×10+12 4.2 ×103 5.000861
    95 7.61×105 10.0 1.13×10+12 4.75×104 12.067666
    110 7.61×105 10.0 1.13×10+12 9.30×105 22.244950
    40 15 80 5.76×105 4.5 1.37×10+12 2.4 ×103 2.034147
    50 4.70×105 5.7 1.17×10+12 2.4 ×103 3.430185
    60 4.32×105 6.9 1.05×10+12 2.4 ×103 4.307865
    70 9.09×105 8.0 1.24×10+12 2.4 ×103 5.308744
    80 4.82×105 9.2 1.14×10+12 2.4 ×103 6.327198
    70 12 90 2.33×105 7.3 1.02×10+12 8.18×104 8.646889
    15 9.09×105 8.0 1.24×10+12 8.18×104 8.534781
    18 8.80×105 8.5 1.17×10+12 8.18×104 8.898832
    21 4.10×105 8.8 1.20×10+12 8.18×104 8.901062
    24 9.30×105 9.0 1.26×10+12 8.18×104 8.835825
    [14] 4.79×106

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical solution in the domain [0,1] and τ=1 obtained using Talbot's contour Γ2.
    α=1.8, α1=1.7, α2=1.6, c=0.5
    N n M L c κ errorest(Γ2) C.TIME(sec)
    70 12 10 3.51×101 7.3 1.02×10+12 2.02×106 0.201190
    12 3.98×102 7.3 1.02×10+12 1.47×107 0.202626
    14 4.20×103 7.3 1.02×10+12 1.06×108 0.213661
    16 4.16×104 7.3 1.02×10+12 7.76×1010 0.203777
    18 4.99×105 7.3 1.02×10+12 5.64×1011 0.204327
    30 15 20 2.97×105 3.4 1.00×10+12 4.10×1012 0.203400
    40 5.99×105 4.5 1.37×10+12 4.10×1012 0.212062
    50 4.82×105 5.7 1.17×10+12 4.10×1012 0.210793
    70 9.41×105 8.0 1.24×10+12 4.10×1012 0.222508
    90 8.14×105 10.4 1.07×10+12 4.10×1012 0.217812
    80 12 20 7.56×105 8.3 1.15×10+12 4.10×1012 0.211422
    14 7.53×105 9.0 1.04×10+12 4.10×1012 0.218499
    16 6.68×105 9.4 1.17×10+12 4.10×1012 0.224207
    18 4.87×105 9.8 1.02×10+12 4.10×1012 0.226878
    20 4.67×105 10.0 1.13×10+12 4.10×1012 0.239581
    [14] 5.69×105

     | Show Table
    DownLoad: CSV
    Figure 2.  In (a) the spacetime plot shows the exact solutions, in (b) the spacetime plot shows the numerical solution, the parameters used are α=1.5,α1=1.4,α2=1.3,c=0.5, N=70,n=12, and M=90, along the hyperbolic contour Γ1.
    Figure 3.  In (a) the Absolute error and error_est(Γ1) are shown corresponding to problem 1 using N=40,n=15,α=1.8,α1=1.7,α2=1.6,c=0.5, the results confirms a good agreement between them. In (b) the error functions for different α, and αj on [0,1] are shown using the hyperbolic contour Γ1.

    As a second test problem we consider the following linear fractional equation

    DατU(χ,τ)+Dα1τU(χ,τ)D2χU(χ,τ)=f(χ,τ), (5.4)

    where

    f(χ,τ)=(6τ3αΓ(4α)+6τ3α1Γ(4α1)+π2τ3)sin(πχ).

    The exact solution of the problem is

    U(χ,τ)=sin(πχ)τ3.

