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[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 |
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 2−D 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),forχ∈Ω,K∈Rτ>0, | (2.1) |
where
Pα,α1,α2,...,αm(Dτ)=Dατ+m∑j=1djDαjτ, |
1<αm<...<α1<α<2, and dj≥0,j=1,2,...,m,m∈N are constants. Dαjτ is a Caputo derivative of order αj defined by
Dαjτf(τ)=1Γ(n−αj)∫τaf(n)(ν)dν(τ−ν)αj+1−n,forn−1<αj<n,n∈N, | (2.2) |
also for n=2, we have
Dαjτf(τ)=1Γ(n−αj)∫τa∂2f(ν)∂ν2dν(τ−ν)αj−1,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),χ∈∂Ω, | (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+...+sαmI−KL)−1, |
and
ˆg(χ,s)=sα−1U0(χ)+sα−2U1(χ)+sα1−1U0(χ)+sα1−2U1(χ)+...+sαm−1U0(χ)+sαm−2U1(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)=12πi∫ΓesτˆU(χ,s)ds, | (2.10) |
where, for Res≥ω 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={s≠0:|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+ℶ(1−sin(η−ιξ)),ξ∈R,(Γ1) | (2.11) |
where,
ℶ>0,0<η<ϕ−π2,andΥ>0. | (2.12) |
By writing s=x+ιy, we observe that Γ1 is the left branch of the hyperbola
(x−Υ−ℶℶsinη)2−(yℶcosη)2=1, | (2.13) |
the asymptotes for (2.13) are y=±(x−Υ−ℶ)cotη, and x-intercept at s=Υ+ℶ(1−sinη). The condition (2.12) confirms that Γ1 lies in the sector ΣΥϕ=Υ+Σϕ⊂Σϕ, and grows into the left half plane. From (2.11) and (2.10), we have the following integral
U(χ,τ)=12πi∫∞−∞es(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)=k2πiM∑j=−MesjτˆU(χ,sj)ˊsj,forξj=jk,sj=s(ξj),s′j=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,−π≤ξ≤π,(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)=12πi∫π−πes(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)=1MiM∑j=1esjτˆU(χ,sj)ˊsj,forξj=−π+(j−12)k,sj=s(ξj),s′j=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,whered≥1 the approximate function for ˆU(χ) using local meshless method has the form,
ˆU(χi)=∑χj∈Ωiλijϕ(‖χi−χij‖), | (2.19) |
where λi={λij}nj=1 is the expansion coefficients vector, ϕ(r) is a kernel function, r=‖χi−χj‖ 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(‖χi1−χi1‖)ϕ(‖χi1−χi2‖)...ϕ(‖χi1−χin‖)ϕ(‖χi2−χi1‖)ϕ(‖χi2−χi2‖)...ϕ(‖χi2−χin‖)............ϕ(‖χin−χi1‖)ϕ(‖χin−χi2‖)...ϕ(‖χin−χin‖))(λi1λi2...λin),i=1,2,...,N, | (2.20) |
which can be written as,
ˆUi=Φiλi,1≤i≤N, | (2.21) |
the matrix Φi contains elements in the form bikj=ϕ(‖χik−χij‖),whereχik,χij∈Ωi, 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)=∑χj∈ΩiλijLϕ(‖χi−χij‖), | (2.22) |
the above Eq (2.22) can be expressed as a dot product
LˆU(χi)=λi⋅νi, | (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(‖χi−χij‖),χij∈Ωi, | (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
LˆU≡DˆU, | (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 N−n 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 Γ⊂Ωr⊂ΣΥϕ, and define b>0 by coshb=1θτ1sin(η), where τ1=t0T, 0<τ0<T, 0<θ<1.0 and let ℶ=θ¯rMbT. Then for Eq (2.15), with k=bM≤¯rlog2, we have |U(χ,τ)−Uk(χ,τ)|≤CQeΥτ1l(ρrM)e−μM(‖U0‖+‖ˆf(χ,τ)‖ΣΥϕ), for μ=¯r(1−θ)b, ρr=θ¯rτ1sin(η−r1)b, ¯r=2πr1, r1>0, τ0≤τ≤T, 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)e−μM). |
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(e−1.358M). |
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=supˆU≠0‖ˆU‖‖MˆU‖, | (4.2) |
for any discrete norm ‖.‖ defined on RN the constant C is finite. From (4.2) we may write
‖M‖−1≤‖ˆU‖‖MˆU‖≤C, | (4.3) |
Similarly for the pseudoinverse M† of M, we can write
‖M†‖=supH≠0‖M†H‖‖H‖. | (4.4) |
Thus we have
‖M†‖≥supH=MˆH≠0‖M†MˆU‖‖MˆU‖=supˆU≠0‖ˆU‖‖MˆU‖=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 ‖M−1‖∞ 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 1.00≤C≤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.
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 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)‖∞=max1≤j≤N(|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)=(6τ3−αΓ(4−α)+6τ3−α1Γ(4−α1)+6τ3−α2Γ(4−α2))1cosh(χ−c)−(2τ3cosech(c−χ)2cosh(c−χ)3−τ3cosh(c−χ)). |
The exact solution of the problem is
U(χ,τ)=τ3cosh(χ−c),c∈R. |
The boundary and initial conditions are
U(−10,τ)=τ3cosh(−10−c),U(10,τ)=τ3cosh(10−c), | (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+(j−12)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.
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 | 7.77×10−5 | 10.0 | 1.13×10+12 | 3.14×10−1 | 0.343148 |
55 | 7.61×10−5 | 10.0 | 1.13×10+12 | 3.64×10−2 | 1.222487 | ||
75 | 7.61×10−5 | 10.0 | 1.13×10+12 | 4.2 ×10−3 | 5.000861 | ||
95 | 7.61×10−5 | 10.0 | 1.13×10+12 | 4.75×10−4 | 12.067666 | ||
110 | 7.61×10−5 | 10.0 | 1.13×10+12 | 9.30×10−5 | 22.244950 | ||
40 | 15 | 80 | 5.76×10−5 | 4.5 | 1.37×10+12 | 2.4 ×10−3 | 2.034147 |
50 | 4.70×10−5 | 5.7 | 1.17×10+12 | 2.4 ×10−3 | 3.430185 | ||
60 | 4.32×10−5 | 6.9 | 1.05×10+12 | 2.4 ×10−3 | 4.307865 | ||
70 | 9.09×10−5 | 8.0 | 1.24×10+12 | 2.4 ×10−3 | 5.308744 | ||
80 | 4.82×10−5 | 9.2 | 1.14×10+12 | 2.4 ×10−3 | 6.327198 | ||
70 | 12 | 90 | 2.33×10−5 | 7.3 | 1.02×10+12 | 8.18×10−4 | 8.646889 |
15 | 9.09×10−5 | 8.0 | 1.24×10+12 | 8.18×10−4 | 8.534781 | ||
18 | 8.80×10−5 | 8.5 | 1.17×10+12 | 8.18×10−4 | 8.898832 | ||
21 | 4.10×10−5 | 8.8 | 1.20×10+12 | 8.18×10−4 | 8.901062 | ||
24 | 9.30×10−5 | 9.0 | 1.26×10+12 | 8.18×10−4 | 8.835825 | ||
[14] | 4.79×10−6 |
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 | 3.51×10−1 | 7.3 | 1.02×10+12 | 2.02×10−6 | 0.201190 |
12 | 3.98×10−2 | 7.3 | 1.02×10+12 | 1.47×10−7 | 0.202626 | ||
14 | 4.20×10−3 | 7.3 | 1.02×10+12 | 1.06×10−8 | 0.213661 | ||
16 | 4.16×10−4 | 7.3 | 1.02×10+12 | 7.76×10−10 | 0.203777 | ||
18 | 4.99×10−5 | 7.3 | 1.02×10+12 | 5.64×10−11 | 0.204327 | ||
30 | 15 | 20 | 2.97×10−5 | 3.4 | 1.00×10+12 | 4.10×10−12 | 0.203400 |
40 | 5.99×10−5 | 4.5 | 1.37×10+12 | 4.10×10−12 | 0.212062 | ||
50 | 4.82×10−5 | 5.7 | 1.17×10+12 | 4.10×10−12 | 0.210793 | ||
70 | 9.41×10−5 | 8.0 | 1.24×10+12 | 4.10×10−12 | 0.222508 | ||
90 | 8.14×10−5 | 10.4 | 1.07×10+12 | 4.10×10−12 | 0.217812 | ||
80 | 12 | 20 | 7.56×10−5 | 8.3 | 1.15×10+12 | 4.10×10−12 | 0.211422 |
14 | 7.53×10−5 | 9.0 | 1.04×10+12 | 4.10×10−12 | 0.218499 | ||
16 | 6.68×10−5 | 9.4 | 1.17×10+12 | 4.10×10−12 | 0.224207 | ||
18 | 4.87×10−5 | 9.8 | 1.02×10+12 | 4.10×10−12 | 0.226878 | ||
20 | 4.67×10−5 | 10.0 | 1.13×10+12 | 4.10×10−12 | 0.239581 | ||
[14] | 5.69×10−5 |
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)=(6τ3−αΓ(4−α)+6τ3−α1Γ(4−α1)+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.
