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Research article

Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria

  • Received: 27 October 2020 Accepted: 31 December 2020 Published: 06 January 2021
  • MSC : 26A33, 34K37

  • This paper inspects the issues of synchronization stability and robust synchronization stability for fractional order coupled complex interconnected Cohen-Grossberg neural networks under linear coupling delays. For investigation of synchronization stability results, the comparison theorem for multiple delayed fractional order linear system is derived at first. Then, by means of given fractional comparison principle, some inequality methods, Kronecker product technique and classical Lyapunov-functional, several asymptotical synchronization stability criteria are addressed in the voice of linear matrix inequality (LMI) for the proposed model. Moreover, when parameter uncertainty exists, we also the investigate on the robust synchronization stability criteria for complex structure on linear coupling delayed Cohen-Grossberg type neural networks. At last, the validity of the proposed analytical results are performed by two computer simulations.

    Citation: Pratap Anbalagan, Evren Hincal, Raja Ramachandran, Dumitru Baleanu, Jinde Cao, Chuangxia Huang, Michal Niezabitowski. Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria[J]. AIMS Mathematics, 2021, 6(3): 2844-2873. doi: 10.3934/math.2021172

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  • This paper inspects the issues of synchronization stability and robust synchronization stability for fractional order coupled complex interconnected Cohen-Grossberg neural networks under linear coupling delays. For investigation of synchronization stability results, the comparison theorem for multiple delayed fractional order linear system is derived at first. Then, by means of given fractional comparison principle, some inequality methods, Kronecker product technique and classical Lyapunov-functional, several asymptotical synchronization stability criteria are addressed in the voice of linear matrix inequality (LMI) for the proposed model. Moreover, when parameter uncertainty exists, we also the investigate on the robust synchronization stability criteria for complex structure on linear coupling delayed Cohen-Grossberg type neural networks. At last, the validity of the proposed analytical results are performed by two computer simulations.



    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=sup (4.2)

    for any discrete norm \|.\| defined on R^{N} the constant \mathcal{C} is finite. From (4.2) we may write

    \begin{equation} \|\mathit{\boldsymbol{\mathcal{M}}}\|^{-1} \leq \frac{\|\mathit{\boldsymbol{\hat{\mathcal{U}}}}\|}{\|\mathit{\boldsymbol{\mathcal{M}}}\mathit{\boldsymbol{\hat{\mathcal{U}}}}\| }\leq \mathcal{C}, \end{equation} (4.3)

    Similarly for the pseudoinverse \mathit{\boldsymbol{\mathcal{M}}}^{†} of \mathit{\boldsymbol{\mathcal{M}}} , we can write

    \begin{equation} \|\mathit{\boldsymbol{\mathcal{M}}}^{†}\| = \sup\limits_{\mathit{\boldsymbol{\mathcal{H}}} \neq 0} \frac{\|\mathit{\boldsymbol{\mathcal{M}}}^{†} \mathit{\boldsymbol{\mathcal{H}}}\|}{\|\mathit{\boldsymbol{\mathcal{H}}}\|}. \end{equation} (4.4)

    Thus we have

    \begin{equation} \|\mathit{\boldsymbol{\mathcal{M}}}^{†}\| \geq \sup\limits_{\mathit{\boldsymbol{\mathcal{H}}} = \mathit{\boldsymbol{\mathcal{M}}}\hat{\mathit{\boldsymbol{\mathcal{H}}}}\neq 0}\frac{\|\mathit{\boldsymbol{\mathcal{M}}}^{†}\mathit{\boldsymbol{\mathcal{M}}}\hat{\mathit{\boldsymbol{\mathcal{U}}}}\| }{\|\mathit{\boldsymbol{\mathcal{M}}}\mathit{\boldsymbol{\hat{\mathcal{U}}}}\| } = \sup\limits_{\mathit{\boldsymbol{\hat{\mathcal{U}}}} \neq 0} \frac{\|\mathit{\boldsymbol{\hat{\mathcal{U}}}}\| }{\|\mathit{\boldsymbol{\mathcal{M}}}\hat{\mathit{\boldsymbol{\mathcal{U}}}}\| } = \mathcal{C}. \end{equation} (4.5)

    We can see that Eqs (4.3) and (4.5) confirms the bounds for the stability constant \mathcal{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 \|\mathit{\boldsymbol{\mathcal{M}}}^{-1}\|_{\infty} in case of square systems, thus we have

    \begin{equation} \mathcal{C} = \frac{condest(\mathit{\boldsymbol{\mathcal{M}}}')}{\|\mathit{\boldsymbol{\mathcal{M}}}\|_{\infty}} \end{equation} (4.6)

    This work well with less number of computations for our sparse differentiation matrix \mathit{\boldsymbol{\mathcal{M}}} . Figure 1(a) shows the bounds for the constant \mathcal{C} of our system (2.8) and (2.9) for Problem 1 using the Talbot's contour \Gamma_2 . Selecting N = 80 , M = 80 , n = 15 , and \alpha = 1.8, \; \alpha_1 = 1.7, \; \alpha_2 = 1.6, \; c = 0.6 at \tau = 1 , we have 1.00\leq \mathcal{C} \leq 4.5501. 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 \bm{\mathcal{M}} corresponding to Problem 1, using the Talbot's contour \Gamma_2. 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 1-D and 2-D linear multi term wave-diffusion equations. In our numerical examples we have utilized the multi-quadrics(MQ) kernel function \phi (r, \varepsilon) = (1+(\varepsilon r)^2)^{\frac{1}{2}} . We have used the uncertainty principal due to [47] for optimization of the shape parameter. The accuracy of the method is measured using L_{\infty} error defined by