    This equation is considered on [0,1] with boundary conditions

    U(0,τ)=U(1,τ)=0 (5.5)

    and initial conditions

    U0(χ)=U1(χ)=0. (5.6)

    In this experiment we have utilized both the contours with the same set of optimal parameters. The numerical experiments are performed with different nodes N in the global domain n in the sub-domain. The results obtained for fractional orders α, and α1. are displayed in Table 3 along the path Γ1, and in Table 4 along the path Γ2. The approximate and exact spacetime solutions are displayed in Figures 4(a) and Figure 4(b). The plot of absolute error and error estimate is displayed in Figure 5(a). Figure 5(b) shows the plot of error functions for various values of α, and α1. The results verifies the accuracy, stability and efficiency of the proposed local meshless scheme for multi-term time fractional wave-diffusion equations.

    Table 3.  Numerical solution in the domain [0,1] and τ=1 hyperbolic contour Γ1.
    α=1.9, α1=1.3
    N n M L c κ errorest(Γ1) C.TIME(sec)
    60 22 40 8.44×105 7.6 1.18×10+12 1.83×101 0.667872
    60 6.62×105 7.6 1.18×10+12 2.12×102 1.580686
    80 8.62×105 7.6 1.18×10+12 2.4 ×103 7.362587
    90 8.62×105 7.6 1.18×10+12 8.18×104 10.843996
    100 8.62×105 7.6 1.18×10+12 2.76×104 15.784191
    70 10 100 5.52×105 6.4 1.15×10+12 2.76×104 17.932164
    12 2.74×105 7.3 1.02×10+12 2.76×104 17.859075
    14 3.44×105 7.8 1.21×10+12 2.76×104 18.129870
    18 5.21×105 8.5 1.17×10+12 2.76×104 18.278882
    22 7.64×105 8.9 1.15×10+12 2.76×103 18.479720
    50 25 90 9.85×105 6.5 1.01×10+12 8.18×104 8.730629
    60 1.03×104 7.8 1.09×10+12 8.18×104 11.049126
    70 5.40×105 9.1 1.14×10+12 8.18×104 13.349187
    80 6.57×105 10.4 1.19×10+12 8.18×104 15.124607
    90 6.17×105 11.8 1.02×10+12 8.18×104 17.134190
    [14] 7.0080×104

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical solution in the domain [0,1] and τ=1 obtained using Talbot's contour Γ2.
    α=1.7, α1=1.2
    N n M L c κ errorest(Γ2) C.TIME(sec)
    70 12 10 2.56×101 7.3 1.02×10+12 2.02×106 0.202184
    12 2.91×102 7.3 1.02×10+12 1.47×107 0.197720
    14 3.10×103 7.3 1.02×10+12 1.06×108 0.200589
    16 3.25×104 7.3 1.02×10+12 7.76×1010 0.202710
    18 5.65×105 7.3 1.02×10+12 5.64×1011 0.194221
    30 15 20 3.74×105 3.4 1.00×10+12 4.10×1012 0.202098
    40 6.97×105 4.5 1.37×10+12 4.10×1012 0.201783
    50 6.68×105 5.7 1.17×10+12 4.10×1012 0.204199
    70 1.00×104 8.0 1.24×10+12 4.10×1012 0.205864
    90 1.36×104 10.4 1.07×10+12 4.10×1012 0.216570
    80 12 20 9.70×105 8.3 1.15×10+12 4.10×1012 0.207446
    14 4.34×105 9.0 1.04×10+12 4.10×1012 0.206234
    16 8.47×105 9.4 1.17×10+12 4.10×1012 0.216176
    18 4.76×105 9.8 1.02×10+12 4.10×1012 0.215397
    20 5.66×105 10.0 1.13×10+12 4.10×1012 0.244942
    [14] 1.39×104

     | Show Table
    DownLoad: CSV
    Figure 4.  In (a) The spacetime plot shows the exact solution. In (b) the spacetime plot shows the numerical solution, the parameters used are α=1.9,α1=1.7,N=70,n=12,M=90 on [5,5], using the hyperbolic contour Γ1.
    Figure 5.  In (a) Absolute error and error_est for problem 2 are presented using N=50,n=10,α=1.7,α1=1.2, the results confirms a good agreement between them. In (b) Error functions for different α, and αj on [0,1], are shown using the hyperbolic contour Γ1. The figure shows that the error decreases with increasing the values of fractional orders α, and α1.