α=1.9, | α1=1.3 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
60 | 22 | 40 | 8.44×10−5 | 7.6 | 1.18×10+12 | 1.83×10−1 | 0.667872 |
60 | 6.62×10−5 | 7.6 | 1.18×10+12 | 2.12×10−2 | 1.580686 | ||
80 | 8.62×10−5 | 7.6 | 1.18×10+12 | 2.4 ×10−3 | 7.362587 | ||
90 | 8.62×10−5 | 7.6 | 1.18×10+12 | 8.18×10−4 | 10.843996 | ||
100 | 8.62×10−5 | 7.6 | 1.18×10+12 | 2.76×10−4 | 15.784191 | ||
70 | 10 | 100 | 5.52×10−5 | 6.4 | 1.15×10+12 | 2.76×10−4 | 17.932164 |
12 | 2.74×10−5 | 7.3 | 1.02×10+12 | 2.76×10−4 | 17.859075 | ||
14 | 3.44×10−5 | 7.8 | 1.21×10+12 | 2.76×10−4 | 18.129870 | ||
18 | 5.21×10−5 | 8.5 | 1.17×10+12 | 2.76×10−4 | 18.278882 | ||
22 | 7.64×10−5 | 8.9 | 1.15×10+12 | 2.76×10−3 | 18.479720 | ||
50 | 25 | 90 | 9.85×10−5 | 6.5 | 1.01×10+12 | 8.18×10−4 | 8.730629 |
60 | 1.03×10−4 | 7.8 | 1.09×10+12 | 8.18×10−4 | 11.049126 | ||
70 | 5.40×10−5 | 9.1 | 1.14×10+12 | 8.18×10−4 | 13.349187 | ||
80 | 6.57×10−5 | 10.4 | 1.19×10+12 | 8.18×10−4 | 15.124607 | ||
90 | 6.17×10−5 | 11.8 | 1.02×10+12 | 8.18×10−4 | 17.134190 | ||
[14] | 7.0080×10−4 |
α=1.7, | α1=1.2 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
70 | 12 | 10 | 2.56×10−1 | 7.3 | 1.02×10+12 | 2.02×10−6 | 0.202184 |
12 | 2.91×10−2 | 7.3 | 1.02×10+12 | 1.47×10−7 | 0.197720 | ||
14 | 3.10×10−3 | 7.3 | 1.02×10+12 | 1.06×10−8 | 0.200589 | ||
16 | 3.25×10−4 | 7.3 | 1.02×10+12 | 7.76×10−10 | 0.202710 | ||
18 | 5.65×10−5 | 7.3 | 1.02×10+12 | 5.64×10−11 | 0.194221 | ||
30 | 15 | 20 | 3.74×10−5 | 3.4 | 1.00×10+12 | 4.10×10−12 | 0.202098 |
40 | 6.97×10−5 | 4.5 | 1.37×10+12 | 4.10×10−12 | 0.201783 | ||
50 | 6.68×10−5 | 5.7 | 1.17×10+12 | 4.10×10−12 | 0.204199 | ||
70 | 1.00×10−4 | 8.0 | 1.24×10+12 | 4.10×10−12 | 0.205864 | ||
90 | 1.36×10−4 | 10.4 | 1.07×10+12 | 4.10×10−12 | 0.216570 | ||
80 | 12 | 20 | 9.70×10−5 | 8.3 | 1.15×10+12 | 4.10×10−12 | 0.207446 |
14 | 4.34×10−5 | 9.0 | 1.04×10+12 | 4.10×10−12 | 0.206234 | ||
16 | 8.47×10−5 | 9.4 | 1.17×10+12 | 4.10×10−12 | 0.216176 | ||
18 | 4.76×10−5 | 9.8 | 1.02×10+12 | 4.10×10−12 | 0.215397 | ||
20 | 5.66×10−5 | 10.0 | 1.13×10+12 | 4.10×10−12 | 0.244942 | ||
[14] | 1.39×10−4 |
We consider the following fractional equation
DατU(χ,τ)+Dα1τU(χ,τ)−D2χU(χ,τ)=f(χ,τ), | (5.7) |
where
f(χ,τ)=(2τ2−αcos(χ)Γ(3−α)+2τ2−α1cos(χ)Γ(3−α1))−(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.
α=1.9, | α1=1.8, | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
80 | 25 | 50 | 4.20×10−5 | 10.4 | 1.19×10+12 | 6.25×10−2 | 2.249386 |
60 | 3.85×10−5 | 10.4 | 1.19×10+12 | 2.12×10−2 | 4.114330 | ||
70 | 3.86×10−5 | 10.4 | 1.19×10+12 | 7.2 ×10−3 | 7.301508 | ||
90 | 3.86×10−5 | 10.4 | 1.19×10+12 | 8.18×10−4 | 15.044453 | ||
100 | 3.86×10−5 | 10.4 | 1.19×10+12 | 2.76×10−4 | 21.588645 | ||
60 | 27 | 90 | 9.51×10−5 | 7.9 | 1.06×10+12 | 8.18×10−4 | 11.338375 |
70 | 8.41×10−5 | 9.2 | 1.16×10+12 | 8.18×10−4 | 13.565846 | ||
80 | 1.15×10−4 | 10.6 | 1.01×10+12 | 8.18×10−4 | 15.034979 | ||
90 | 9.66×10−5 | 11.9 | 1.10×10+12 | 8.18×10−4 | 17.556509 | ||
100 | 7.93×10−5 | 13.2 | 1.16×10+12 | 8.18×10−4 | 21.605433 | ||
85 | 20 | 95 | 8.45×10−5 | 10.6 | 1.21×10+12 | 4.75×10−4 | 18.828414 |
22 | 5.73×10−5 | 10.9 | 1.01×10+12 | 4.75×10−4 | 19.612251 | ||
24 | 1.68×10−4 | 11.0 | 1.16×10+12 | 4.75×10−4 | 19.670357 | ||
27 | 4.26×10−5 | 11.2 | 1.16×10+12 | 4.75×10−4 | 19.563284 | ||
30 | 5.36×10−5 | 11.4 | 1.09×10+12 | 4.75×10−4 | 20.041057 | ||
[14] | 8.81×10−5 |
α=1.9, | α1=1.8, | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
70 | 12 | 10 | 3.54×10−2 | 7.3 | 1.02×10+12 | 2.02×10−6 | 0.142848 |
12 | 3.40×10−3 | 7.3 | 1.02×10+12 | 1.47×10−7 | 0.133224 | ||
14 | 2.83×10−4 | 7.3 | 1.02×10+12 | 1.06×10−8 | 0.134380 | ||
16 | 4.64×10−5 | 7.3 | 1.02×10+12 | 7.76×10−10 | 0.133445 | ||
18 | 5.19×10−5 | 7.3 | 1.02×10+12 | 5.64×10−11 | 0.136420 | ||
30 | 15 | 20 | 1.15×10−4 | 3.4 | 1.00×10+12 | 4.10×10−12 | 0.135380 |
40 | 5.10×10−5 | 4.5 | 1.37×10+12 | 4.10×10−12 | 0.142579 | ||
50 | 1.07×10−4 | 5.7 | 1.17×10+12 | 4.10×10−12 | 0.137483 | ||
70 | 1.44×10−4 | 8.0 | 1.24×10+12 | 4.10×10−12 | 0.145392 | ||
90 | 1.49×10−4 | 10.4 | 1.07×10+12 | 4.10×10−12 | 0.151340 | ||
80 | 12 | 20 | 1.29×10−4 | 8.3 | 1.15×10+12 | 4.10×10−12 | 0.144167 |
14 | 1.22×10−4 | 9.0 | 1.04×10+12 | 4.10×10−12 | 0.139500 | ||
16 | 8.31×10−5 | 9.4 | 1.17×10+12 | 4.10×10−12 | 0.148407 | ||
18 | 4.78×10−5 | 9.8 | 1.02×10+12 | 4.10×10−12 | 0.156982 | ||
20 | 9.51×10−5 | 10.0 | 1.13×10+12 | 4.10×10−12 | 0.166119 | ||
[14] | 8.81×10−5 |
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.