    L_{\infty} = \|\mathcal{U}(\mathit{\boldsymbol{\chi}},\tau)-\mathcal{U}_{k}(\mathit{\boldsymbol{\chi}},\tau)\|_{\infty} = \max\limits_{1 \leq j \leq N}(|\mathcal{U}(\mathit{\boldsymbol{\chi}},\tau)-\mathcal{U}_{k}(\mathit{\boldsymbol{\chi}},\tau)|)

    is used. Here \mathcal{U}_{k} and \mathcal{U} are the numerical and exact solutions respectively.

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

    \begin{align} &D^{\alpha}_{\tau} \mathcal{U}(\chi,\tau)+D^{\alpha_1}_{\tau} \mathcal{U}(\chi,\tau)+D^{\alpha_2}_{\tau} \mathcal{U}(\chi,\tau)-D^{2}_{\chi} \mathcal{U}(\chi,\tau) = f(\chi,\tau), \end{align} (5.1)

    where

    \begin{align*} f(\chi,\tau)& = \left(\frac{6\tau^{3-\alpha}}{\Gamma(4-\alpha)}+\frac{6\tau^{3-\alpha_{1}}}{\Gamma(4-\alpha_{1})}+ \frac{6\tau^{3-\alpha_{2}}}{\Gamma(4-\alpha_{2})}\right)\frac{1}{\cosh(\chi-c)}\\&- \left(\frac{2\tau^{3}}{cosech(c-\chi)^{2}\cosh(c-\chi)^{3}}-\frac{\tau^{3}}{\cosh(c-\chi)}\right). \end{align*}

    The exact solution of the problem is

    \begin{equation*} \mathcal{U}(\chi,\tau) = \frac{\tau^{3}}{\cosh(\chi-c)},\; c\in \mathbb{R}. \end{equation*}

    The boundary and initial conditions are

    \begin{align} \mathcal{U}(-10,\tau)& = \frac{\tau^{3}}{\cosh(-10-c)}, \; \mathcal{U}(10,\tau) = \frac{\tau^{3}}{\cosh(10-c)}, \end{align} (5.2)

    and

    \begin{equation} \mathcal{U}_{0}(\chi) = \mathcal{U}_{1}(\chi) = 0. \end{equation} (5.3)

    The points along the hyperbolic contour \Gamma_1 are calculated using the statement \xi = -M:k:M, and along Talbot's contour \Gamma_2 using the relation \xi_j = -\pi+(j-\frac{1}{2})k, \; \text{where}\; j = 1:M, \; \text{and}\; k = \frac{2\pi}{M}. The parameters used in our computations for the contour \Gamma_1 are \theta = 0.10, \; \eta = 0.15410, \; \tau_1 = \frac{t_0}{T}, \; r_1 = 0.13870, \; \bar{r} = 2r_1 \pi, \; \Upsilon = 2.0 The results obtained for the parameters \alpha, \; \alpha_1, \; \alpha_2, and c along the contour \Gamma_1 are displayed in Table 1, and along \Gamma_2 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 \alpha_j. 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 \tau = 1 obtained using hyperbolic contour \Gamma_1 .
    \alpha=1.5 , \alpha_{1}=1.4 , \alpha_{2}=1.3 , c=0.5
    N n M L_{\infty} c \kappa error_{est}(\Gamma_1) C.TIME(sec)
    80 20 35 7.77 \times10^{-5} 10.0 1.13 \times10^{+12} 3.14 \times10^{-1} 0.343148
    55 7.61 \times10^{-5} 10.0 1.13 \times10^{+12} 3.64 \times10^{-2} 1.222487
    75 7.61 \times10^{-5} 10.0 1.13 \times10^{+12} 4.2 \times10^{-3} 5.000861
    95 7.61 \times10^{-5} 10.0 1.13 \times10^{+12} 4.75 \times10^{-4} 12.067666
    110 7.61 \times10^{-5} 10.0 1.13 \times10^{+12} 9.30 \times10^{-5} 22.244950
    40 15 80 5.76 \times10^{-5} 4.5 1.37 \times10^{+12} 2.4 \times10^{-3} 2.034147
    50 4.70 \times10^{-5} 5.7 1.17 \times10^{+12} 2.4 \times10^{-3} 3.430185
    60 4.32 \times10^{-5} 6.9 1.05 \times10^{+12} 2.4 \times10^{-3} 4.307865
    70 9.09 \times10^{-5} 8.0 1.24 \times10^{+12} 2.4 \times10^{-3} 5.308744
    80 4.82 \times10^{-5} 9.2 1.14 \times10^{+12} 2.4 \times10^{-3} 6.327198
    70 12 90 2.33 \times10^{-5} 7.3 1.02 \times10^{+12} 8.18 \times10^{-4} 8.646889
    15 9.09 \times10^{-5} 8.0 1.24 \times10^{+12} 8.18 \times10^{-4} 8.534781
    18 8.80 \times10^{-5} 8.5 1.17 \times10^{+12} 8.18 \times10^{-4} 8.898832
    21 4.10 \times10^{-5} 8.8 1.20 \times10^{+12} 8.18 \times10^{-4} 8.901062
    24 9.30 \times10^{-5} 9.0 1.26 \times10^{+12} 8.18 \times10^{-4} 8.835825
    [14] 4.79 \times10^{-6}