    We consider the following fractional equation

    DατU(χ,τ)+Dα1τU(χ,τ)D2χU(χ,τ)=f(χ,τ), (5.7)

    where

    f(χ,τ)=(2τ2αcos(χ)Γ(3α)+2τ2α1cos(χ)Γ(3α1))(τ2cos(χ)+2τ2cosec(χ)2cos(χ)3).

    The exact solution of the problem is

    U(χ,τ)=τ2cos(χ).

    This equation is considered on [0,1] with boundary conditions

    U(0,τ)=τ2,U(1,τ)=τ2cos(1) (5.8)

    and initial conditions

    U0(χ)=0,U1(χ)=0. (5.9)

    The results obtained for third test problem with fractional orders α, and α1 along the hyperbolic contour Γ1 are displayed in Tables 5, and along the Talbots contour are displayed in Table 6. From the Tables it can be seen the method has good results in accuracy. Figures 6(a) shows the exact spacetime solution and Figure 6(b) shows the numerical spacetime solution. Figure 7(a), and Figure 7(b) absolute error and error estimate for the contour Γ1 and Γ2 respectively.

    Table 5.  Numerical solution in the domain [0,1] and τ=1 hyperbolic contour Γ1.
    α=1.9, α1=1.8,
    N n M L c κ errorest(Γ1) C.TIME(sec)
    80 25 50 4.20×105 10.4 1.19×10+12 6.25×102 2.249386
    60 3.85×105 10.4 1.19×10+12 2.12×102 4.114330
    70 3.86×105 10.4 1.19×10+12 7.2 ×103 7.301508
    90 3.86×105 10.4 1.19×10+12 8.18×104 15.044453
    100 3.86×105 10.4 1.19×10+12 2.76×104 21.588645
    60 27 90 9.51×105 7.9 1.06×10+12 8.18×104 11.338375
    70 8.41×105 9.2 1.16×10+12 8.18×104 13.565846
    80 1.15×104 10.6 1.01×10+12 8.18×104 15.034979
    90 9.66×105 11.9 1.10×10+12 8.18×104 17.556509
    100 7.93×105 13.2 1.16×10+12 8.18×104 21.605433
    85 20 95 8.45×105 10.6 1.21×10+12 4.75×104 18.828414
    22 5.73×105 10.9 1.01×10+12 4.75×104 19.612251
    24 1.68×104 11.0 1.16×10+12 4.75×104 19.670357
    27 4.26×105 11.2 1.16×10+12 4.75×104 19.563284
    30 5.36×105 11.4 1.09×10+12 4.75×104 20.041057
    [14] 8.81×105

     | Show Table
    DownLoad: CSV
    Table 6.  Numerical solution in the domain [0,1] and τ=1 obtained using Talbot's contour Γ2.
    α=1.9, α1=1.8,
    N n M L c κ errorest(Γ2) C.TIME(sec)
    70 12 10 3.54×102 7.3 1.02×10+12 2.02×106 0.142848
    12 3.40×103 7.3 1.02×10+12 1.47×107 0.133224
    14 2.83×104 7.3 1.02×10+12 1.06×108 0.134380
    16 4.64×105 7.3 1.02×10+12 7.76×1010 0.133445
    18 5.19×105 7.3 1.02×10+12 5.64×1011 0.136420
    30 15 20 1.15×104 3.4 1.00×10+12 4.10×1012 0.135380
    40 5.10×105 4.5 1.37×10+12 4.10×1012 0.142579
    50 1.07×104 5.7 1.17×10+12 4.10×1012 0.137483
    70 1.44×104 8.0 1.24×10+12 4.10×1012 0.145392
    90 1.49×104 10.4 1.07×10+12 4.10×1012 0.151340
    80 12 20 1.29×104 8.3 1.15×10+12 4.10×1012 0.144167
    14 1.22×104 9.0 1.04×10+12 4.10×1012 0.139500
    16 8.31×105 9.4 1.17×10+12 4.10×1012 0.148407
    18 4.78×105 9.8 1.02×10+12 4.10×1012 0.156982
    20 9.51×105 10.0 1.13×10+12 4.10×1012 0.166119
    [14] 8.81×105