α=1.3, | α1=1.1 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
900 | 14 | 70 | 4.26×10−2 | 1.1 | 1.22×10+14 | 7.20×10−3 | 755.137804 |
16 | 1.12×10−2 | 1.7 | 3.71×10+12 | 7.20×10−3 | 756.918480 | ||
18 | 5.00×10−3 | 1.9 | 1.26×10+12 | 7.20×10−3 | 751.184383 | ||
20 | 9.22×10−4 | 2.4 | 1.48×10+12 | 7.20×10−3 | 748.664343 | ||
576 | 20 | 90 | 5.15×10−4 | 1.9 | 1.51×10+12 | 8.18×10−4 | 286.863124 |
676 | 5.08×10−4 | 2.0 | 2.19×10+12 | 8.18×10−4 | 421.440624 | ||
784 | 8.58×10−4 | 2.2 | 1.77×10+12 | 8.18×10−4 | 608.764593 | ||
900 | 9.22×10−4 | 2.4 | 1.48×10+12 | 8.18×10−4 | 861.916452 | ||
729 | 20 | 20 | 1.10×10−3 | 2.1 | 1.96×10+12 | 1.55×10+0 | 22.119245 |
30 | 7.80×10−4 | 2.1 | 1.96×10+12 | 5.73×10−1 | 48.770754 | ||
50 | 7.57×10−4 | 2.1 | 1.96×10+12 | 6.25×10−2 | 138.310373 | ||
80 | 7.57×10−4 | 2.1 | 1.96×10+12 | 2.40×10−3 | 386.725304 |
α=1.3, | α1=1.1 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
900 | 20 | 16 | 1.98×10−1 | 2.4 | 1.48×10+12 | 7.76×10−6 | 9.513650 |
18 | 2.15×10−2 | 2.4 | 1.48×10+12 | 5.64×10−11 | 10.640051 | ||
20 | 2.00×10−3 | 2.4 | 1.48×10+12 | 4.10×10−12 | 11.661281 | ||
22 | 8.40×10−4 | 2.4 | 1.48×10+12 | 2.97×10−13 | 12.824995 | ||
24 | 9.12×10−4 | 2.4 | 1.48×10+12 | 2.16×10−14 | 13.560164 | ||
900 | 14 | 24 | 4.28×10−2 | 1.1 | 1.22×10+14 | 2.16×10−14 | 13.026362 |
16 | 1.12×10−2 | 1.7 | 3.31×10+12 | 2.16×10−14 | 13.463156 | ||
18 | 5.00×10−3 | 1.9 | 1.26×10+12 | 2.16×10−14 | 13.591828 | ||
20 | 9.12×10−4 | 2.4 | 1.48×10+12 | 2.16×10−14 | 13.601463 | ||
576 | 20 | 22 | 5.11×10−4 | 1.9 | 1.51×10+12 | 2.97×10−13 | 4.451590 |
676 | 5.17×10−4 | 2.0 | 2.19×10+12 | 2.97×10−13 | 6.401350 | ||
784 | 8.14×10−4 | 2.2 | 1.77×10+12 | 2.97×10−13 | 9.076440 | ||
900 | 8.40×10−4 | 2.4 | 1.48×10+12 | 2.97×10−13 | 12.560123 | ||
961 | 8.06×10−4 | 2.4 | 2.19×10+12 | 2.97×10−13 | 14.919706 |
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.
α=1.3, | α1=1.1 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
950 | 60 | 15 | 8.30×10−3 | 4.3 | 3.11×10+12 | 2.63×10+0 | 38.116660 |
20 | 1.20×10−3 | 4.3 | 3.11×10+12 | 1.55×10+0 | 62.890208 | ||
30 | 9.23×10−4 | 4.3 | 3.11×10+12 | 5.37×10−1 | 133.694751 | ||
40 | 9.17×10−4 | 4.3 | 3.11×10+12 | 1.83×10−2 | 369.149122 | ||
900 | 30 | 50 | 1.30×10−3 | 3.1 | 2.82×10+12 | 6.25×10−2 | 361.783503 |
40 | 2.10×10−3 | 3.5 | 5.12×10+12 | 6.25×10−2 | 366.883195 | ||
50 | 2.20×10−3 | 4.1 | 2.43×10+12 | 6.25×10−2 | 363.312789 | ||
60 | 9.17×10−4 | 4.3 | 3.11×10+12 | 6.25×10−2 | 367.839901 | ||
300 | 59 | 60 | 9.77×10−4 | 4.3 | 3.06×10+12 | 2.12×10−2 | 553.852222 |
550 | 9.77×10−4 | 4.3 | 3.06×10+12 | 2.12×10−2 | 447.562447 | ||
800 | 9.77×10−4 | 4.3 | 3.06×10+12 | 2.12×10−2 | 531.883062 | ||
1100 | 9.77×10−4 | 4.3 | 3.06×10+12 | 2.12×10−2 | 531.921143 |
α=1.5, | α1=1.3 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
950 | 50 | 18 | 5.87×10−2 | 4.1 | 2.43×10+12 | 5.64×10−11 | 9.258343 |
20 | 7.60×10−3 | 4.1 | 2.43×10+12 | 4.10×10−12 | 9.959348 | ||
22 | 2.40×10−3 | 4.1 | 2.43×10+12 | 2.97×10−13 | 10.597861 | ||
24 | 1.90×10−3 | 4.1 | 2.43×10+12 | 2.16×10−14 | 11.241704 | ||
1050 | 10 | 26 | 9.43×10−2 | 1.2 | 7.70×10+13 | 1.57×10−15 | 9.389988 |
30 | 1.20×10−3 | 3.1 | 2.82×10+12 | 1.57×10−15 | 10.100778 | ||
50 | 1.90×10−3 | 4.1 | 2.43×10+12 | 1.57×10−15 | 11.882140 | ||
60 | 8.84×10−4 | 4.3 | 3.11×10+12 | 1.57×10−15 | 13.157724 | ||
750 | 59 | 28 | 9.36×10−4 | 4.3 | 3.06×10+12 | 1.14×10−16 | 13.662250 |
1150 | 9.36×10−4 | 4.3 | 3.06×10+12 | 1.14×10−16 | 13.759729 | ||
1250 | 9.36×10−4 | 4.3 | 3.06×10+12 | 1.14×10−16 | 13.572409 |
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.
α=1.5, | α1=1.3 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
851 | 50 | 40 | 4.70×10−3 | 4.0 | 1.13×10+12 | 1.83×10−1 | 50.524737 |
50 | 6.75×10−4 | 4.0 | 1.13×10+12 | 6.25×10−2 | 79.388630 | ||
60 | 6.74×10−4 | 4.0 | 1.13×10+12 | 2.12×10−2 | 116.511151 | ||
80 | 6.74×10−4 | 4.0 | 1.13×10+12 | 2.40×10−3 | 227.719695 | ||
852 | 30 | 70 | 1.30×10−3 | 2.9 | 6.40×10+12 | 7.20×10−3 | 147.655977 |
40 | 1.20×10−3 | 3.4 | 5.83×10+12 | 7.20×10−3 | 158.960295 | ||
50 | 9.34×10−4 | 3.9 | 1.82×10+12 | 7.20×10−3 | 219.043351 | ||
60 | 9.47×10−4 | 4.2 | 1.92×10+12 | 7.20×10−3 | 268.501711 | ||
457 | 60 | 60 | 6.97×10−4 | 3.0 | 3.07×10+12 | 2.12×10−2 | 68.420275 |
542 | 4.39×10−4 | 3.3 | 1.62×10+12 | 2.12×10−2 | 89.095051 | ||
643 | 4.86×10−4 | 3.6 | 1.79×10+12 | 2.12×10−2 | 117.459116 | ||
851 | 7.29×10−4 | 4.2 | 1.90×10+12 | 2.12×10−2 | 184.241136 |
α=1.5, | α1=1.3 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
752 | 30 | 24 | 1.10×10−3 | 2.7 | 3.89×10+12 | 2.16×10−14 | 3.188624 |
850 | 1.80×10−3 | 2.9 | 7.98×10+12 | 2.16×10−14 | 3.613122 | ||
921 | 1.50×10−3 | 3.0 | 2.48×10+13 | 2.16×10−14 | 4.028301 | ||
974 | 9.61×10−4 | 3.2 | 5.64×10+12 | 2.16×10−14 | 4.297978 | ||
1020 | 28 | 22 | 2.90×10−3 | 2.9 | 3.79×10+13 | 2.97×10−13 | 4.261794 |
40 | 1.90×10−3 | 3.7 | 2.62×10+13 | 2.97×10−13 | 5.485432 | ||
50 | 1.00×10−3 | 4.2 | 1.13×10+13 | 2.97×10−13 | 6.750489 | ||
60 | 9.32×10−4 | 4.6 | 6.25×10+12 | 2.97×10−13 | 8.366275 | ||
1095 | 70 | 18 | 6.47×10−2 | 5.1 | 2.15×10+12 | 5.64×10−11 | 10.452683 |
20 | 6.60×10−3 | 5.1 | 2.15×10+12 | 4.10×10−12 | 10.778798 | ||
22 | 9.01×10−4 | 5.1 | 2.15×10+12 | 2.97×10−13 | 11.238818 | ||