     | Show Table
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    Table 2.  Numerical solution in the domain [0, 1] and \tau = 1 obtained using Talbot's contour \Gamma_2 .
    \alpha=1.8 , \alpha_{1}=1.7 , \alpha_{2}=1.6 , c=0.5
    N n M L_{\infty} c \kappa error_{est}(\Gamma_2) C.TIME(sec)
    70 12 10 3.51 \times10^{-1} 7.3 1.02 \times10^{+12} 2.02 \times10^{-6} 0.201190
    12 3.98 \times10^{-2} 7.3 1.02 \times10^{+12} 1.47 \times10^{-7} 0.202626
    14 4.20 \times10^{-3} 7.3 1.02 \times10^{+12} 1.06 \times10^{-8} 0.213661
    16 4.16 \times10^{-4} 7.3 1.02 \times10^{+12} 7.76 \times10^{-10} 0.203777
    18 4.99 \times10^{-5} 7.3 1.02 \times10^{+12} 5.64 \times10^{-11} 0.204327
    30 15 20 2.97 \times10^{-5} 3.4 1.00 \times10^{+12} 4.10 \times10^{-12} 0.203400
    40 5.99 \times10^{-5} 4.5 1.37 \times10^{+12} 4.10 \times10^{-12} 0.212062
    50 4.82 \times10^{-5} 5.7 1.17 \times10^{+12} 4.10 \times10^{-12} 0.210793
    70 9.41 \times10^{-5} 8.0 1.24 \times10^{+12} 4.10 \times10^{-12} 0.222508
    90 8.14 \times10^{-5} 10.4 1.07 \times10^{+12} 4.10 \times10^{-12} 0.217812
    80 12 20 7.56 \times10^{-5} 8.3 1.15 \times10^{+12} 4.10 \times10^{-12} 0.211422
    14 7.53 \times10^{-5} 9.0 1.04 \times10^{+12} 4.10 \times10^{-12} 0.218499
    16 6.68 \times10^{-5} 9.4 1.17 \times10^{+12} 4.10 \times10^{-12} 0.224207
    18 4.87 \times10^{-5} 9.8 1.02 \times10^{+12} 4.10 \times10^{-12} 0.226878
    20 4.67 \times10^{-5} 10.0 1.13 \times10^{+12} 4.10 \times10^{-12} 0.239581
    [14] 5.69 \times10^{-5}

     | 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 \alpha = 1.5, \; \alpha_1 = 1.4, \; \alpha_2 = 1.3, \; c = 0.5 , N = 70, \; n = 12 , and M = 90 , along the hyperbolic contour \Gamma_1 .
    Figure 3.  In (a) the Absolute error and error_est (\Gamma_1) are shown corresponding to problem 1 using N = 40, n = 15, \alpha = 1.8, \alpha_1 = 1.7, \alpha_2 = 1.6, \; c = 0.5, the results confirms a good agreement between them. In (b) the error functions for different \alpha, and \alpha_j on [0, 1] are shown using the hyperbolic contour \Gamma_1 .

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

    \begin{eqnarray} D^{\alpha}_{\tau}\mathcal{U}(\chi,\tau)+ D^{\alpha_{1}}_{\tau}\mathcal{U}(\chi,\tau)-D^{2}_{\chi} \mathcal{U}(\chi,\tau) = f(\chi,\tau), \end{eqnarray} (5.4)

    where

    \begin{eqnarray*} f(\chi,\tau) = \left(\frac{6\tau^{3-\alpha}}{\Gamma(4-\alpha)}+\frac{6\tau^{3-\alpha_{1}}}{\Gamma(4-\alpha_{1})}+\pi^{2}\tau^{3}\right)\sin(\pi \chi). \end{eqnarray*}

    The exact solution of the problem is

    \begin{equation*} \mathcal{U}(\chi,\tau) = \sin(\pi \chi)\tau^{3}. \end{equation*}