     | Show Table
    DownLoad: CSV
    Figure 6.  In (a) The spacetime plot shows the exact solution. In (b) The spacetime plot shows the numerical solution, the parameters used are α=1.9,α1=1.8,N=70,n=15,M=80 on [0,1], using the hyperbolic contour Γ1.
    Figure 7.  In (a) absolute error and errorest(Γ1) are shown corresponding to problem 3 using N=50,n=10,α=1.9,α1=1.8, the results confirms a good agreement between them. In (b) absolute error and errorest(Γ2) are shown for the parameter values α=1.9,α1=1.8,N=70,n=12 on [0,1].

    We consider the two dimensional multi-term time fractional wave-diffusion equation

    DατU(χ,ϑ,τ)+Dα1τU(χ,ϑ,τ)ΔU(χ,ϑ,τ)=f(χ,ϑ,τ), (5.10)

    subject to zero initial conditions and the boundary conditions are generated from the exact solution

    U(χ,ϑ,τ)=eχ+ϑτ2+α+α1

    the given 2D test problem is solved with regular nodal points in rectangular, circular and complex domains.

    The rectangular domain [0,1]2 is descretized with N uniformly distributed points. For this problem also we have used the hyperbolic contour Γ1 and Talbot's contour Γ2 with the same set of optimal parameters used for Problem 1. The uniform nodes distribution with boundary stencil red and interior stencil green are shown in Figure 8. The graphs of exact and approximate solutions for the parameters α=1.3,α1=1.1, at τ=1 are shown in the Figure 9(a) and Figure 9(b). The results obtained for various values of N,n, and M along the path Γ1 and Γ2 are depicted in Table 7 and Table 8 respectively. From the results one can see that with large number of nodes the proposed method produced accurate results.

    Figure 8.  The regular nodes distribution in rectangular domain with boundary stencil red and interior stencil green.
    Figure 9.  In (a) The plot shows the exact solution. In (b) the plot shows the numerical solution, the parameters used are α=1.3,α1=1.1.
    Table 7.  Numerical solution in the rectangular domain [0,1]2 for α=1.3,α1=1.1, and τ=1 obtained using hyperbolic contour Γ1.
    α=1.3, α1=1.1
    N n M L c κ errorest(Γ1) C.TIME(sec)
    900 14 70 4.26×102 1.1 1.22×10+14 7.20×103 755.137804
    16 1.12×102 1.7 3.71×10+12 7.20×103 756.918480
    18 5.00×103 1.9 1.26×10+12 7.20×103 751.184383
    20 9.22×104 2.4 1.48×10+12 7.20×103 748.664343
    576 20 90 5.15×104 1.9 1.51×10+12 8.18×104 286.863124
    676 5.08×104 2.0 2.19×10+12 8.18×104 421.440624
    784 8.58×104 2.2 1.77×10+12 8.18×104 608.764593
    900 9.22×104 2.4 1.48×10+12 8.18×104 861.916452
    729 20 20 1.10×103 2.1 1.96×10+12 1.55×10+0 22.119245
    30 7.80×104 2.1 1.96×10+12 5.73×101 48.770754
    50 7.57×104 2.1 1.96×10+12 6.25×102 138.310373
    80 7.57×104 2.1 1.96×10+12 2.40×103 386.725304