24 | 9.97×10−4 | 5.1 | 2.15×10+12 | 2.16×10−14 | 11.540931 |
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] |
González-Fernández C, Sialve B, Bernet N, et al. (2012) Impact of microalgae characteristics on their conversion to biofuel. Part I: focus on cultivation and biofuel production. Biofuels Bioprod Biorefin 6: 105-113. doi: 10.1002/bbb.338
![]() |
[2] |
Fradique M, Batista AP, Nunes MC, et al. (2010) Incorporation of Chlorella vulgaris and Spirulina maxima biomass in pasta products. Part 1: preparation and evaluation. J Sci Food Agric 90: 1656-1664. doi: 10.1002/jsfa.3999
![]() |
[3] | Becker W (2004) Microalgae in human and animal nutrition. Handbook of Microalgal Culture: Biotechnology and Applied Phycology Oxford: Blackwell, 312-351. |
[4] |
Borowitzka MA (1999) Commercial production of microalgae: ponds, tanks, tubes and fermenters. J Biotechnol 70: 313-321. doi: 10.1016/S0168-1656(99)00083-8
![]() |
[5] |
Pulz O, Scheibenbogen K (1998) Photobioreactors: design and performance with respect to light energy input. Bioprocess and Algae Reactor Technology, Apoptosis Heidelberg: Springer, 123-151. doi: 10.1007/BFb0102298
![]() |
[6] |
Chaumont D (1993) Biotechnology of algal biomass production: a review of systems for outdoor mass culture. J Appl Phycol 5: 593-604. doi: 10.1007/BF02184638
![]() |
[7] |
Day JG, Gong YC, Hu Q (2017) Microzooplanktonic grazers—A potentially devastating threat to the commercial success of microalgal mass culture. Algal Res 27: 356-365. doi: 10.1016/j.algal.2017.08.024
![]() |
[8] | Iwamoto H (2004) Industrial production of microalgal cell-mass and secondary products—major industrial species: Chlorella. Handbook of Microalgal Culture: Biotechnology and Applied Phycology Oxford: Blackwell, 253-263. |
[9] |
Pulz O, Gross W (2004) Valuable products from biotechnology of microalgae. Appl Microbiol Biotechnol 65: 635-648. doi: 10.1007/s00253-004-1647-x
![]() |
[10] |
Lee YK (2001) Microalgal mass culture systems and methods: Their limitation and potential. J Appl Phycol 13: 307-315. doi: 10.1023/A:1017560006941
![]() |
[11] |
Souza Queiroz J, Barbosa CMV, da Rocha MC, et al. (2012) Chlorella vulgaris treatment ameliorates the suppressive effects of single and repeated stressors on hematopoiesis. Brain Behav Immun 29: 39-50. doi: 10.1016/j.bbi.2012.12.001
![]() |
[12] | Morris HJ, Carillo OV, Almarales Á, et al. (2009) Protein hydrolysates from the alga Chlorella vulgaris 87/1 with potentialities in immunonutrition. Biotechnol Appl 26: 162-165. |
[13] |
Justo GZ, Silva MR, Queiroz ML (2001) Effects of the green algae Chlorella vulgaris on the response of the host hematopoietic system to intraperitoneal ehrlich ascites tumor transplantation in mice. Immunopharmacol Immunotoxicol 23: 119-132. doi: 10.1081/IPH-100102573
![]() |
[14] |
Morimoto T, Nagatsu A, Murakami N, et al. (1995) Anti-tumour-promoting glyceroglycolipids from the green alga Chlorella vulgaris. Phytochemistry 40: 1433-1437. doi: 10.1016/0031-9422(95)00458-J
![]() |
[15] | Sano T, Tanaka Y (1987) Effect of dried, powdered Chlorella vulgaris on experimental atherosclerosis and alimentary hypercholesterolemia in cholesterol-fed rabbits. Artery 14: 76-84. |
[16] |
Borowitzka MA (2013) High-value products from microalgae—their development and commercialisation. J Appl Phycol 25: 743-756. doi: 10.1007/s10811-013-9983-9
![]() |
[17] |
Ozkan A, Kinney K, Katz L, et al. (2012) Reduction of water and energy requirement of algae cultivation using an algae biofilm photobioreactor. Bioresour Technol 114: 542-548. doi: 10.1016/j.biortech.2012.03.055
![]() |
[18] |
Johnson MB, Wen Z (2010) Development of an attached microalgal growth system for biofuel production. Appl Microbiol Biotechnol 85: 525-534. doi: 10.1007/s00253-009-2133-2
![]() |
[19] |
Sekar R, Venugopalan V, Satpathy K, et al. (2004) Laboratory studies on adhesion of microalgae to hard substrates. Hydrobiologia 512: 109-116. doi: 10.1023/B:HYDR.0000020315.40349.38
![]() |
[20] |
Ozkan A, Berberoglu H (2013) Adhesion of algal cells to surfaces. Biofouling 29: 469-482. doi: 10.1080/08927014.2013.782397
![]() |
[21] |
Gao F, Yang Z-H, Li C, et al. (2015) A novel algal biofilm membrane photobioreactor for attached microalgae growth and nutrients removal from secondary effluent. Bioresour Technol 179: 8-12. doi: 10.1016/j.biortech.2014.11.108
![]() |
[22] |
Melo M, Fernandes S, Caetano N, et al. (2018) Chlorella vulgaris (SAG 211-12) biofilm formation capacity and proposal of a rotating flat plate photobioreactor for more sustainable biomass production. J Appl Phycol 30: 887-899. doi: 10.1007/s10811-017-1290-4
![]() |
[23] |
Shen Y, Yang T, Zhu W, et al. (2017) Wastewater treatment and biofuel production through attached culture of Chlorella vulgaris in a porous substratum biofilm reactor. J Appl Phycol 29: 833-841. doi: 10.1007/s10811-016-0981-6
![]() |
[24] |
Shen Y, Zhu W, Chen C, et al. (2016) Biofilm formation in attached microalgal reactors. Bioproc Biosyst Eng 39: 1281-1288. doi: 10.1007/s00449-016-1606-9
![]() |
[25] |
Großerhode C, Wehlage D, Grothe D, et al. (2017) Investigation of microalgae growth on electrospun nanofiber mats. AIMS Bioengineering 4: 376-385. doi: 10.3934/bioeng.2017.3.376