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

    \begin{equation} \mathcal{U}(0,\tau) = \mathcal{U}(1,\tau) = 0 \end{equation} (5.5)

    and initial conditions

    \begin{equation} \mathcal{U}_{0}(\chi) = \mathcal{U}_{1}(\chi) = 0. \end{equation} (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 \alpha , and \; \alpha_1. are displayed in Table 3 along the path \Gamma_1 , and in Table 4 along the path \Gamma_2 . 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 \alpha, and \alpha_1 . 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 \tau = 1 hyperbolic contour \Gamma_1 .
    \alpha=1.9 , \alpha_{1}=1.3
    N n M L_{\infty} c \kappa error_{est}(\Gamma_1) C.TIME(sec)
    60 22 40 8.44 \times10^{-5} 7.6 1.18 \times10^{+12} 1.83 \times10^{-1} 0.667872
    60 6.62 \times10^{-5} 7.6 1.18 \times10^{+12} 2.12 \times10^{-2} 1.580686
    80 8.62 \times10^{-5} 7.6 1.18 \times10^{+12} 2.4 \times10^{-3} 7.362587
    90 8.62 \times10^{-5} 7.6 1.18 \times10^{+12} 8.18 \times10^{-4} 10.843996
    100 8.62 \times10^{-5} 7.6 1.18 \times10^{+12} 2.76 \times10^{-4} 15.784191
    70 10 100 5.52 \times10^{-5} 6.4 1.15 \times10^{+12} 2.76 \times10^{-4} 17.932164
    12 2.74 \times10^{-5} 7.3 1.02 \times10^{+12} 2.76 \times10^{-4} 17.859075
    14 3.44 \times10^{-5} 7.8 1.21 \times10^{+12} 2.76 \times10^{-4} 18.129870
    18 5.21 \times10^{-5} 8.5 1.17 \times10^{+12} 2.76 \times10^{-4} 18.278882
    22 7.64 \times10^{-5} 8.9 1.15 \times10^{+12} 2.76 \times10^{-3} 18.479720
    50 25 90 9.85 \times10^{-5} 6.5 1.01 \times10^{+12} 8.18 \times10^{-4} 8.730629
    60 1.03 \times10^{-4} 7.8 1.09 \times10^{+12} 8.18 \times10^{-4} 11.049126
    70 5.40 \times10^{-5} 9.1 1.14 \times10^{+12} 8.18 \times10^{-4} 13.349187
    80 6.57 \times10^{-5} 10.4 1.19 \times10^{+12} 8.18 \times10^{-4} 15.124607
    90 6.17 \times10^{-5} 11.8 1.02 \times10^{+12} 8.18 \times10^{-4} 17.134190
    [14] 7.0080 \times10^{-4}

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical solution in the domain [0, 1] and \tau = 1 obtained using Talbot's contour \Gamma_2 .
    \alpha=1.7 , \alpha_{1}=1.2
    N n M L_{\infty} c \kappa error_{est}(\Gamma_2) C.TIME(sec)
    70 12 10 2.56 \times10^{-1} 7.3 1.02 \times10^{+12} 2.02 \times10^{-6} 0.202184
    12 2.91 \times10^{-2} 7.3 1.02 \times10^{+12} 1.47 \times10^{-7} 0.197720
    14 3.10 \times10^{-3} 7.3 1.02 \times10^{+12} 1.06 \times10^{-8} 0.200589
    16 3.25 \times10^{-4} 7.3 1.02 \times10^{+12} 7.76 \times10^{-10} 0.202710
    18 5.65 \times10^{-5} 7.3 1.02 \times10^{+12} 5.64 \times10^{-11} 0.194221
    30 15 20 3.74 \times10^{-5} 3.4 1.00 \times10^{+12} 4.10 \times10^{-12} 0.202098
    40 6.97 \times10^{-5} 4.5 1.37 \times10^{+12} 4.10 \times10^{-12} 0.201783
    50 6.68 \times10^{-5} 5.7 1.17 \times10^{+12} 4.10 \times10^{-12} 0.204199
    70 1.00 \times10^{-4} 8.0 1.24 \times10^{+12} 4.10 \times10^{-12} 0.205864
    90 1.36 \times10^{-4} 10.4 1.07 \times10^{+12} 4.10 \times10^{-12} 0.216570
    80 12 20 9.70 \times10^{-5} 8.3 1.15 \times10^{+12} 4.10 \times10^{-12} 0.207446
    14 4.34 \times10^{-5} 9.0 1.04 \times10^{+12} 4.10 \times10^{-12} 0.206234
    16 8.47 \times10^{-5} 9.4 1.17 \times10^{+12} 4.10 \times10^{-12} 0.216176
    18 4.76 \times10^{-5} 9.8 1.02 \times10^{+12} 4.10 \times10^{-12} 0.215397
    20 5.66 \times10^{-5} 10.0 1.13 \times10^{+12} 4.10 \times10^{-12} 0.244942
    [14] 1.39 \times10^{-4}

     | 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 \alpha = 1.9, \; \alpha_1 = 1.7, \; N = 70, \; n = 12, \; M = 90 on [-5, 5], using the hyperbolic contour \Gamma_1 .
    Figure 5.  In (a) Absolute error and error_est for problem 2 are presented using N = 50, n = 10, \alpha = 1.7, \alpha_1 = 1.2 , the results confirms a good agreement between them. In (b) Error functions for different \alpha, and \alpha_j on [0, 1], are shown using the hyperbolic contour \Gamma_1 . The figure shows that the error decreases with increasing the values of fractional orders \alpha, and \alpha_1 .

    We consider the following fractional equation

    \begin{eqnarray} D^{\alpha}_{\tau}\mathcal{U}(\chi,\tau)+ D^{\alpha_{1}}_{\tau}\mathcal{U}(\chi,\tau)-D^{2}_{\chi} \mathcal{U}(\chi,\tau) = f(\chi,\tau), \end{eqnarray} (5.7)

    where

    \begin{eqnarray*} f(\chi,\tau)& = &\left(\frac{2 \tau^{2-\alpha}}{\cos(\chi)\Gamma(3-\alpha)}+\frac{2\tau^{2-\alpha_{1}}}{\cos(\chi)\Gamma(3-\alpha_{1})}\right) -\left(\frac{\tau^{2}}{\cos(\chi)}+\frac{2\tau^{2}}{cosec(\chi)^{2}\cos(\chi)^{3}} \right). \end{eqnarray*}

    The exact solution of the problem is

    \begin{equation*} \mathcal{U}(\chi,\tau) = \frac{\tau^{2}}{\cos(\chi)}. \end{equation*}

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

    \begin{equation} \mathcal{U}(0,\tau) = \tau^{2}, \; \mathcal{U}(1,\tau) = \frac{\tau^{2}}{\cos(1)} \end{equation} (5.8)

    and initial conditions

    \begin{equation} \mathcal{U}_{0}(\chi) = 0,\; \mathcal{U}_{1}(\chi) = 0. \end{equation} (5.9)

    The results obtained for third test problem with fractional orders \alpha, and \alpha_1 along the hyperbolic contour \Gamma_1 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 \Gamma_1 and \Gamma_2 respectively.

    Table 5.  Numerical solution in the domain [0, 1] and \tau = 1 hyperbolic contour \Gamma_1 .
    \alpha=1.9 , \alpha_{1}=1.8 ,
    N n M L_{\infty} c \kappa error_{est}(\Gamma_1) C.TIME(sec)
    80 25 50 4.20 \times10^{-5} 10.4 1.19 \times10^{+12} 6.25 \times10^{-2} 2.249386
    60 3.85 \times10^{-5} 10.4 1.19 \times10^{+12} 2.12 \times10^{-2} 4.114330
    70 3.86 \times10^{-5} 10.4 1.19 \times10^{+12} 7.2 \times10^{-3} 7.301508
    90 3.86 \times10^{-5} 10.4 1.19 \times10^{+12} 8.18 \times10^{-4} 15.044453
    100 3.86 \times10^{-5} 10.4 1.19 \times10^{+12} 2.76 \times10^{-4} 21.588645
    60 27 90 9.51 \times10^{-5} 7.9 1.06 \times10^{+12} 8.18 \times10^{-4} 11.338375
    70 8.41 \times10^{-5} 9.2 1.16 \times10^{+12} 8.18 \times10^{-4} 13.565846
    80 1.15 \times10^{-4} 10.6 1.01 \times10^{+12} 8.18 \times10^{-4} 15.034979
    90 9.66 \times10^{-5} 11.9 1.10 \times10^{+12} 8.18 \times10^{-4} 17.556509
    100 7.93 \times10^{-5} 13.2 1.16 \times10^{+12} 8.18 \times10^{-4} 21.605433
    85 20 95 8.45 \times10^{-5} 10.6 1.21 \times10^{+12} 4.75 \times10^{-4} 18.828414
    22 5.73 \times10^{-5} 10.9 1.01 \times10^{+12} 4.75 \times10^{-4} 19.612251
    24 1.68 \times10^{-4} 11.0 1.16 \times10^{+12} 4.75 \times10^{-4} 19.670357
    27 4.26 \times10^{-5} 11.2 1.16 \times10^{+12} 4.75 \times10^{-4} 19.563284
    30 5.36 \times10^{-5} 11.4 1.09 \times10^{+12} 4.75 \times10^{-4} 20.041057
    [14] 8.81 \times10^{-5}

     | Show Table
    DownLoad: CSV
    Table 6.  Numerical solution in the domain [0, 1] and \tau = 1 obtained using Talbot's contour \Gamma_2 .
    \alpha=1.9 , \alpha_{1}=1.8 ,
    N n M L_{\infty} c \kappa error_{est}(\Gamma_2) C.TIME(sec)
    70 12 10 3.54 \times10^{-2} 7.3 1.02 \times10^{+12} 2.02 \times10^{-6} 0.142848
    12 3.40 \times10^{-3} 7.3 1.02 \times10^{+12} 1.47 \times10^{-7} 0.133224
    14 2.83 \times10^{-4} 7.3 1.02 \times10^{+12} 1.06 \times10^{-8} 0.134380
    16 4.64 \times10^{-5} 7.3 1.02 \times10^{+12} 7.76 \times10^{-10} 0.133445
    18 5.19 \times10^{-5} 7.3 1.02 \times10^{+12} 5.64 \times10^{-11} 0.136420
    30 15 20 1.15 \times10^{-4} 3.4 1.00 \times10^{+12} 4.10 \times10^{-12} 0.135380
    40 5.10 \times10^{-5} 4.5 1.37 \times10^{+12} 4.10 \times10^{-12} 0.142579
    50 1.07 \times10^{-4} 5.7 1.17 \times10^{+12} 4.10 \times10^{-12} 0.137483
    70 1.44 \times10^{-4} 8.0 1.24 \times10^{+12} 4.10 \times10^{-12} 0.145392
    90 1.49 \times10^{-4} 10.4 1.07 \times10^{+12} 4.10 \times10^{-12} 0.151340
    80 12 20 1.29 \times10^{-4} 8.3 1.15 \times10^{+12} 4.10 \times10^{-12} 0.144167
    14 1.22 \times10^{-4} 9.0 1.04 \times10^{+12} 4.10 \times10^{-12} 0.139500
    16 8.31 \times10^{-5} 9.4 1.17 \times10^{+12} 4.10 \times10^{-12} 0.148407
    18 4.78 \times10^{-5} 9.8 1.02 \times10^{+12} 4.10 \times10^{-12} 0.156982
    20 9.51 \times10^{-5} 10.0 1.13 \times10^{+12} 4.10 \times10^{-12} 0.166119
    [14] 8.81 \times10^{-5}

     | 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 \alpha = 1.9, \; \alpha_1 = 1.8, \; N = 70, \; n = 15, \; M = 80 on [0, 1] , using the hyperbolic contour \Gamma_1 .
    Figure 7.  In (a) absolute error and error_{est}(\Gamma_1) are shown corresponding to problem 3 using N = 50, n = 10, \alpha = 1.9, \alpha_1 = 1.8 , the results confirms a good agreement between them. In (b) absolute error and error_{est}(\Gamma_2) are shown for the parameter values \alpha = 1.9, \; \alpha_1 = 1.8, \; N = 70, \; n = 12 on [0, 1] .

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

    \begin{eqnarray} D^{\alpha}_{\tau}\mathcal{U}(\chi,\vartheta,\tau)+D^{\alpha_{1}}_{\tau}\mathcal{U}(\chi,\vartheta,\tau)-\Delta \mathcal{U}(\chi,\vartheta,\tau) = f(\chi,\vartheta,\tau), \end{eqnarray} (5.10)

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

    \mathcal{U}(\chi,\vartheta,\tau) = e^{\chi+\vartheta}\tau^{2+\alpha+\alpha_{1}}

    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 \Gamma_1 and Talbot's contour \Gamma_2 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 \alpha = 1.3, \; \alpha_1 = 1.1, at \tau = 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 \Gamma_1 and \Gamma_2 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 \alpha = 1.3, \; \alpha_1 = 1.1 .
    Table 7.  Numerical solution in the rectangular domain [0, 1]^2 for \alpha = 1.3, \; \alpha_1 = 1.1, and \tau = 1 obtained using hyperbolic contour \Gamma_1 .
    \alpha=1.3 , \alpha_{1}=1.1
    N n M L_{\infty} c \kappa error_{est}(\Gamma_1) C.TIME(sec)
    900 14 70 4.26 \times10^{-2} 1.1 1.22 \times10^{+14} 7.20 \times10^{-3} 755.137804
    16 1.12 \times10^{-2} 1.7 3.71 \times10^{+12} 7.20 \times10^{-3} 756.918480
    18 5.00 \times10^{-3} 1.9 1.26 \times10^{+12} 7.20 \times10^{-3} 751.184383
    20 9.22 \times10^{-4} 2.4 1.48 \times10^{+12} 7.20 \times10^{-3} 748.664343
    576 20 90 5.15 \times10^{-4} 1.9 1.51 \times10^{+12} 8.18 \times10^{-4} 286.863124
    676 5.08 \times10^{-4} 2.0 2.19 \times10^{+12} 8.18 \times10^{-4} 421.440624
    784 8.58 \times10^{-4} 2.2 1.77 \times10^{+12} 8.18 \times10^{-4} 608.764593
    900 9.22 \times10^{-4} 2.4 1.48 \times10^{+12} 8.18 \times10^{-4} 861.916452
    729 20 20 1.10 \times10^{-3} 2.1 1.96 \times10^{+12} 1.55 \times10^{+0} 22.119245
    30 7.80 \times10^{-4} 2.1 1.96 \times10^{+12} 5.73 \times10^{-1} 48.770754
    50 7.57 \times10^{-4} 2.1 1.96 \times10^{+12} 6.25 \times10^{-2} 138.310373
    80 7.57 \times10^{-4} 2.1 1.96 \times10^{+12} 2.40 \times10^{-3} 386.725304

     | Show Table
    DownLoad: CSV
    Table 8.  Numerical solution in the rectangular domain [0, 1]^2 for \alpha = 1.3, \; \alpha_1 = 1.1, and \tau = 1 obtained using Talbot's contour \Gamma_2 .
    \alpha=1.3 , \alpha_{1}=1.1
    N n M L_{\infty} c \kappa error_{est}(\Gamma_2) C.TIME(sec)
    900 20 16 1.98 \times10^{-1} 2.4 1.48 \times10^{+12} 7.76 \times10^{-6} 9.513650
    18 2.15 \times10^{-2} 2.4 1.48 \times10^{+12} 5.64 \times10^{-11} 10.640051
    20 2.00 \times10^{-3} 2.4 1.48 \times10^{+12} 4.10 \times10^{-12} 11.661281
    22 8.40 \times10^{-4} 2.4 1.48 \times10^{+12} 2.97 \times10^{-13} 12.824995
    24 9.12 \times10^{-4} 2.4 1.48 \times10^{+12} 2.16 \times10^{-14} 13.560164
    900 14 24 4.28 \times10^{-2} 1.1 1.22 \times10^{+14} 2.16 \times10^{-14} 13.026362
    16 1.12 \times10^{-2} 1.7 3.31 \times10^{+12} 2.16 \times10^{-14} 13.463156
    18 5.00 \times10^{-3} 1.9 1.26 \times10^{+12} 2.16 \times10^{-14} 13.591828
    20 9.12 \times10^{-4} 2.4 1.48 \times10^{+12} 2.16 \times10^{-14} 13.601463
    576 20 22 5.11 \times10^{-4} 1.9 1.51 \times10^{+12} 2.97 \times10^{-13} 4.451590
    676 5.17 \times10^{-4} 2.0 2.19 \times10^{+12} 2.97 \times10^{-13} 6.401350
    784 8.14 \times10^{-4} 2.2 1.77 \times10^{+12} 2.97 \times10^{-13} 9.076440
    900 8.40 \times10^{-4} 2.4 1.48 \times10^{+12} 2.97 \times10^{-13} 12.560123
    961 8.06 \times10^{-4} 2.4 2.19 \times10^{+12} 2.97 \times10^{-13} 14.919706

     | Show Table
    DownLoad: CSV

    Here we solve the given problem in unit circle with center at (\chi, \vartheta) = (0.5, 0.5) . The domain is descretized with N uniform nodes. The computational results for different values of N, \; n, and M along \Gamma_1 and \Gamma_2 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 \alpha = 1.3, \; \alpha_1 = 1.1, and \tau = 1 obtained using hyperbolic contour \Gamma_1 .
    \alpha=1.3 , \alpha_{1}=1.1
    N n M L_{\infty} c \kappa error_{est}(\Gamma_1) C.TIME(sec)
    950 60 15 8.30 \times10^{-3} 4.3 3.11 \times10^{+12} 2.63 \times10^{+0} 38.116660
    20 1.20 \times10^{-3} 4.3 3.11 \times10^{+12} 1.55 \times10^{+0} 62.890208
    30 9.23 \times10^{-4} 4.3 3.11 \times10^{+12} 5.37 \times10^{-1} 133.694751
    40 9.17 \times10^{-4} 4.3 3.11 \times10^{+12} 1.83 \times10^{-2} 369.149122
    900 30 50 1.30 \times10^{-3} 3.1 2.82 \times10^{+12} 6.25 \times10^{-2} 361.783503
    40 2.10 \times10^{-3} 3.5 5.12 \times10^{+12} 6.25 \times10^{-2} 366.883195
    50 2.20 \times10^{-3} 4.1 2.43 \times10^{+12} 6.25 \times10^{-2} 363.312789
    60 9.17 \times10^{-4} 4.3 3.11 \times10^{+12} 6.25 \times10^{-2} 367.839901
    300 59 60 9.77 \times10^{-4} 4.3 3.06 \times10^{+12} 2.12 \times10^{-2} 553.852222
    550 9.77 \times10^{-4} 4.3 3.06 \times10^{+12} 2.12 \times10^{-2} 447.562447
    800 9.77 \times10^{-4} 4.3 3.06 \times10^{+12} 2.12 \times10^{-2} 531.883062
    1100 9.77 \times10^{-4} 4.3 3.06 \times10^{+12} 2.12 \times10^{-2} 531.921143

     | Show Table
    DownLoad: CSV
    Table 10.  Numerical solution in the circular domain for \alpha = 1.5, \; \alpha_1 = 1.3, and \tau = 1 obtained using Talbot's contour \Gamma_2 .
    \alpha=1.5 , \alpha_{1}=1.3
    N n M L_{\infty} c \kappa error_{est}(\Gamma_2) C.TIME(sec)
    950 50 18 5.87 \times10^{-2} 4.1 2.43 \times10^{+12} 5.64 \times10^{-11} 9.258343
    20 7.60 \times10^{-3} 4.1 2.43 \times10^{+12} 4.10 \times10^{-12} 9.959348
    22 2.40 \times10^{-3} 4.1 2.43 \times10^{+12} 2.97 \times10^{-13} 10.597861
    24 1.90 \times10^{-3} 4.1 2.43 \times10^{+12} 2.16 \times10^{-14} 11.241704
    1050 10 26 9.43 \times10^{-2} 1.2 7.70 \times10^{+13} 1.57 \times10^{-15} 9.389988
    30 1.20 \times10^{-3} 3.1 2.82 \times10^{+12} 1.57 \times10^{-15} 10.100778
    50 1.90 \times10^{-3} 4.1 2.43 \times10^{+12} 1.57 \times10^{-15} 11.882140
    60 8.84 \times10^{-4} 4.3 3.11 \times10^{+12} 1.57 \times10^{-15} 13.157724
    750 59 28 9.36 \times10^{-4} 4.3 3.06 \times10^{+12} 1.14 \times10^{-16} 13.662250
    1150 9.36 \times10^{-4} 4.3 3.06 \times10^{+12} 1.14 \times10^{-16} 13.759729
    1250 9.36 \times10^{-4} 4.3 3.06 \times10^{+12} 1.14 \times10^{-16} 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 \alpha = 1.7, \; \alpha_1 = 1.5, N = 900, \; n = 50, and M = 90 along the hyperbolic contour \Gamma_1 .
    Figure 11.  In (a) The plot shows the exact solution. In (b) the plot shows the numerical solution, the parameters used are \alpha = 1.5, \; \alpha_1 = 1.3 .

    In the last test problem we have considered the complex shape domain. The domain is generated by r_d = \frac{1}{d}[1+2d+d^2-(d+1)\cos(d \theta)], d = 4. In this experiment also we have used the contours \Gamma_1 and \Gamma_2 with the same set of optimal parameters used in Problem 1. The results obtained for fractional orders \alpha = 1.5, \; \alpha_1 = 1.3 , and various nodes N in the global domain and n in the local domain and quadrature points along the contour \Gamma_1 and \Gamma_2 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 \alpha = 1.5, \; \alpha_1 = 1.3, and \tau = 1 obtained using hyperbolic contour \Gamma_1 .
    \alpha=1.5 , \alpha_{1}=1.3
    N n M L_{\infty} c \kappa error_{est}(\Gamma_1) C.TIME(sec)
    851 50 40 4.70 \times10^{-3} 4.0 1.13 \times10^{+12} 1.83 \times10^{-1} 50.524737
    50 6.75 \times10^{-4} 4.0 1.13 \times10^{+12} 6.25 \times10^{-2} 79.388630
    60 6.74 \times10^{-4} 4.0 1.13 \times10^{+12} 2.12 \times10^{-2} 116.511151
    80 6.74 \times10^{-4} 4.0 1.13 \times10^{+12} 2.40 \times10^{-3} 227.719695
    852 30 70 1.30 \times10^{-3} 2.9 6.40 \times10^{+12} 7.20 \times10^{-3} 147.655977
    40 1.20 \times10^{-3} 3.4 5.83 \times10^{+12} 7.20 \times10^{-3} 158.960295
    50 9.34 \times10^{-4} 3.9 1.82 \times10^{+12} 7.20 \times10^{-3} 219.043351
    60 9.47 \times10^{-4} 4.2 1.92 \times10^{+12} 7.20 \times10^{-3} 268.501711
    457 60 60 6.97 \times10^{-4} 3.0 3.07 \times10^{+12} 2.12 \times10^{-2} 68.420275
    542 4.39 \times10^{-4} 3.3 1.62 \times10^{+12} 2.12 \times10^{-2} 89.095051
    643 4.86 \times10^{-4} 3.6 1.79 \times10^{+12} 2.12 \times10^{-2} 117.459116
    851 7.29 \times10^{-4} 4.2 1.90 \times10^{+12} 2.12 \times10^{-2} 184.241136

     | Show Table
    DownLoad: CSV
    Table 12.  Numerical solution in the circular domain for \alpha = 1.5, \; \alpha_1 = 1.3, and \tau = 1 obtained using Talbot's contour \Gamma_2 .
    \alpha=1.5 , \alpha_{1}=1.3
    N n M L_{\infty} c \kappa error_{est}(\Gamma_2) C.TIME(sec)
    752 30 24 1.10 \times10^{-3} 2.7 3.89 \times10^{+12} 2.16 \times10^{-14} 3.188624
    850 1.80 \times10^{-3} 2.9 7.98 \times10^{+12} 2.16 \times10^{-14} 3.613122
    921 1.50 \times10^{-3} 3.0 2.48 \times10^{+13} 2.16 \times10^{-14} 4.028301
    974 9.61 \times10^{-4} 3.2 5.64 \times10^{+12} 2.16 \times10^{-14} 4.297978
    1020 28 22 2.90 \times10^{-3} 2.9 3.79 \times10^{+13} 2.97 \times10^{-13} 4.261794
    40 1.90 \times10^{-3} 3.7 2.62 \times10^{+13} 2.97 \times10^{-13} 5.485432
    50 1.00 \times10^{-3} 4.2 1.13 \times10^{+13} 2.97 \times10^{-13} 6.750489
    60 9.32 \times10^{-4} 4.6 6.25 \times10^{+12} 2.97 \times10^{-13} 8.366275
    1095 70 18 6.47 \times10^{-2} 5.1 2.15 \times10^{+12} 5.64 \times10^{-11} 10.452683
    20 6.60 \times10^{-3} 5.1 2.15 \times10^{+12} 4.10 \times10^{-12} 10.778798
    22 9.01 \times10^{-4} 5.1 2.15 \times10^{+12} 2.97 \times10^{-13} 11.238818
    24 9.97 \times10^{-4} 5.1 2.15 \times10^{+12} 2.16 \times10^{-14} 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 \alpha = 1.5, \; \alpha_1 = 1.3 .
    Figure 13.  The plot shows absolute error obtained using Talbot's contour \Gamma_2 for the parameter values N = 993, \; n = 30, \; M = 24, \; \alpha = 1.5, and \; \alpha_1 = 1.3, at \tau = 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.



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