     | Show Table
    DownLoad: CSV
    Table 8.  Numerical solution in the rectangular domain [0,1]2 for α=1.3,α1=1.1, and τ=1 obtained using Talbot's contour Γ2.
    α=1.3, α1=1.1
    N n M L c κ errorest(Γ2) C.TIME(sec)
    900 20 16 1.98×101 2.4 1.48×10+12 7.76×106 9.513650
    18 2.15×102 2.4 1.48×10+12 5.64×1011 10.640051
    20 2.00×103 2.4 1.48×10+12 4.10×1012 11.661281
    22 8.40×104 2.4 1.48×10+12 2.97×1013 12.824995
    24 9.12×104 2.4 1.48×10+12 2.16×1014 13.560164
    900 14 24 4.28×102 1.1 1.22×10+14 2.16×1014 13.026362
    16 1.12×102 1.7 3.31×10+12 2.16×1014 13.463156
    18 5.00×103 1.9 1.26×10+12 2.16×1014 13.591828
    20 9.12×104 2.4 1.48×10+12 2.16×1014 13.601463
    576 20 22 5.11×104 1.9 1.51×10+12 2.97×1013 4.451590
    676 5.17×104 2.0 2.19×10+12 2.97×1013 6.401350
    784 8.14×104 2.2 1.77×10+12 2.97×1013 9.076440
    900 8.40×104 2.4 1.48×10+12 2.97×1013 12.560123
    961 8.06×104 2.4 2.19×10+12 2.97×1013 14.919706

     | Show Table
    DownLoad: CSV

    Here we solve the given problem in unit circle with center at (χ,ϑ)=(0.5,0.5). The domain is descretized with N uniform nodes. The computational results for different values of N,n, and M along Γ1 and Γ2 are depicted in Table 9 and Table 10 respectively. Figure 10(a) shows the uniform nodes in circular domain, whereas Figure 10(b) shows the absolute error computed along the hyperbolic path. The exact and approximate solutions are presented in Figures 11(a) and Figure 11(b). The proposed method produced results with good accuracy in circular domain.

    Table 9.  Numerical solution in the circular domain for α=1.3,α1=1.1, and τ=1 obtained using hyperbolic contour Γ1.
    α=1.3, α1=1.1
    N n M L c κ errorest(Γ1) C.TIME(sec)
    950 60 15 8.30×103 4.3 3.11×10+12 2.63×10+0 38.116660
    20 1.20×103 4.3 3.11×10+12 1.55×10+0 62.890208
    30 9.23×104 4.3 3.11×10+12 5.37×101 133.694751
    40 9.17×104 4.3 3.11×10+12 1.83×102 369.149122
    900 30 50 1.30×103 3.1 2.82×10+12 6.25×102 361.783503
    40 2.10×103 3.5 5.12×10+12 6.25×102 366.883195
    50 2.20×103 4.1 2.43×10+12 6.25×102 363.312789
    60 9.17×104 4.3 3.11×10+12 6.25×102 367.839901
    300 59 60 9.77×104 4.3 3.06×10+12 2.12×102 553.852222
    550 9.77×104 4.3 3.06×10+12 2.12×102 447.562447
    800 9.77×104 4.3 3.06×10+12 2.12×102 531.883062
    1100 9.77×104 4.3 3.06×10+12 2.12×102 531.921143

     | Show Table
    DownLoad: CSV
    Table 10.  Numerical solution in the circular domain for α=1.5,α1=1.3, and τ=1 obtained using Talbot's contour Γ2.
    α=1.5, α1=1.3
    N n M L c κ errorest(Γ2) C.TIME(sec)
    950 50 18 5.87×102 4.1 2.43×10+12 5.64×1011 9.258343
    20 7.60×103 4.1 2.43×10+12 4.10×1012 9.959348
    22 2.40×103 4.1 2.43×10+12 2.97×1013 10.597861
    24 1.90×103 4.1 2.43×10+12 2.16×1014 11.241704
    1050 10 26 9.43×102 1.2 7.70×10+13 1.57×1015 9.389988
    30 1.20×103 3.1 2.82×10+12 1.57×1015 10.100778
    50 1.90×103 4.1 2.43×10+12 1.57×1015 11.882140
    60 8.84×104 4.3 3.11×10+12 1.57×1015 13.157724
    750 59 28 9.36×104 4.3 3.06×10+12 1.14×1016 13.662250
    1150 9.36×104 4.3 3.06×10+12 1.14×1016 13.759729
    1250 9.36×104 4.3 3.06×10+12 1.14×1016 13.572409