![]() |
[26] |
Böttjer R, Storck JL, Vahle D, et al. (2019) Influence of textile and environmental parameters on plant growth on vertically mounted knitted fabrics. Tekstilec 62: 200-207. doi: 10.14502/Tekstilec2019.62.200-207
![]() |
[27] |
Storck JL, Böttjer R, Vahle D, et al. (2019) Seed germination and seedling growth on knitted fabrics as new substrates for hydroponic systems. Horticulturae 5: 73. doi: 10.3390/horticulturae5040073
![]() |
[28] |
Gorman DS, Levine RP (1965) Cytochrome f and plastocyanin: their sequence in the photosynthetic electron transport chain of Chlamydomonas reinhardi. Proc Natl Acad Sci USA 54: 1665-1669. doi: 10.1073/pnas.54.6.1665
![]() |
[29] |
Latala A, Nędzi M, Stepnowski P (2009) Acute toxicity assessment of perfluorinated carboxylic acids towards the Baltic microalgae. Environment Toxicol Pharmacol 28: 167-171. doi: 10.1016/j.etap.2009.03.010
![]() |
[30] |
Jacobi A, Steinweg C, Sastre RR, et al. (2012) Advanced photobioreactor LED illumination system: Scale-down approach to study microalgal growth kinetics. Eng Life Sci 12: 621-630. doi: 10.1002/elsc.201200004
![]() |
[31] |
Sebök S, Brockhagen B, Storck JL, et al. (2020) Growth of marine macroalgae Ectocarpus sp. on various textile substrates. doi: 10.1080/09593330.2020.1829086
![]() |
[32] |
Atta M, Idris A, Bukhari A, et al. (2013) Intensity of blue LED light: A potential stimulus for biomass and lipid content in fresh water microalgae Chlorella vulgaris. Bioresour Technol 148: 373-378. doi: 10.1016/j.biortech.2013.08.162
![]() |
[33] |
Bhola V, Desikan R, Santosh SH, et al. (2011) Effects of parameters affecting biomass yield and thermal behaviour of Chlorella vulgaris. J Biosci Bioeng 111: 377-382. doi: 10.1016/j.jbiosc.2010.11.006
![]() |
[34] |
Sasi D, Mitra P, Vigueras A, et al. (2011) Growth kinetics and lipid production using Chlorella vulgaris in a circulating loop photobioreactor. J Chem Technol Biotechnol 86: 875-880. doi: 10.1002/jctb.2603
![]() |
[35] |
Zhang Q, Yu ZG, Jin SP, et al. (2019) Lignocellulosic residue as bio-carrier for algal biofilm growth: Effects of carrier physicochemical proprieties and toxicity on algal biomass production and composition. Bioresour Technol 293: 122091. doi: 10.1016/j.biortech.2019.122091
![]() |
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 | |
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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 | |
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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 |
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 | 7.77×10−5 | 10.0 | 1.13×10+12 | 3.14×10−1 | 0.343148 |
55 | 7.61×10−5 | 10.0 | 1.13×10+12 | 3.64×10−2 | 1.222487 | ||
75 | 7.61×10−5 | 10.0 | 1.13×10+12 | 4.2 ×10−3 | 5.000861 | ||
95 | 7.61×10−5 | 10.0 | 1.13×10+12 | 4.75×10−4 | 12.067666 | ||
110 | 7.61×10−5 | 10.0 | 1.13×10+12 | 9.30×10−5 | 22.244950 | ||
40 | 15 | 80 | 5.76×10−5 | 4.5 | 1.37×10+12 | 2.4 ×10−3 | 2.034147 |
50 | 4.70×10−5 | 5.7 | 1.17×10+12 | 2.4 ×10−3 | 3.430185 | ||
60 | 4.32×10−5 | 6.9 | 1.05×10+12 | 2.4 ×10−3 | 4.307865 | ||
70 | 9.09×10−5 | 8.0 | 1.24×10+12 | 2.4 ×10−3 | 5.308744 | ||
80 | 4.82×10−5 | 9.2 | 1.14×10+12 | 2.4 ×10−3 | 6.327198 | ||
70 | 12 | 90 | 2.33×10−5 | 7.3 | 1.02×10+12 | 8.18×10−4 | 8.646889 |
15 | 9.09×10−5 | 8.0 | 1.24×10+12 | 8.18×10−4 | 8.534781 | ||
18 | 8.80×10−5 | 8.5 | 1.17×10+12 | 8.18×10−4 | 8.898832 | ||
21 | 4.10×10−5 | 8.8 | 1.20×10+12 | 8.18×10−4 | 8.901062 | ||
24 | 9.30×10−5 | 9.0 | 1.26×10+12 | 8.18×10−4 | 8.835825 | ||
[14] | 4.79×10−6 |
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 | 3.51×10−1 | 7.3 | 1.02×10+12 | 2.02×10−6 | 0.201190 |
12 | 3.98×10−2 | 7.3 | 1.02×10+12 | 1.47×10−7 | 0.202626 | ||
14 | 4.20×10−3 | 7.3 | 1.02×10+12 | 1.06×10−8 | 0.213661 | ||
16 | 4.16×10−4 | 7.3 | 1.02×10+12 | 7.76×10−10 | 0.203777 | ||
18 | 4.99×10−5 | 7.3 | 1.02×10+12 | 5.64×10−11 | 0.204327 | ||
30 | 15 | 20 | 2.97×10−5 | 3.4 | 1.00×10+12 | 4.10×10−12 | 0.203400 |
40 | 5.99×10−5 | 4.5 | 1.37×10+12 | 4.10×10−12 | 0.212062 | ||
50 | 4.82×10−5 | 5.7 | 1.17×10+12 | 4.10×10−12 | 0.210793 | ||
70 | 9.41×10−5 | 8.0 | 1.24×10+12 | 4.10×10−12 | 0.222508 | ||
90 | 8.14×10−5 | 10.4 | 1.07×10+12 | 4.10×10−12 | 0.217812 | ||
80 | 12 | 20 | 7.56×10−5 | 8.3 | 1.15×10+12 | 4.10×10−12 | 0.211422 |
14 | 7.53×10−5 | 9.0 | 1.04×10+12 | 4.10×10−12 | 0.218499 | ||
16 | 6.68×10−5 | 9.4 | 1.17×10+12 | 4.10×10−12 | 0.224207 | ||
18 | 4.87×10−5 | 9.8 | 1.02×10+12 | 4.10×10−12 | 0.226878 | ||
20 | 4.67×10−5 | 10.0 | 1.13×10+12 | 4.10×10−12 | 0.239581 | ||
[14] | 5.69×10−5 |
α=1.9, | α1=1.3 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
60 | 22 | 40 | 8.44×10−5 | 7.6 | 1.18×10+12 | 1.83×10−1 | 0.667872 |
60 | 6.62×10−5 | 7.6 | 1.18×10+12 | 2.12×10−2 | 1.580686 | ||
80 | 8.62×10−5 | 7.6 | 1.18×10+12 | 2.4 ×10−3 | 7.362587 | ||
90 | 8.62×10−5 | 7.6 | 1.18×10+12 | 8.18×10−4 | 10.843996 | ||
100 | 8.62×10−5 | 7.6 | 1.18×10+12 | 2.76×10−4 | 15.784191 | ||
70 | 10 | 100 | 5.52×10−5 | 6.4 | 1.15×10+12 | 2.76×10−4 | 17.932164 |
12 | 2.74×10−5 | 7.3 | 1.02×10+12 | 2.76×10−4 | 17.859075 | ||
14 | 3.44×10−5 | 7.8 | 1.21×10+12 | 2.76×10−4 | 18.129870 | ||
18 | 5.21×10−5 | 8.5 | 1.17×10+12 | 2.76×10−4 | 18.278882 | ||
22 | 7.64×10−5 | 8.9 | 1.15×10+12 | 2.76×10−3 | 18.479720 | ||
50 | 25 | 90 | 9.85×10−5 | 6.5 | 1.01×10+12 | 8.18×10−4 | 8.730629 |
60 | 1.03×10−4 | 7.8 | 1.09×10+12 | 8.18×10−4 | 11.049126 | ||
70 | 5.40×10−5 | 9.1 | 1.14×10+12 | 8.18×10−4 | 13.349187 | ||
80 | 6.57×10−5 | 10.4 | 1.19×10+12 | 8.18×10−4 | 15.124607 | ||
90 | 6.17×10−5 | 11.8 | 1.02×10+12 | 8.18×10−4 | 17.134190 | ||
[14] | 7.0080×10−4 |
α=1.7, | α1=1.2 | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
70 | 12 | 10 | 2.56×10−1 | 7.3 | 1.02×10+12 | 2.02×10−6 | 0.202184 |
12 | 2.91×10−2 | 7.3 | 1.02×10+12 | 1.47×10−7 | 0.197720 | ||
14 | 3.10×10−3 | 7.3 | 1.02×10+12 | 1.06×10−8 | 0.200589 | ||
16 | 3.25×10−4 | 7.3 | 1.02×10+12 | 7.76×10−10 | 0.202710 | ||
18 | 5.65×10−5 | 7.3 | 1.02×10+12 | 5.64×10−11 | 0.194221 | ||
30 | 15 | 20 | 3.74×10−5 | 3.4 | 1.00×10+12 | 4.10×10−12 | 0.202098 |
40 | 6.97×10−5 | 4.5 | 1.37×10+12 | 4.10×10−12 | 0.201783 | ||
50 | 6.68×10−5 | 5.7 | 1.17×10+12 | 4.10×10−12 | 0.204199 | ||
70 | 1.00×10−4 | 8.0 | 1.24×10+12 | 4.10×10−12 | 0.205864 | ||
90 | 1.36×10−4 | 10.4 | 1.07×10+12 | 4.10×10−12 | 0.216570 | ||
80 | 12 | 20 | 9.70×10−5 | 8.3 | 1.15×10+12 | 4.10×10−12 | 0.207446 |
14 | 4.34×10−5 | 9.0 | 1.04×10+12 | 4.10×10−12 | 0.206234 | ||
16 | 8.47×10−5 | 9.4 | 1.17×10+12 | 4.10×10−12 | 0.216176 | ||
18 | 4.76×10−5 | 9.8 | 1.02×10+12 | 4.10×10−12 | 0.215397 | ||
20 | 5.66×10−5 | 10.0 | 1.13×10+12 | 4.10×10−12 | 0.244942 | ||
[14] | 1.39×10−4 |
α=1.9, | α1=1.8, | ||||||
N | n | M | L∞ | c | κ | errorest(Γ1) | C.TIME(sec) |
80 | 25 | 50 | 4.20×10−5 | 10.4 | 1.19×10+12 | 6.25×10−2 | 2.249386 |
60 | 3.85×10−5 | 10.4 | 1.19×10+12 | 2.12×10−2 | 4.114330 | ||
70 | 3.86×10−5 | 10.4 | 1.19×10+12 | 7.2 ×10−3 | 7.301508 | ||
90 | 3.86×10−5 | 10.4 | 1.19×10+12 | 8.18×10−4 | 15.044453 | ||
100 | 3.86×10−5 | 10.4 | 1.19×10+12 | 2.76×10−4 | 21.588645 | ||
60 | 27 | 90 | 9.51×10−5 | 7.9 | 1.06×10+12 | 8.18×10−4 | 11.338375 |
70 | 8.41×10−5 | 9.2 | 1.16×10+12 | 8.18×10−4 | 13.565846 | ||
80 | 1.15×10−4 | 10.6 | 1.01×10+12 | 8.18×10−4 | 15.034979 | ||
90 | 9.66×10−5 | 11.9 | 1.10×10+12 | 8.18×10−4 | 17.556509 | ||
100 | 7.93×10−5 | 13.2 | 1.16×10+12 | 8.18×10−4 | 21.605433 | ||
85 | 20 | 95 | 8.45×10−5 | 10.6 | 1.21×10+12 | 4.75×10−4 | 18.828414 |
22 | 5.73×10−5 | 10.9 | 1.01×10+12 | 4.75×10−4 | 19.612251 | ||
24 | 1.68×10−4 | 11.0 | 1.16×10+12 | 4.75×10−4 | 19.670357 | ||
27 | 4.26×10−5 | 11.2 | 1.16×10+12 | 4.75×10−4 | 19.563284 | ||
30 | 5.36×10−5 | 11.4 | 1.09×10+12 | 4.75×10−4 | 20.041057 | ||
[14] | 8.81×10−5 |
α=1.9, | α1=1.8, | ||||||
N | n | M | L∞ | c | κ | errorest(Γ2) | C.TIME(sec) |
70 | 12 | 10 | 3.54×10−2 | 7.3 | 1.02×10+12 | 2.02×10−6 | 0.142848 |
12 | 3.40×10−3 | 7.3 | 1.02×10+12 | 1.47×10−7 | 0.133224 | ||
14 | 2.83×10−4 | 7.3 | 1.02×10+12 | 1.06×10−8 | 0.134380 | ||
16 | 4.64×10−5 | 7.3 | 1.02×10+12 | 7.76×10−10 | 0.133445 | ||
18 | 5.19×10−5 | 7.3 | 1.02×10+12 | 5.64×10−11 | 0.136420 | ||
30 | 15 | 20 | 1.15×10−4 | 3.4 | 1.00×10+12 | 4.10×10−12 | 0.135380 |
40 | 5.10×10−5 | 4.5 | 1.37×10+12 | 4.10×10−12 | 0.142579 | ||
50 | 1.07×10−4 | 5.7 | 1.17×10+12 | 4.10 | 0.137483 | ||
70 | 1.44 | 8.0 | 1.24 | 4.10 | 0.145392 | ||
90 | 1.49 | 10.4 | 1.07 | 4.10 | 0.151340 | ||
80 | 12 | 20 | 1.29 | 8.3 | 1.15 | 4.10 | 0.144167 |
14 | 1.22 | 9.0 | 1.04 | 4.10 | 0.139500 | ||
16 | 8.31 | 9.4 | 1.17 | 4.10 | 0.148407 | ||
18 | 4.78 | 9.8 | 1.02 | 4.10 | 0.156982 | ||
20 | 9.51 | 10.0 | 1.13 | 4.10 | 0.166119 | ||
[14] | 8.81 |
, | |||||||
C.TIME(sec) | |||||||
900 | 14 | 70 | 4.26 | 1.1 | 1.22 | 7.20 | 755.137804 |
16 | 1.12 | 1.7 | 3.71 | 7.20 | 756.918480 | ||
18 | 5.00 | 1.9 | 1.26 | 7.20 | 751.184383 | ||
20 | 9.22 | 2.4 | 1.48 | 7.20 | 748.664343 | ||
576 | 20 | 90 | 5.15 | 1.9 | 1.51 | 8.18 | 286.863124 |
676 | 5.08 | 2.0 | 2.19 | 8.18 | 421.440624 | ||
784 | 8.58 | 2.2 | 1.77 | 8.18 | 608.764593 | ||
900 | 9.22 | 2.4 | 1.48 | 8.18 | 861.916452 | ||
729 | 20 | 20 | 1.10 | 2.1 | 1.96 | 1.55 | 22.119245 |
30 | 7.80 | 2.1 | 1.96 | 5.73 | 48.770754 | ||
50 | 7.57 | 2.1 | 1.96 | 6.25 | 138.310373 | ||
80 | 7.57 | 2.1 | 1.96 | 2.40 | 386.725304 |
, | |||||||
C.TIME(sec) | |||||||
900 | 20 | 16 | 1.98 | 2.4 | 1.48 | 7.76 | 9.513650 |
18 | 2.15 | 2.4 | 1.48 | 5.64 | 10.640051 | ||
20 | 2.00 | 2.4 | 1.48 | 4.10 | 11.661281 | ||
22 | 8.40 | 2.4 | 1.48 | 2.97 | 12.824995 | ||
24 | 9.12 | 2.4 | 1.48 | 2.16 | 13.560164 | ||
900 | 14 | 24 | 4.28 | 1.1 | 1.22 | 2.16 | 13.026362 |
16 | 1.12 | 1.7 | 3.31 | 2.16 | 13.463156 | ||
18 | 5.00 | 1.9 | 1.26 | 2.16 | 13.591828 | ||
20 | 9.12 | 2.4 | 1.48 | 2.16 | 13.601463 | ||
576 | 20 | 22 | 5.11 | 1.9 | 1.51 | 2.97 | 4.451590 |
676 | 5.17 | 2.0 | 2.19 | 2.97 | 6.401350 | ||
784 | 8.14 | 2.2 | 1.77 | 2.97 | 9.076440 | ||
900 | 8.40 | 2.4 | 1.48 | 2.97 | 12.560123 | ||
961 | 8.06 | 2.4 | 2.19 | 2.97 | 14.919706 |
, | |||||||
C.TIME(sec) | |||||||
950 | 60 | 15 | 8.30 | 4.3 | 3.11 | 2.63 | 38.116660 |
20 | 1.20 | 4.3 | 3.11 | 1.55 | 62.890208 | ||
30 | 9.23 | 4.3 | 3.11 | 5.37 | 133.694751 | ||
40 | 9.17 | 4.3 | 3.11 | 1.83 | 369.149122 | ||
900 | 30 | 50 | 1.30 | 3.1 | 2.82 | 6.25 | 361.783503 |
40 | 2.10 | 3.5 | 5.12 | 6.25 | 366.883195 | ||
50 | 2.20 | 4.1 | 2.43 | 6.25 | 363.312789 | ||
60 | 9.17 | 4.3 | 3.11 | 6.25 | 367.839901 | ||
300 | 59 | 60 | 9.77 | 4.3 | 3.06 | 2.12 | 553.852222 |
550 | 9.77 | 4.3 | 3.06 | 2.12 | 447.562447 | ||
800 | 9.77 | 4.3 | 3.06 | 2.12 | 531.883062 | ||
1100 | 9.77 | 4.3 | 3.06 | 2.12 | 531.921143 |
, | |||||||
C.TIME(sec) | |||||||
950 | 50 | 18 | 5.87 | 4.1 | 2.43 | 5.64 | 9.258343 |
20 | 7.60 | 4.1 | 2.43 | 4.10 | 9.959348 | ||
22 | 2.40 | 4.1 | 2.43 | 2.97 | 10.597861 | ||
24 | 1.90 | 4.1 | 2.43 | 2.16 | 11.241704 | ||
1050 | 10 | 26 | 9.43 | 1.2 | 7.70 | 1.57 | 9.389988 |
30 | 1.20 | 3.1 | 2.82 | 1.57 | 10.100778 | ||
50 | 1.90 | 4.1 | 2.43 | 1.57 | 11.882140 | ||
60 | 8.84 | 4.3 | 3.11 | 1.57 | 13.157724 | ||
750 | 59 | 28 | 9.36 | 4.3 | 3.06 | 1.14 | 13.662250 |
1150 | 9.36 | 4.3 | 3.06 | 1.14 | 13.759729 | ||
1250 | 9.36 | 4.3 | 3.06 | 1.14 | 13.572409 |
, | |||||||
C.TIME(sec) | |||||||
851 | 50 | 40 | 4.70 | 4.0 | 1.13 | 1.83 | 50.524737 |
50 | 6.75 | 4.0 | 1.13 | 6.25 | 79.388630 | ||
60 | 6.74 | 4.0 | 1.13 | 2.12 | 116.511151 | ||
80 | 6.74 | 4.0 | 1.13 | 2.40 | 227.719695 | ||
852 | 30 | 70 | 1.30 | 2.9 | 6.40 | 7.20 | 147.655977 |
40 | 1.20 | 3.4 | 5.83 | 7.20 | 158.960295 | ||
50 | 9.34 | 3.9 | 1.82 | 7.20 | 219.043351 | ||
60 | 9.47 | 4.2 | 1.92 | 7.20 | 268.501711 | ||
457 | 60 | 60 | 6.97 | 3.0 | 3.07 | 2.12 | 68.420275 |
542 | 4.39 | 3.3 | 1.62 | 2.12 | 89.095051 | ||
643 | 4.86 | 3.6 | 1.79 | 2.12 | 117.459116 | ||
851 | 7.29 | 4.2 | 1.90 | 2.12 | 184.241136 |
, | |||||||
C.TIME(sec) | |||||||
752 | 30 | 24 | 1.10 | 2.7 | 3.89 | 2.16 | 3.188624 |
850 | 1.80 | 2.9 | 7.98 | 2.16 | 3.613122 | ||
921 | 1.50 | 3.0 | 2.48 | 2.16 | 4.028301 | ||
974 | 9.61 | 3.2 | 5.64 | 2.16 | 4.297978 | ||
1020 | 28 | 22 | 2.90 | 2.9 | 3.79 | 2.97 | 4.261794 |
40 | 1.90 | 3.7 | 2.62 | 2.97 | 5.485432 | ||
50 | 1.00 | 4.2 | 1.13 | 2.97 | 6.750489 | ||
60 | 9.32 | 4.6 | 6.25 | 2.97 | 8.366275 | ||
1095 | 70 | 18 | 6.47 | 5.1 | 2.15 | 5.64 | 10.452683 |
20 | 6.60 | 5.1 | 2.15 | 4.10 | 10.778798 | ||
22 | 9.01 | 5.1 | 2.15 | 2.97 | 11.238818 | ||
24 | 9.97 | 5.1 | 2.15 | 2.16 | 11.540931 |
, | , | , | |||||
C.TIME(sec) | |||||||
80 | 20 | 35 | 7.77 | 10.0 | 1.13 | 3.14 | 0.343148 |
55 | 7.61 | 10.0 | 1.13 | 3.64 | 1.222487 | ||
75 | 7.61 | 10.0 | 1.13 | 4.2 | 5.000861 | ||
95 | 7.61 | 10.0 | 1.13 | 4.75 | 12.067666 | ||
110 | 7.61 | 10.0 | 1.13 | 9.30 | 22.244950 | ||
40 | 15 | 80 | 5.76 | 4.5 | 1.37 | 2.4 | 2.034147 |
50 | 4.70 | 5.7 | 1.17 | 2.4 | 3.430185 | ||
60 | 4.32 | 6.9 | 1.05 | 2.4 | 4.307865 | ||
70 | 9.09 | 8.0 | 1.24 | 2.4 | 5.308744 | ||
80 | 4.82 | 9.2 | 1.14 | 2.4 | 6.327198 | ||
70 | 12 | 90 | 2.33 | 7.3 | 1.02 | 8.18 | 8.646889 |
15 | 9.09 | 8.0 | 1.24 | 8.18 | 8.534781 | ||
18 | 8.80 | 8.5 | 1.17 | 8.18 | 8.898832 | ||
21 | 4.10 | 8.8 | 1.20 | 8.18 | 8.901062 | ||
24 | 9.30 | 9.0 | 1.26 | 8.18 | 8.835825 | ||
[14] | 4.79 |
, | , | , | |||||
C.TIME(sec) | |||||||
70 | 12 | 10 | 3.51 | 7.3 | 1.02 | 2.02 | 0.201190 |
12 | 3.98 | 7.3 | 1.02 | 1.47 | 0.202626 | ||
14 | 4.20 | 7.3 | 1.02 | 1.06 | 0.213661 | ||
16 | 4.16 | 7.3 | 1.02 | 7.76 | 0.203777 | ||
18 | 4.99 | 7.3 | 1.02 | 5.64 | 0.204327 | ||
30 | 15 | 20 | 2.97 | 3.4 | 1.00 | 4.10 | 0.203400 |
40 | 5.99 | 4.5 | 1.37 | 4.10 | 0.212062 | ||
50 | 4.82 | 5.7 | 1.17 | 4.10 | 0.210793 | ||
70 | 9.41 | 8.0 | 1.24 | 4.10 | 0.222508 | ||
90 | 8.14 | 10.4 | 1.07 | 4.10 | 0.217812 | ||
80 | 12 | 20 | 7.56 | 8.3 | 1.15 | 4.10 | 0.211422 |
14 | 7.53 | 9.0 | 1.04 | 4.10 | 0.218499 | ||
16 | 6.68 | 9.4 | 1.17 | 4.10 | 0.224207 | ||
18 | 4.87 | 9.8 | 1.02 | 4.10 | 0.226878 | ||
20 | 4.67 | 10.0 | 1.13 | 4.10 | 0.239581 | ||
[14] | 5.69 |
, | |||||||
C.TIME(sec) | |||||||
60 | 22 | 40 | 8.44 | 7.6 | 1.18 | 1.83 | 0.667872 |
60 | 6.62 | 7.6 | 1.18 | 2.12 | 1.580686 | ||
80 | 8.62 | 7.6 | 1.18 | 2.4 | 7.362587 | ||
90 | 8.62 | 7.6 | 1.18 | 8.18 | 10.843996 | ||
100 | 8.62 | 7.6 | 1.18 | 2.76 | 15.784191 | ||
70 | 10 | 100 | 5.52 | 6.4 | 1.15 | 2.76 | 17.932164 |
12 | 2.74 | 7.3 | 1.02 | 2.76 | 17.859075 | ||
14 | 3.44 | 7.8 | 1.21 | 2.76 | 18.129870 | ||
18 | 5.21 | 8.5 | 1.17 | 2.76 | 18.278882 | ||
22 | 7.64 | 8.9 | 1.15 | 2.76 | 18.479720 | ||
50 | 25 | 90 | 9.85 | 6.5 | 1.01 | 8.18 | 8.730629 |
60 | 1.03 | 7.8 | 1.09 | 8.18 | 11.049126 | ||
70 | 5.40 | 9.1 | 1.14 | 8.18 | 13.349187 | ||
80 | 6.57 | 10.4 | 1.19 | 8.18 | 15.124607 | ||
90 | 6.17 | 11.8 | 1.02 | 8.18 | 17.134190 | ||
[14] | 7.0080 |
, | |||||||
C.TIME(sec) | |||||||
70 | 12 | 10 | 2.56 | 7.3 | 1.02 | 2.02 | 0.202184 |
12 | 2.91 | 7.3 | 1.02 | 1.47 | 0.197720 | ||
14 | 3.10 | 7.3 | 1.02 | 1.06 | 0.200589 | ||
16 | 3.25 | 7.3 | 1.02 | 7.76 | 0.202710 | ||
18 | 5.65 | 7.3 | 1.02 | 5.64 | 0.194221 | ||
30 | 15 | 20 | 3.74 | 3.4 | 1.00 | 4.10 | 0.202098 |
40 | 6.97 | 4.5 | 1.37 | 4.10 | 0.201783 | ||
50 | 6.68 | 5.7 | 1.17 | 4.10 | 0.204199 | ||
70 | 1.00 | 8.0 | 1.24 | 4.10 | 0.205864 | ||
90 | 1.36 | 10.4 | 1.07 | 4.10 | 0.216570 | ||
80 | 12 | 20 | 9.70 | 8.3 | 1.15 | 4.10 | 0.207446 |
14 | 4.34 | 9.0 | 1.04 | 4.10 | 0.206234 | ||
16 | 8.47 | 9.4 | 1.17 | 4.10 | 0.216176 | ||
18 | 4.76 | 9.8 | 1.02 | 4.10 | 0.215397 | ||
20 | 5.66 | 10.0 | 1.13 | 4.10 | 0.244942 | ||
[14] | 1.39 |
, | , | ||||||
C.TIME(sec) | |||||||
80 | 25 | 50 | 4.20 | 10.4 | 1.19 | 6.25 | 2.249386 |
60 | 3.85 | 10.4 | 1.19 | 2.12 | 4.114330 | ||
70 | 3.86 | 10.4 | 1.19 | 7.2 | 7.301508 | ||
90 | 3.86 | 10.4 | 1.19 | 8.18 | 15.044453 | ||
100 | 3.86 | 10.4 | 1.19 | 2.76 | 21.588645 | ||
60 | 27 | 90 | 9.51 | 7.9 | 1.06 | 8.18 | 11.338375 |
70 | 8.41 | 9.2 | 1.16 | 8.18 | 13.565846 | ||
80 | 1.15 | 10.6 | 1.01 | 8.18 | 15.034979 | ||
90 | 9.66 | 11.9 | 1.10 | 8.18 | 17.556509 | ||
100 | 7.93 | 13.2 | 1.16 | 8.18 | 21.605433 | ||
85 | 20 | 95 | 8.45 | 10.6 | 1.21 | 4.75 | 18.828414 |
22 | 5.73 | 10.9 | 1.01 | 4.75 | 19.612251 | ||
24 | 1.68 | 11.0 | 1.16 | 4.75 | 19.670357 | ||
27 | 4.26 | 11.2 | 1.16 | 4.75 | 19.563284 | ||
30 | 5.36 | 11.4 | 1.09 | 4.75 | 20.041057 | ||
[14] | 8.81 |
, | , | ||||||
C.TIME(sec) | |||||||
70 | 12 | 10 | 3.54 | 7.3 | 1.02 | 2.02 | 0.142848 |
12 | 3.40 | 7.3 | 1.02 | 1.47 | 0.133224 | ||
14 | 2.83 | 7.3 | 1.02 | 1.06 | 0.134380 | ||
16 | 4.64 | 7.3 | 1.02 | 7.76 | 0.133445 | ||
18 | 5.19 | 7.3 | 1.02 | 5.64 | 0.136420 | ||
30 | 15 | 20 | 1.15 | 3.4 | 1.00 | 4.10 | 0.135380 |
40 | 5.10 | 4.5 | 1.37 | 4.10 | 0.142579 | ||
50 | 1.07 | 5.7 | 1.17 | 4.10 | 0.137483 | ||
70 | 1.44 | 8.0 | 1.24 | 4.10 | 0.145392 | ||
90 | 1.49 | 10.4 | 1.07 | 4.10 | 0.151340 | ||
80 | 12 | 20 | 1.29 | 8.3 | 1.15 | 4.10 | 0.144167 |
14 | 1.22 | 9.0 | 1.04 | 4.10 | 0.139500 | ||
16 | 8.31 | 9.4 | 1.17 | 4.10 | 0.148407 | ||
18 | 4.78 | 9.8 | 1.02 | 4.10 | 0.156982 | ||
20 | 9.51 | 10.0 | 1.13 | 4.10 | 0.166119 | ||
[14] | 8.81 |
, | |||||||
C.TIME(sec) | |||||||
900 | 14 | 70 | 4.26 | 1.1 | 1.22 | 7.20 | 755.137804 |
16 | 1.12 | 1.7 | 3.71 | 7.20 | 756.918480 | ||
18 | 5.00 | 1.9 | 1.26 | 7.20 | 751.184383 | ||
20 | 9.22 | 2.4 | 1.48 | 7.20 | 748.664343 | ||
576 | 20 | 90 | 5.15 | 1.9 | 1.51 | 8.18 | 286.863124 |
676 | 5.08 | 2.0 | 2.19 | 8.18 | 421.440624 | ||
784 | 8.58 | 2.2 | 1.77 | 8.18 | 608.764593 | ||
900 | 9.22 | 2.4 | 1.48 | 8.18 | 861.916452 | ||
729 | 20 | 20 | 1.10 | 2.1 | 1.96 | 1.55 | 22.119245 |
30 | 7.80 | 2.1 | 1.96 | 5.73 | 48.770754 | ||
50 | 7.57 | 2.1 | 1.96 | 6.25 | 138.310373 | ||
80 | 7.57 | 2.1 | 1.96 | 2.40 | 386.725304 |
, | |||||||
C.TIME(sec) | |||||||
900 | 20 | 16 | 1.98 | 2.4 | 1.48 | 7.76 | 9.513650 |
18 | 2.15 | 2.4 | 1.48 | 5.64 | 10.640051 | ||
20 | 2.00 | 2.4 | 1.48 | 4.10 | 11.661281 | ||
22 | 8.40 | 2.4 | 1.48 | 2.97 | 12.824995 | ||
24 | 9.12 | 2.4 | 1.48 | 2.16 | 13.560164 | ||
900 | 14 | 24 | 4.28 | 1.1 | 1.22 | 2.16 | 13.026362 |
16 | 1.12 | 1.7 | 3.31 | 2.16 | 13.463156 | ||
18 | 5.00 | 1.9 | 1.26 | 2.16 | 13.591828 | ||
20 | 9.12 | 2.4 | 1.48 | 2.16 | 13.601463 | ||
576 | 20 | 22 | 5.11 | 1.9 | 1.51 | 2.97 | 4.451590 |
676 | 5.17 | 2.0 | 2.19 | 2.97 | 6.401350 | ||
784 | 8.14 | 2.2 | 1.77 | 2.97 | 9.076440 | ||
900 | 8.40 | 2.4 | 1.48 | 2.97 | 12.560123 | ||
961 | 8.06 | 2.4 | 2.19 | 2.97 | 14.919706 |
, | |||||||
C.TIME(sec) | |||||||
950 | 60 | 15 | 8.30 | 4.3 | 3.11 | 2.63 | 38.116660 |
20 | 1.20 | 4.3 | 3.11 | 1.55 | 62.890208 | ||
30 | 9.23 | 4.3 | 3.11 | 5.37 | 133.694751 | ||
40 | 9.17 | 4.3 | 3.11 | 1.83 | 369.149122 | ||
900 | 30 | 50 | 1.30 | 3.1 | 2.82 | 6.25 | 361.783503 |
40 | 2.10 | 3.5 | 5.12 | 6.25 | 366.883195 | ||
50 | 2.20 | 4.1 | 2.43 | 6.25 | 363.312789 | ||
60 | 9.17 | 4.3 | 3.11 | 6.25 | 367.839901 | ||
300 | 59 | 60 | 9.77 | 4.3 | 3.06 | 2.12 | 553.852222 |
550 | 9.77 | 4.3 | 3.06 | 2.12 | 447.562447 | ||
800 | 9.77 | 4.3 | 3.06 | 2.12 | 531.883062 | ||
1100 | 9.77 | 4.3 | 3.06 | 2.12 | 531.921143 |
, | |||||||
C.TIME(sec) | |||||||
950 | 50 | 18 | 5.87 | 4.1 | 2.43 | 5.64 | 9.258343 |
20 | 7.60 | 4.1 | 2.43 | 4.10 | 9.959348 | ||
22 | 2.40 | 4.1 | 2.43 | 2.97 | 10.597861 | ||
24 | 1.90 | 4.1 | 2.43 | 2.16 | 11.241704 | ||
1050 | 10 | 26 | 9.43 | 1.2 | 7.70 | 1.57 | 9.389988 |
30 | 1.20 | 3.1 | 2.82 | 1.57 | 10.100778 | ||
50 | 1.90 | 4.1 | 2.43 | 1.57 | 11.882140 | ||
60 | 8.84 | 4.3 | 3.11 | 1.57 | 13.157724 | ||
750 | 59 | 28 | 9.36 | 4.3 | 3.06 | 1.14 | 13.662250 |
1150 | 9.36 | 4.3 | 3.06 | 1.14 | 13.759729 | ||
1250 | 9.36 | 4.3 | 3.06 | 1.14 | 13.572409 |
, | |||||||
C.TIME(sec) | |||||||
851 | 50 | 40 | 4.70 | 4.0 | 1.13 | 1.83 | 50.524737 |
50 | 6.75 | 4.0 | 1.13 | 6.25 | 79.388630 | ||
60 | 6.74 | 4.0 | 1.13 | 2.12 | 116.511151 | ||
80 | 6.74 | 4.0 | 1.13 | 2.40 | 227.719695 | ||
852 | 30 | 70 | 1.30 | 2.9 | 6.40 | 7.20 | 147.655977 |
40 | 1.20 | 3.4 | 5.83 | 7.20 | 158.960295 | ||
50 | 9.34 | 3.9 | 1.82 | 7.20 | 219.043351 | ||
60 | 9.47 | 4.2 | 1.92 | 7.20 | 268.501711 | ||
457 | 60 | 60 | 6.97 | 3.0 | 3.07 | 2.12 | 68.420275 |
542 | 4.39 | 3.3 | 1.62 | 2.12 | 89.095051 | ||
643 | 4.86 | 3.6 | 1.79 | 2.12 | 117.459116 | ||
851 | 7.29 | 4.2 | 1.90 | 2.12 | 184.241136 |
, | |||||||
C.TIME(sec) | |||||||
752 | 30 | 24 | 1.10 | 2.7 | 3.89 | 2.16 | 3.188624 |
850 | 1.80 | 2.9 | 7.98 | 2.16 | 3.613122 | ||
921 | 1.50 | 3.0 | 2.48 | 2.16 | 4.028301 | ||
974 | 9.61 | 3.2 | 5.64 | 2.16 | 4.297978 | ||
1020 | 28 | 22 | 2.90 | 2.9 | 3.79 | 2.97 | 4.261794 |
40 | 1.90 | 3.7 | 2.62 | 2.97 | 5.485432 | ||
50 | 1.00 | 4.2 | 1.13 | 2.97 | 6.750489 | ||
60 | 9.32 | 4.6 | 6.25 | 2.97 | 8.366275 | ||
1095 | 70 | 18 | 6.47 | 5.1 | 2.15 | 5.64 | 10.452683 |
20 | 6.60 | 5.1 | 2.15 | 4.10 | 10.778798 | ||
22 | 9.01 | 5.1 | 2.15 | 2.97 | 11.238818 | ||
24 | 9.97 | 5.1 | 2.15 | 2.16 | 11.540931 |