     | Show Table
    DownLoad: CSV
    Figure 10.  In (a) The regular nodes distribution in circular domain are shown. In (b) the plot shows the absolute error for the parameters values α=1.7,α1=1.5,N=900,n=50, and M=90 along the hyperbolic contour Γ1.
    Figure 11.  In (a) The plot shows the exact solution. In (b) the plot shows the numerical solution, the parameters used are α=1.5,α1=1.3.

    In the last test problem we have considered the complex shape domain. The domain is generated by rd=1d[1+2d+d2(d+1)cos(dθ)],d=4. In this experiment also we have used the contours Γ1 and Γ2 with the same set of optimal parameters used in Problem 1. The results obtained for fractional orders α=1.5,α1=1.3, and various nodes N in the global domain and n in the local domain and quadrature points along the contour Γ1 and Γ2 are shown in Table 11 and Table 12 respectively. The regular nodes distribution in the complex domain are shown in Figure 12(a), whereas the approximate and exact solutions are presented in Figures 12(b). Figure 13 shows the absolute error obtained using the Talbots contour. It can be seen that the proposed numerical method produced very accurate and stable results in the complex domain, this confirms the efficiency of the method for such type of equations.

    Table 11.  Numerical solution in the circular domain for α=1.5,α1=1.3, and τ=1 obtained using hyperbolic contour Γ1.
    α=1.5, α1=1.3
    N n M L c κ errorest(Γ1) C.TIME(sec)
    851 50 40 4.70×103 4.0 1.13×10+12 1.83×101 50.524737
    50 6.75×104 4.0 1.13×10+12 6.25×102 79.388630
    60 6.74×104 4.0 1.13×10+12 2.12×102 116.511151
    80 6.74×104 4.0 1.13×10+12 2.40×103 227.719695
    852 30 70 1.30×103 2.9 6.40×10+12 7.20×103 147.655977
    40 1.20×103 3.4 5.83×10+12 7.20×103 158.960295
    50 9.34×104 3.9 1.82×10+12 7.20×103 219.043351
    60 9.47×104 4.2 1.92×10+12 7.20×103 268.501711
    457 60 60 6.97×104 3.0 3.07×10+12 2.12×102 68.420275
    542 4.39×104 3.3 1.62×10+12 2.12×102 89.095051
    643 4.86×104 3.6 1.79×10+12 2.12×102 117.459116
    851 7.29×104 4.2 1.90×10+12 2.12×102 184.241136

     | Show Table
    DownLoad: CSV
    Table 12.  Numerical solution in the circular domain for α=1.5,α1=1.3, and τ=1 obtained using Talbot's contour Γ2.
    α=1.5, α1=1.3
    N n M L c κ errorest(Γ2) C.TIME(sec)
    752 30 24 1.10×103 2.7 3.89×10+12 2.16×1014 3.188624
    850 1.80×103 2.9 7.98×10+12 2.16×1014 3.613122
    921 1.50×103 3.0 2.48×10+13 2.16×1014 4.028301
    974 9.61×104 3.2 5.64×10+12 2.16×1014 4.297978
    1020 28 22 2.90×103 2.9 3.79×10+13 2.97×1013 4.261794
    40 1.90×103 3.7 2.62×10+13 2.97×1013 5.485432
    50 1.00×103 4.2 1.13×10+13 2.97×1013 6.750489
    60 9.32×104 4.6 6.25×10+12 2.97×1013 8.366275
    1095 70 18 6.47×102 5.1 2.15×10+12 5.64×1011 10.452683
    20 6.60×103 5.1 2.15×10+12 4.10×1012 10.778798
    22 9.01×104 5.1 2.15×10+12 2.97×1013 11.238818
    24 9.97×104 5.1 2.15×10+12 2.16×1014 11.540931

     | Show Table
    DownLoad: CSV
    Figure 12.  In (a) The regular nodes distribution in complex domain is shown. In (b) the plot shows the numerical solution and exact solutions, the parameters used are α=1.5,α1=1.3.
    Figure 13.  The plot shows absolute error obtained using Talbot's contour Γ2 for the parameter values N=993,n=30,M=24,α=1.5, and α1=1.3, at τ=1.

    In this work, a local meshless method based on Laplace transform has been utilized for the approximation of the numerical solution of 1D and 2D multi-term time fractional wave diffusion equations. We resolved the issue of time-instability which is the common short coming of time-stepping methods using the Laplace transformation, and the issues of ill-conditioning due to dense differentiation matrices and shape parameter sensitivity with localized meshless method. The stability and convergence of the method are discussed. To verify the theoretical results some test problem in 1D and a test problem in 2D are considered. For the two dimensional problem we have considered rectangular, circular, and complex domains. For numerical inversion of Laplace transform we have utilized two types of contours the hyperbolic and the improved Talbot's contour. The results obtained using these two contours were accurate and stable. However, the results show that the Talbot's contour is more efficient computationally. The benefit of this method is that it can approximate such type equations very efficiently and accurately with less computation time, and without any time instability. The obtained results proves the simplicity in implementation, efficiency, accuracy, and stability of the proposed method.

    This work was supported by “the Construction team project of the introduction and cultivation of young innovative talents in Colleges and universities of Shandong Province of China, 2019 (Project Name: Big data and business intelligence social service innovation team)” and “the Social Science Planning Project of Qingdao, China,2018 (Grant No. QDSKL1801229)”.

    The authors declare that no competing interests exist.

    First and second authors revised the paper, solved the examples and used software to compute and sketch the results. Third author did analysis and wrote the paper. Forth proposed the problem and verified the results.

    [1] Raminosoa T, Torrey DA, El-Refaie AM, et al. (2016) Sinusoidal reluctance machine with DC winding : An attractive non-permanent-magnet option. IEEE T Ind Appl 52: 2129-2137.
    [2] Shi JT, Zhu ZQ (2015) Analysis of novel multi-tooth variable flux reluctance machines with different stator and rotor pole combinations. IEEE T Magn 51: 1-11.
    [3] Liu X, Zhu ZQ, Hasegawa M, et al. (2012) Vibration and noise in novel variable flux reluctance machine with DC-field coil in stator. Proceedings of the 7th International Power Electronics and Motion Control Conference 2: 1100-1107.
    [4] Fukami T, Matsuura Y, Shima K, et al. (2010) Development of a low-speed multi-pole synchronous machine with a field winding on the stator side. The XIX International Conference on Electrical Machines - ICEM 2010 Rome: 1-6.
    [5] Fukami T, Aoki H, Shima H, et al. (2012) Assessment of core losses in a flux-modulating synchronous machine. IEEE T Ind Appl 48: 603-611. doi: 10.1109/TIA.2011.2180286
    [6] Fukami T, Matsuura Y, Shima K, et al. (2012) A multipole synchronous machine with nonoverlapping concentrated armature and field windings on the stator. IEEE T Ind Electron 59: 2583-2591. doi: 10.1109/TIE.2011.2157293
    [7] Fukami T, Ueno Y, Shima K (2015) Magnet arrangement in novel flux-modulating synchronous machines with permanent magnet excitation. IEEE T Magn 51: 1-4.
    [8] Liu X, Zhu ZQ (2013) Electromagnetic performance of novel variable flux reluctance machines with DC-field coil in stator. IEEE T Magn 49: 3020-3028. doi: 10.1109/TMAG.2012.2235182
    [9] Shi JT, Liu X, Wu D, et al. (2014) Influence of stator and rotor pole arcs on electromagnetic torque of variable flux reluctance machines. IEEE T Magn 50: 1-4.
    [10] Liu X, Zhu ZQ (2014) Stator/rotor pole combinations and winding configurations of variable flux reluctance machines. IEEE T Ind Appl 50: 3675-3684. doi: 10.1109/TIA.2014.2315505
    [11] Zhu ZQ, Liu X (2015) Novel stator electrically field excited synchronous machines without rare-earth magnet. IEEE T Magn 51: 8103609.
    [12] Azar Z, Zhu ZQ (2014) Performance analysis of synchronous reluctance machines having nonoverlapping concentrated winding and sinusoidal bipolar with DC bias excitation. IEEE T Ind Appl 50: 3346-3356. doi: 10.1109/TIA.2014.2311503
    [13] Tang YQ (1998) In: Electromagnetic fields in Electrical machines-Second edition. Beijing, CN: Science Press, 341-355.
    [14] Ou J, Doppelbauer M (2016) Torque analysis and comparison of the switched reluctance machine and the doubly-salient permanent magnet machine, 2016 18th European Conference on Power Electronics and Applications (EPE'16 ECCE Europe) Karlsruhe: 1-11.
    [15] Williams FC, Mamak RS (1962) Electromagnetic forces in slotted structures. Proceeding of the IET-Part C: Monographs 109: 11-17. doi: 10.1049/pi-c.1962.0002
    [16] Robert Pohl (1946) Theory of pulsating-field machines. Journal of the Institution of Electrical Engineers-Part II: Power Engineering 93: 37-47. doi: 10.1049/ji-2.1946.0006
  • This article has been cited by:

    1. Kamran Kamran, Zahir Shah, Poom Kumam, Nasser Aedh Alreshidi, A Meshless Method Based on the Laplace Transform for the 2D Multi-Term Time Fractional Partial Integro-Differential Equation, 2020, 8, 2227-7390, 1972, 10.3390/math8111972
    2. Abdul Ghafoor, Nazish Khan, Manzoor Hussain, Rahman Ullah, A hybrid collocation method for the computational study of multi-term time fractional partial differential equations, 2022, 128, 08981221, 130, 10.1016/j.camwa.2022.10.005
    3. Siraj Ahmad, Kamal Shah, Thabet Abdeljawad, Bahaaeldin Abdalla, On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods, 2023, 135, 1526-1506, 2743, 10.32604/cmes.2023.023705
    4. A. Ahmadian, M. Salimi, S. Salahshour, Local RBF Method for Transformed Three Dimensional Sub-Diffusion Equations, 2022, 8, 2349-5103, 10.1007/s40819-022-01338-w
    5. Monireh Nosrati Sahlan, Hojjat Afshari, Jehad Alzabut, Ghada Alobaidi, Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions, 2021, 5, 2504-3110, 212, 10.3390/fractalfract5040212
    6. Ujala Gul, Zareen A. Khan, Salma Haque, Nabil Mlaiki, A Local Radial Basis Function Method for Numerical Approximation of Multidimensional Multi-Term Time-Fractional Mixed Wave-Diffusion and Subdiffusion Equation Arising in Fluid Mechanics, 2024, 8, 2504-3110, 639, 10.3390/fractalfract8110639
    7. Aisha Subhan, Kamal Shah, Suhad Subhi Aiadi, Nabil Mlaiki, Fahad M. Alotaibi, Abdellatif Ben Makhlouf, Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator, 2023, 2023, 1099-0526, 1, 10.1155/2023/7210126
    8. Tao Hu, Cheng Huang, Sergiy Reutskiy, Jun Lu, Ji Lin, A Novel Accurate Method for Multi-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains, 2024, 138, 1526-1506, 1521, 10.32604/cmes.2023.030449
  • Reader Comments
  • © 2017 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(5269) PDF downloads(1277) Cited by(2)

Figures and Tables

Figures(17)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog