
Based on ecological significance, a delayed diffusive predator-prey system with food-limited and nonlinear harvesting subject to the Neumann boundary conditions is investigated in this paper. Firstly, the sufficient conditions of the stability of nonnegative constant steady state solutions of system are derived. The existence of Hopf bifurcation is obtained by analyzing the associated characteristic equation and the conditions of Turing instability are derived when the system has no delay. Furthermore, the occurrence conditions the Hopf bifurcation are discussed by regarding delay expressing the gestation time of the predator as the bifurcation parameter. Secondly, by using upper-lower solution method, the global asymptotical stability of a unique positive constant steady state solution of system is investigated. Moreover, we also give the detailed formulas to determine the direction, stability of Hopf bifurcation by applying the normal form theory and center manifold reduction. Finally, numerical simulations are carried out to demonstrate our theoretical results.
Citation: Guangxun Sun, Binxiang Dai. Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3520-3552. doi: 10.3934/mbe.2020199
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Based on ecological significance, a delayed diffusive predator-prey system with food-limited and nonlinear harvesting subject to the Neumann boundary conditions is investigated in this paper. Firstly, the sufficient conditions of the stability of nonnegative constant steady state solutions of system are derived. The existence of Hopf bifurcation is obtained by analyzing the associated characteristic equation and the conditions of Turing instability are derived when the system has no delay. Furthermore, the occurrence conditions the Hopf bifurcation are discussed by regarding delay expressing the gestation time of the predator as the bifurcation parameter. Secondly, by using upper-lower solution method, the global asymptotical stability of a unique positive constant steady state solution of system is investigated. Moreover, we also give the detailed formulas to determine the direction, stability of Hopf bifurcation by applying the normal form theory and center manifold reduction. Finally, numerical simulations are carried out to demonstrate our theoretical results.
The importance of fractional differential equations (FDE) needs no emphasis as they are considered to be fundamental in the analysis and modeling of numerous physical phenomena of science and engineering such as heat conduction, viscoelasticity, polarization, diffusion waves, electromagnetic waves, signal processing and control theory ADDIN EN.CITE.DATA [1,2,3,4,5]. Fractional dynamic systems that contains a single differential operator in the equations are sometimes found insufficient to accurately and realistically address the problems effectively, therefore, at times it is necessary to use multi-term fractional differential equations (MFDE) instead. For example, to illustrate the motion of real physical systems, the modeling of the motion of a rigid plate immersed in a Newtonian fluid and a gas in a fluid, respectively. Also the asymptotic behavior for low and high frequencies in the modeling of frequency-dependent damping materials are well described by the fractional differential equations of
ω1D2ϕ(x)+ω2D1.5ϕ(x)+ω3Dϕ(x)+ω4D0.5ϕ(x)+ω5ϕ(x)=g(x), 0<x⩽X | (1) |
where,
In the past few years, many heuristic and meta-heuristic algorithm have been proposed to solve various global optimization problems. By far the majority of new meta-heuristic algorithms are inspired by nature among which, the largest fraction of these nature inspired algorithms are based on some successful characteristics of a biological system. The bio-inspired meta-heuristic algorithms particularly those based on the swarm intelligence are an essential part of the modern global optimization algorithms. Over the past few decade, the swarm intelligence based (SIB) algorithms have gained strong attention of scientists and researchers due to its remarkable advantages as compared to the traditional optimization techniques used earlier [12,13,14]. The SIB algorithm which have been developed by drawing inspiration from the swarm intelligence system in nature includes the firefly algorithm (FFA) which is based on the flashing pattern and behavior of swarming fireflies [15], while the cuckoo search algorithm (CSA) uses the parasitic strategy of cuckoo species [16], particle swarm optimization (PSO) and its improved version known as the accelerated particle swarm optimization algorithm (APSOA) are based on the swarming behavior of birds and fishes schooling in nature [17,18], the ant algorithm (AA) mimics the behavior of social insects (i.e., ants) [19], the bat algorithm (BATA) which is one of the most powerful and extensively used meta-heuristic algorithm based on the echolocation behavior of foraging bats [20,21,22] and many more. On the other hand there are some meta-heuristic algorithms which are bio-inspired but are not based on the swarm intelligence such as the genetic algorithm (GA) [23], the differential evolution algorithm (DEA) [24,25] and the flower pollination algorithm (FPA) [26].
The BATA developed by Yang is one of the latest bio-inspired meta-heuristic algorithm which is based on the echolocation behavior of foraging microbats. The echolocation capability of microbats is very fascinating, which is used to detect prey and avoid obstacles even in complete darkness. The microbats emit loud sound pulse and sense the echo that is reflected by the surrounding objects, which enables them to determine the environment around them. This interesting process of emitting and reflecting sound waves is mimicked in Figure 1. These bats build a 3D image of the surrounding in their brains by using the deferral time of the response, time difference between the two ears and the loudness variation of echoes. The varying pulse rate of emission and loudness provides a mechanism for auto-zooming into the region where promising solutions are found. To further enhance the performance of this highly efficient and powerful algorithm many new variants of BATA algorithm have been produced which are found in the literature ADDIN EN.CITE.DATA [20,21,22,27].
In the present study a framework is provided to solve MFDEs subject to four different variants of the two point boundary conditions, which are commonly used to solve real world problems. The residual power series method (RPSM) introduced by Arqub [28] in 2013 provides an efficient and practical approach for determining the analytical solution of classical differential equations (DEs). This numerical technique (RPSM) based on the generalized Taylor series formula was later modified and successfully applied to solve various types of FDEs ADDIN EN.CITE.DATA [29,30,31,32]. The new residual optimization technique (NROT) suggested in this framework is an amalgamation of the BATA with the principal idea for constructing the solution of the FDEs offered in RPSM. Moreover, the validity and accuracy of the NROT is further ratified by incorporating the idea of the RPSM with two other popular and powerful bio-inspired meta-heuristic algorithms, i.e., APSOA and DEA.
The silent features of the employed technique are summarized as:
● Developing a non-interrupted and non-iterative NROT by using the fractional approach in Caputo sense to solve MFDEs subject to a set of four elementary boundary conditions.
● Optimization of the constructed residual function to attain the optimal values of the unknown weights in the considered power series solution.
● Utilization of the BATA to achieve the desired fitness of the residual function along with two other influential and proficient optimization techniques i.e., APSOA and DEA, to ratify the accuracy and consistency of the presented scheme.
● Framing the designed methodology of the presented scheme along with the relevant parameters detailing.
● Performing numerical simulation and statistical analysis to analyze the performance indices.
A brief overview of the paper is as follows: In Section 2 fundamental definitions and theorems are presented. The designed methodology is projected in Section 3 with the learning solver described in Section 4. The mathematical formulation of the considered performance indices is displayed in Section 5 with the discussion of the attained numerical and statistical results given in Section 6. Lastly, the summary and highlight remarks are listed in Section 7.
This section comprises of some essential definitions and results of fractional calculus that are going to be used in the further discussion.
Definition 2.1. For a continuously differential function
cDγ0f(x)={1Γ(τ−γ)x∫0(x−t)τ−γ−1fτ(t)dt,τ−1<γ<τ,dτf(x)dxτ,γ=τ∈ℵ, | (2) |
While, the direct implementation of Caputo fractional derivative on a power function
cDγ0xϖ={Γ(ϖ+1)xϖ−γΓ(ϖ−γ+1),ϖ>τ−1,0,ϖ≤τ−1, | (3) |
Where,
Definition 2.2. A power series (PS) expansion about
∞∑i=0wi(x−ˉx0)iγ=w0+w1(x−ˉx0)γ+w2(x−ˉx0)2γ+..., | (4) |
where, all constant coefficient
Theorem 2.3. Suppose that a FPS representation of a function
f(x)=∞∑i=0wi(x−ˉx0)iγ, 0⩽τ−1<γ<τ, ˉx0⩽x<ˉx0+R∗ | (5) |
If
wi=Diγf(ˉx0)Γ(iγ+1), i=1,2,... | (6) |
where,
Consider a non-linear multi-term fractional differential equation expressed as:
Drϕ(x)=f(x,ϕ(x),Dα1ϕ(x),Dα2ϕ(x),...,Dαjϕ(x)) x∈Γ=[0,X] | (7) |
subject to the initial conditions written as:
ϕm(x0)=ϕm(0)=cm,0⩽x0<X,m=0,1,2,...,r−1, | (8) |
and the boundary conditions at
ϕl(xn)=dl,0<xn⩽X,l=0,1,2,...,r−1, | (9) |
where,
Now for the proposed technique we assume the weighted trial solution of Eq. (7) is of the form:
ϕ(x)=∞∑i=0wi(x−ˉx0)iΓ(1+i) | (10) |
where,
Rkϕ,k(x)=Drϕ(x)−f(x,ϕ(x),Dα1ϕ(x),Dα2ϕ(x),...,Dαjϕ(x)) | (11) |
with
limk→∞Rkϕ,k(x)=0 ∀x>0, | (12) |
where,
w0=cm−F(x0,w1,w2,w3,...,wk) | (13) |
wk=dl−F(xn,w0,w1,w2,...,wk−1) | (14) |
By using Eq. (13) and Eq. (14) in Eq. (10), the kth truncated series of the refined trial solution is given by:
ϕ(x)=w0+wk(x−ˉx0)kΓ(k+1)+k−1∑i=1wi(x−ˉx0)iΓ(1+i) | (15) |
where,
Rk−2ϕ,k(x)=Drϕ(x)−f(x,ϕ(x),Dα1ϕ(x),Dα2ϕ(x),...,Dαjϕ(x)), | (16) |
E(x)=min1νν∑i=1(Rk−2ϕ,k(xi))2, | (17) |
where, xi=ih∗, with the step size
Lemma 3.1. For problem (7) subject to the two point boundary conditions, if the trial solutions
Lemma 3.2. Consider a continuous and integrable function
Theorem 3.3. Suppose ϕ(x) be an integrable and continuous function defined in [0,X] with (x−ˉx0)i≤M for all i=0,1,2,⋯,k and the optimal values of k−2 unknown weights achieved by the considered meta-heuristic techniques such that |wi|≤M∗;i=1,2,⋯,k−1 then the refined series solution (15) converges as k→∞.
The proof easily follows by utilizing the aforementioned Lemma 3.2.
Bat-inspired algorithm (BATA) is a relatively new and powerful meta-heuristic algorithm based on the successful characteristics of biological system (i.e., echolocation behavior of the microbats). They emit loud sound and listen for the echo that is reflected back by the surrounding objects. Studies from the past have proved that these bats have the ability to locate and identify the target, the type of target and even the moving speed of it. Moreover, the wing flutter rate of the prey induces a Doppler effect which helps these foraging bats to discriminate between the targets. In BATA the travelling range of the emitted pulses can be adjusted by either adjusting the wavelengths
● All bats use their echolocation capability to avoid obstacles and detect prey in the dark.
● Bats fly randomly with the velocity
● While the loudness can vary in many ways, from a minimum value
After initializing the dimension
˜f=˜fmin+(˜fmax−˜fmin)∗ℏ, ˜vj+1i=˜vji+(˜xji−x∗)˜fi, ˜xj+1i=˜xji+˜vj+1i, | (18) |
where,
˜Aj+1i=θ1˜Aji, and ˜rj+1i=˜r0i[1−e−jθ2], | (19) |
with
BATA | APSOA | DEA | |||||
Parameter | Value | Parameter | Value | Parameter | Value | ||
|
0 | Maximum swarm size | 80 | Cross Probability | 0.5 | ||
|
2 | Maximum number of flights | 2000 | Initial Points | Default | ||
Maximum bat population size | 40 | Lower bound | -2 | Penelty Function | Default | ||
|
1 | Upper bound | 2 | Post Process | Default | ||
Loudness | 1 | Accelaration constant |
0.7 | Random Seed | 0 | ||
Pulse rate | 0.05 | Accelaration constant |
0.5 | Scaling Factor | 0.6 | ||
Maximum number of generations, |
2000 | Step size | 0.1 | Step size | 0.1 | ||
Step size | 0.1 | Search Points | Default | ||||
Tolerance | 0.001 |
To further ratify the correctness and efficiency of the NROT the solutions are also constructed by using the algorithm APSOA defined in [18] and the built-in MATHEMATICA codes for the DEA used in the software MATHEMATICA 11.0 for the optimization of the fitness function defined as Eq. (17).
Performance indices based on the root mean square error
In reference to the solution obtained by the designed methodology, the definition of
PMAD=1KK∑i=1|y∗i−yei|, | (20) |
PRMS=√1KK∑i=1|yei−y∗i|2, | (21) |
The mathematical definition of Nash Sutcliffe efficiency,
PNSE=1−K∑i=1(y∗i−yei)2K∑i=1(⌢yei−yei)2, ⌢yei=1KK∑i=1yei | (22) |
and
PENSE=1−PNSE. | (23) |
A global operator based on the mean fitness value is written as:
GMF=1ˆK⌢K∑r=1E(x)|μ | (24) |
where,
In this section, numerical results of the suggested scheme (NROT) optimized using BATA are presented here for the assessment of linear and nonlinear non-homogenous MFDEs subject to four different variants of boundary conditions that cope a reasonable spectrum of possible cases. Furthermore, to validate the correctness and effectiveness of the deliberated scheme the results are also constructed by using the present scheme NROT in conjunction with two other powerful and efficient meta-heuristic techniques i.e., APSOA and DEA. The parameter setting for BATA, APSOA and DEA are defined in Table 1. In addition, comparative study is also conducted on the basis of performance indices formulated in the above Section 5. The reliability and accuracy of the suggested scheme NROT using the meta-heuristic solver BATA is further analyzed through the statistical results based on the large number of independent runs.
The boundary value problems for these four variants are given below:
Case Study I
Consider the following fractional non-homogenous MFDE
D2ϕ(x)+c0D1.5ϕ(x)+ϕ(x)=2+√16xπ+x2, | (25) |
subject to the Neumann boundary conditions given as:
ϕ′(0)=0,ϕ′(xn)=2xn, | (26) |
Here, the exact solution is
Rϕ,k(x)=D2ϕ(x)+c0D1.5ϕ(x)+ϕ(x)−2−√16xπ−x2, | (27) |
which automatically leads to a fitness function formulated as:
E(x)=min1rr∑i=1(D2ϕ(xi)+c0D1.5ϕ(xi)+ϕ(xi)−2−√16xiπ−x2i)2, | (28) |
Case Study Ⅱ
Consider the following fractional order MFDE
D2ϕ(x)+c0D1.5ϕ(x)+ϕ(x)=2+2(x1.5Γ2.5+x22)+(x0.5Γ1.5+x), | (29) |
with the Dirichlet boundary conditions written as:
ϕ(0)=0,ϕ(xn)=x2n−xn, | (30) |
The exact solution of the above formulated MFDE is
Rϕ,k(x)=D2ϕ(x)+c0D1.5ϕ(x)+ϕ(x)−2−2(x1.5Γ2.5+x22)−(x0.5Γ1.5+x), | (31) |
with the related fitness function given as:
E(x)=min1rr∑i=1(D2ϕ(x)+c0D1.5ϕ(x)+ϕ(x)−2−2(x1.5Γ2.5+x22)−(x0.5Γ1.5+x))2, | (32) |
Case Study Ⅲ
The proposed scheme is now tested on another form of MFDE with the two point boundary conditions given as:
D2ϕ(x)+0.5c0D0.5ϕ(x)+ϕ(x)=3+(x1.5Γ2.5+x2)+x2, | (33) |
ϕ′(0)=0,ϕ(xn)=x2n+1, | (34) |
The exact solution of the Eq. (33) is
Rϕ.k(x)=D2ϕ(x)+0.5c0D0.5ϕ(x)+ϕ(x)−3−(x1.5Γ2.5+x2)−x2, | (35) |
which automatically leads to a fitness function written as:
E(x)=min1rr∑i=1(D2ϕ(x)+0.5c0D0.5ϕ(x)+ϕ(x)−3−(x1.5Γ2.5+x2)−x2)2, | (36) |
Case Study Ⅳ
Consider the following non-linear MFDE,
D2ϕ(x)+c0D1.455ϕ(x)+(c0D0.555ϕ(x))2+ϕ3(x)=2x+2x1.545Γ2.545+(2x2.455Γ3.445)2+x927, | (37) |
with two point boundary conditions given as:
ϕ(0)=0,ϕ′(xn)=x2n, | (38) |
Here, the exact solution of Eq. (37) is
Rϕ,k=D2ϕ(x)+c0D1.455ϕ(x)+(c0D0.555ϕ(x))2+ϕ3(x)−2x−2x1.545Γ2.545−(2x2.455Γ3.445)2−x927, | (39) |
By using the proposed scheme Eq (39) automatically leads to a fitness function presented as:
E(x)=min1rr∑i=1(D2ϕ(x)+c0D1.455ϕ(x)+(c0D0.555ϕ(x))2+ϕ3(x)−2x−2x1.545Γ2.545−(2x2.455Γ3.445)2−x927)2 | (40) |
Minimization of the above given fitness functions (28), (32), (36) and (40) defined for the respective four test cases of the MFDEs is carried out by applying the proposed design methodology (NROT) using BATA, APSOA and DEA, with the parameter settings provided in Table 1. One of the best set of weights learned stochastically by using BATA, APSOA and DEA with
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0.1 | 0.01 | 0.01 | 0.009988 | 0.012820 | 1.02349×10-16 | 1.17878×10-5 | 2.82030×10-3 |
0.2 | 0.04 | 0.04 | 0.039989 | 0.042810 | 7.63278×10-17 | 1.09495×10-5 | 2.81007×10-3 |
0.3 | 0.09 | 0.09 | 0.089999 | 0.092786 | 4.16334×10-17 | 1.00600×10-5 | 2.78584×10-3 |
0.4 | 0.16 | 0.16 | 0.159991 | 0.162745 | 5.55112×10-17 | 9.35407×10-6 | 2.74546×10-3 |
0.5 | 0.25 | 0.25 | 0.249991 | 0.252690 | 5.55112×10-17 | 8.93795×10-6 | 2.68994×10-3 |
0.6 | 0.36 | 0.36 | 0.359991 | 0.362623 | 5.55112×10-17 | 8.81746×10-6 | 2.62319×10-3 |
0.7 | 0.49 | 0.49 | 0.489991 | 0.492552 | 1.11022×10-16 | 8.92606×10-6 | 2.55189×10-3 |
0.8 | 0.64 | 0.64 | 0.639991 | 0.642485 | 1.11022×10-16 | 9.15278×10-6 | 2.48528×10-3 |
0.9 | 0.81 | 0.81 | 0.809991 | 0.812435 | 2.22045×10-16 | 9.37028×10-6 | 2.43497×10-3 |
1.0 | 1.00 | 1.00 | 0.999991 | 1.002410 | 0.000000 | 9.46283×10-6 | 2.41478×10-3 |
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0.1 | -0.09 | -0.09 | -0.089999 | -0.090088 | 7.49401×10-16 | 3.34267×10-7 | 8.76839×10-5 |
0.2 | -0.16 | -0.16 | -0.159999 | -0.160208 | 2.77556×10-15 | 8.73920×10-7 | 2.07961×10-4 |
0.3 | -0.21 | -0.21 | -0.209999 | -0.210359 | 4.41314×10-15 | 1.29197×10-6 | 3.58686×10-4 |
0.4 | -0.24 | -0.24 | -0.239998 | -0.240513 | 4.91274×10-15 | 1.60640×10-6 | 5.13425×10-4 |
0.5 | -0.25 | -0.25 | -0.249998 | -0.250633 | 4.27436×10-15 | 1.99844×10-6 | 6.33424×10-4 |
0.6 | -0.24 | -0.24 | -0.239997 | -0.240680 | 2.83107×10-15 | 2.63088×10-6 | 6.79560×10-4 |
0.7 | -0.21 | -0.21 | -0.209997 | -0.210624 | 1.33227×10-15 | 3.46638×10-6 | 6.24306×10-4 |
0.8 | -0.16 | -0.16 | -0.159996 | -0.160464 | 3.05311×10-16 | 4.08576×10-6 | 4.63690×10-4 |
0.9 | -0.09 | -0.09 | -0.089996 | -0.090229 | 9.71445×10-17 | 3.50629×10-6 | 2.29250×10-4 |
1.0 | 0.00 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
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0.1 | 1.01 | 1.01 | 1.009980 | 1.008052 | 6.48762×10-8 | 1.95158×10-5 | 1.94787×10-3 |
0.2 | 1.04 | 1.04 | 1.039982 | 1.038052 | 6.12838×10-8 | 1.81608×10-5 | 1.92087×10-3 |
0.3 | 1.09 | 1.09 | 1.089984 | 1.088150 | 5.55299×10-8 | 1.61406×10-5 | 1.84995×10-3 |
0.4 | 1.16 | 1.16 | 1.159986 | 1.158281 | 4.80703×10-8 | 1.37066×10-5 | 1.71900×10-3 |
0.5 | 1.25 | 1.25 | 1.249989 | 1.248481 | 3.95009×10-8 | 1.11155×10-5 | 1.51917×10-3 |
0.6 | 1.36 | 1.36 | 1.359991 | 1.358749 | 3.04728×10-8 | 8.58836×10-6 | 1.25134×10-3 |
0.7 | 1.49 | 1.49 | 1.489994 | 1.489071 | 2.16083×10-8 | 6.26947×10-6 | 9.28685×10-4 |
0.8 | 1.64 | 1.64 | 1.639996 | 1.639421 | 1.34155×10-8 | 4.18586×10-6 | 5.79129×10-4 |
0.9 | 1.81 | 1.81 | 1.809998 | 1.809752 | 6.20399×10-9 | 2.20620×10-6 | 2.47895×10-4 |
1.0 | 2.00 | 2.00 | 2.000000 | 2.000000 | 0.000000 | 2.22045×10-16 | 0.000000 |
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0.1 | 0.000333 | 0.000333 | 0.000331 | 0.000324 | 3.84892×10-17 | 2.27034×10-6 | 9.00070×10-6 |
0.2 | 0.002667 | 0.002666 | 0.002663 | 0.002716 | 5.98480×10-17 | 3.60640×10-6 | 4.90556×10-5 |
0.3 | 0.009000 | 0.009000 | 0.008995 | 0.009168 | 7.80626×10-17 | 4.76408×10-6 | 1.67919×10-4 |
0.4 | 0.021333 | 0.021333 | 0.021327 | 0.021671 | 1.00614×10-16 | 6.03412×10-6 | 3.37354×10-4 |
0.5 | 0.041667 | 0.041666 | 0.041659 | 0.042210 | 1.17961×10-16 | 7.38289×10-6 | 5.43291×10-4 |
0.6 | 0.072000 | 0.072000 | 0.071991 | 0.072768 | 1.52656×10-16 | 8.59316×10-6 | 7.67981×10-4 |
0.7 | 0.114333 | 0.114333 | 0.114324 | 0.115323 | 1.66533×10-16 | 9.40486×10-6 | 9.90150×10-4 |
0.8 | 0.170667 | 0.170667 | 0.170657 | 0.171852 | 1.94289×10-16 | 9.65586×10-6 | 1.18515×10-3 |
0.9 | 0.243000 | 0.243000 | 0.242991 | 0.244325 | 1.94289×10-16 | 9.42277×10-6 | 1.32513×10-3 |
1.0 | 0.333333 | 0.333333 | 0.333324 | 0.334712 | 2.22045×10-16 | 9.16170×10-6 | 1.37915×10-3 |
Cases | Algorithm | FV | |
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Case I | DEA | 2.53316×10-29 | 1.00615×10-16 | 8.60394×10-17 | 1.000000 | 9.66226×10-32 |
BATA | 1.67350×10-9 | 9.72613×10-6 | 9.90934×10-6 | 1.000000 | 1.48160×10-8 | |
APSOA | 3.10631×10-6 | 2.64036×10-3 | 1.22854×10-3 | 0.999986 | 1.40408×10-5 | |
Case Ⅱ | DEA | 1.42109×10-14 | 2.82911×10-15 | 2.63426×10-15 | 1.000000 | 9.32853×10-28 |
BATA | 6.07290×10-9 | 2.38838×10-6 | 1.79948×10-6 | 1.000000 | 6.64846×10-10 | |
APSOA | 4.70803×10-5 | 4.43119×10-4 | 3.45271×10-4 | 0.999977 | 2.22885×10-5 | |
Case Ⅲ | DEA | 4.64252×10-14 | 4.06611×10-8 | 3.70035×10-8 | 1.000000 | 1.76247×10-14 |
BATA | 1.87730×10-9 | 1.19083×10-5 | 1.08990×10-5 | 1.000000 | 1.53321×10-9 | |
APSOA | 3.86227×10-5 | 1.37820×10-3 | 1.26506×10-3 | 0.999981 | 1.92306×10-5 | |
Case Ⅳ | DEA | 1.13794×10-16 | 1.45270×10-16 | 1.11022×10-16 | 1.000000 | 1.65656×10-30 |
BATA | 2.22120×10-9 | 7.48351×10-6 | 6.39056×10-6 | 1.000000 | 4.39610×10-9 | |
APSOA | 5.51747×10-5 | 8.40980×10-4 | 6.14017×10-4 | 0.999944 | 5.55517×10-5 |
Cases | |
|
|
|
Case I | 1.23390×10-4 | 9.86141×10-5 | 1.000000 | 2.47705×10-10 |
Case Ⅱ | 1.62394×10-3 | 1.41170×10-3 | 1.000000 | 6.45741×10-8 |
Case Ⅲ | 1.36834×10-4 | 1.18751×10-4 | 1.000000 | 3.04623×10-10 |
Case Ⅳ | 1.64242×10-2 | 1.03170×10-2 | 9.999980 | 1.68620×10-6 |
The above discussion validates about the correctness and accuracy of the NROT incorporated with BATA, APSOA and DEA. The numerical results also interprets about the outclass performance of the BATA in terms of accuracy and competency which outstands the APSOA. Whereas, the DEA computed results outperforms both [36,37]. Moreover, the deliberated scheme (NROT) is better than all the former numerical and approximate techniques defined in the literature, as it requires no linearization, integration or transformation to successfully solve the MFDEs subject to different possible variants of boundary conditions. Accuracy, competency and computational intelligence are the main attributes of the suggested scheme, which remains valid even for a larger span of the computational domain.
Statistical analysis
This section contains the results of statistical analysis in terms of mean, standard deviation and mean deviation for the fitness achieved by the execution of 100 independent runs to solve the above defined cases of the MFDEs. The reliability and accuracy of the recommended scheme are validated by this analysis.
To optimize the fitness functions (28), (32), (36) and (40) defined for the test cases Ⅰ–Ⅳ respectively, 100 independent runs are carried out by using BATA for
Cases | k | MINF | MAXF | GMF | STDF | MDF |
Case I | 5 | 1.67350×10-9 | 2.60170×10-3 | 1.00588×10-4 | 3.71797×10-4 | 1.31929×10-4 |
6 | 7.42850×10-8 | 2.60550×10-4 | 1.56482×10-5 | 3.22757×10-5 | 1.59137×10-5 | |
7 | 3.01530×10-8 | 1.20840×10-3 | 9.44284×10-5 | 2.14560×10-4 | 1.24477×10-4 | |
Case Ⅱ | 5 | 6.07290×10-9 | 6.99340×10-2 | 8.57081×10-4 | 1.37830×10-4 | 9.04938×10-4 |
6 | 9.53340×10-8 | 3.57130×10-2 | 1.71610×10-3 | 4.16909×10-3 | 1.74473×10-3 | |
7 | 5.20710×10-8 | 7.52620×10-2 | 2.34578×10-3 | 9.42239×10-3 | 3.65243×10-3 | |
Case Ⅲ | 5 | 1.87730×10-9 | 3.62650×10-4 | 5.02668×10-5 | 6.38566×10-5 | 4.46301×10-5 |
6 | 3.21290×10-8 | 3.50730×10-2 | 1.11512×10-3 | 4.56102×10-3 | 1.75846×10-3 | |
7 | 4.30290×10-8 | 1.39250×10-2 | 7.87759×10-4 | 2.10378×10-3 | 1.00833×10-3 | |
Case Ⅳ | 5 | 2.22120×10-9 | 6.18400×10-2 | 2.87805×10-3 | 1.13173×10-2 | 5.28277×10-3 |
6 | 5.75010×10-8 | 5.39330×10-2 | 2.61573×10-3 | 1.06274×10-2 | 4.69668×10-3 | |
7 | 5.86660×10-8 | 5.87900×10-2 | 6.89697×10-3 | 1.57383×10-2 | 1.12698×10-2 |
Cases | Case I | Case Ⅱ | Case Ⅲ | Case Ⅳ | ||||||||||||||
% Run with fitness ≤ | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | ||||||
10-3 | 100 | 100 | 100 | 100 | 97 | 95 | 100 | 96 | 99 | 94 | 95 | 83 | ||||||
10-4 | 97 | 100 | 98 | 100 | 85 | 81 | 100 | 90 | 86 | 93 | 92 | 81 | ||||||
10-5 | 86 | 97 | 81 | 75 | 56 | 37 | 84 | 59 | 43 | 80 | 72 | 67 | ||||||
10-6 | 34 | 61 | 43 | 30 | 15 | 14 | 30 | 19 | 13 | 38 | 28 | 34 | ||||||
10-7 | 10 | 13 | 9 | 8 | 6 | 5 | 10 | 5 | 6 | 14 | 8 | 8 |
In this endeavor, a novel bio-inspired meta-heuristic approach based on the idea of RPSM with the BATA, is successfully applied to solve linear and non-linear MFDEs subject to different possible sets of two point boundary conditions. The competency and effectiveness of the NROT is validated by conducting numerical experiments consuming the offered scheme in combination with two other meta-heuristic techniques. A concrete inference on the performance of the offered technique in terms of accuracy and convergence is drawn by the statistical analysis based on the hundred independent runs. The contribution of the suggested scheme can be briefly narrated in form of the salient features given as:
● A competitive bio-inspired meta-heuristic solver is presented for the assessment of the MFDEs with fractional derivative defined in the Caputo sense by using BATA.
● The design methodology is well suited to solve the MFDEs for different possible variants of two point boundary conditions, even for a larger span.
● The numerical results constructed by the two other popular optimization techniques i.e., DEA and APSOA for the learning of
● Comparative study of the results computed using the NROT with the available exact solution evidences the outstanding performance of the suggested scheme in terms of accuracy, computational cost and simplicity.
● Accuracy and reliability of the deliberated scheme is validated by the accomplished near optimal values of the error indices based on the root mean square error
● Statistical analysis show that consistently accurate results are obtained for more than 80% of the independent runs of BATA for all the test cases based on the criterion
In the future, one may explore the dynamics of various physical phenomena of real life interpreted in terms of the fractional partial differential equations by the implementation of the recommended technique incorporated with another new and powerful meta-heuristic optimization technique.
The authors declare no conflict of interest.
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2. | Bashir Ahmad, Manal Alblewi, Sotiris K. Ntouyas, Ahmed Alsaedi, Existence results for a coupled system of nonlinear multi‐term fractional differential equations with anti‐periodic type coupled nonlocal boundary conditions, 2021, 0170-4214, 10.1002/mma.7301 | |
3. | Moa’ath N. Oqielat, Tareq Eriqat, Zeyad Al-Zhour, Osama Ogilat, Ahmad El-Ajou, Ishak Hashim, Construction of fractional series solutions to nonlinear fractional reaction–diffusion for bacteria growth model via Laplace residual power series method, 2022, 2195-268X, 10.1007/s40435-022-01001-8 | |
4. | Bashir Ahmad, Ymnah Alruwaily, Ahmed Alsaedi, Sotiris K. Ntouyas, Sequential fractional differential equations with nonlocal integro-multipoint boundary conditions , 2021, Accepted, 1450-5444, 10.30755/NSJOM.12668 | |
5. | Najeeb Alam Khan, Samreen Ahmed, Finite Difference Method with Metaheuristic Orientation for Exploration of Time Fractional Partial Differential Equations, 2021, 7, 2349-5103, 10.1007/s40819-021-01061-y |
BATA | APSOA | DEA | |||||
Parameter | Value | Parameter | Value | Parameter | Value | ||
|
0 | Maximum swarm size | 80 | Cross Probability | 0.5 | ||
|
2 | Maximum number of flights | 2000 | Initial Points | Default | ||
Maximum bat population size | 40 | Lower bound | -2 | Penelty Function | Default | ||
|
1 | Upper bound | 2 | Post Process | Default | ||
Loudness | 1 | Accelaration constant |
0.7 | Random Seed | 0 | ||
Pulse rate | 0.05 | Accelaration constant |
0.5 | Scaling Factor | 0.6 | ||
Maximum number of generations, |
2000 | Step size | 0.1 | Step size | 0.1 | ||
Step size | 0.1 | Search Points | Default | ||||
Tolerance | 0.001 |
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0.1 | 0.01 | 0.01 | 0.009988 | 0.012820 | 1.02349×10-16 | 1.17878×10-5 | 2.82030×10-3 |
0.2 | 0.04 | 0.04 | 0.039989 | 0.042810 | 7.63278×10-17 | 1.09495×10-5 | 2.81007×10-3 |
0.3 | 0.09 | 0.09 | 0.089999 | 0.092786 | 4.16334×10-17 | 1.00600×10-5 | 2.78584×10-3 |
0.4 | 0.16 | 0.16 | 0.159991 | 0.162745 | 5.55112×10-17 | 9.35407×10-6 | 2.74546×10-3 |
0.5 | 0.25 | 0.25 | 0.249991 | 0.252690 | 5.55112×10-17 | 8.93795×10-6 | 2.68994×10-3 |
0.6 | 0.36 | 0.36 | 0.359991 | 0.362623 | 5.55112×10-17 | 8.81746×10-6 | 2.62319×10-3 |
0.7 | 0.49 | 0.49 | 0.489991 | 0.492552 | 1.11022×10-16 | 8.92606×10-6 | 2.55189×10-3 |
0.8 | 0.64 | 0.64 | 0.639991 | 0.642485 | 1.11022×10-16 | 9.15278×10-6 | 2.48528×10-3 |
0.9 | 0.81 | 0.81 | 0.809991 | 0.812435 | 2.22045×10-16 | 9.37028×10-6 | 2.43497×10-3 |
1.0 | 1.00 | 1.00 | 0.999991 | 1.002410 | 0.000000 | 9.46283×10-6 | 2.41478×10-3 |
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0.1 | -0.09 | -0.09 | -0.089999 | -0.090088 | 7.49401×10-16 | 3.34267×10-7 | 8.76839×10-5 |
0.2 | -0.16 | -0.16 | -0.159999 | -0.160208 | 2.77556×10-15 | 8.73920×10-7 | 2.07961×10-4 |
0.3 | -0.21 | -0.21 | -0.209999 | -0.210359 | 4.41314×10-15 | 1.29197×10-6 | 3.58686×10-4 |
0.4 | -0.24 | -0.24 | -0.239998 | -0.240513 | 4.91274×10-15 | 1.60640×10-6 | 5.13425×10-4 |
0.5 | -0.25 | -0.25 | -0.249998 | -0.250633 | 4.27436×10-15 | 1.99844×10-6 | 6.33424×10-4 |
0.6 | -0.24 | -0.24 | -0.239997 | -0.240680 | 2.83107×10-15 | 2.63088×10-6 | 6.79560×10-4 |
0.7 | -0.21 | -0.21 | -0.209997 | -0.210624 | 1.33227×10-15 | 3.46638×10-6 | 6.24306×10-4 |
0.8 | -0.16 | -0.16 | -0.159996 | -0.160464 | 3.05311×10-16 | 4.08576×10-6 | 4.63690×10-4 |
0.9 | -0.09 | -0.09 | -0.089996 | -0.090229 | 9.71445×10-17 | 3.50629×10-6 | 2.29250×10-4 |
1.0 | 0.00 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
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0.1 | 1.01 | 1.01 | 1.009980 | 1.008052 | 6.48762×10-8 | 1.95158×10-5 | 1.94787×10-3 |
0.2 | 1.04 | 1.04 | 1.039982 | 1.038052 | 6.12838×10-8 | 1.81608×10-5 | 1.92087×10-3 |
0.3 | 1.09 | 1.09 | 1.089984 | 1.088150 | 5.55299×10-8 | 1.61406×10-5 | 1.84995×10-3 |
0.4 | 1.16 | 1.16 | 1.159986 | 1.158281 | 4.80703×10-8 | 1.37066×10-5 | 1.71900×10-3 |
0.5 | 1.25 | 1.25 | 1.249989 | 1.248481 | 3.95009×10-8 | 1.11155×10-5 | 1.51917×10-3 |
0.6 | 1.36 | 1.36 | 1.359991 | 1.358749 | 3.04728×10-8 | 8.58836×10-6 | 1.25134×10-3 |
0.7 | 1.49 | 1.49 | 1.489994 | 1.489071 | 2.16083×10-8 | 6.26947×10-6 | 9.28685×10-4 |
0.8 | 1.64 | 1.64 | 1.639996 | 1.639421 | 1.34155×10-8 | 4.18586×10-6 | 5.79129×10-4 |
0.9 | 1.81 | 1.81 | 1.809998 | 1.809752 | 6.20399×10-9 | 2.20620×10-6 | 2.47895×10-4 |
1.0 | 2.00 | 2.00 | 2.000000 | 2.000000 | 0.000000 | 2.22045×10-16 | 0.000000 |
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0.1 | 0.000333 | 0.000333 | 0.000331 | 0.000324 | 3.84892×10-17 | 2.27034×10-6 | 9.00070×10-6 |
0.2 | 0.002667 | 0.002666 | 0.002663 | 0.002716 | 5.98480×10-17 | 3.60640×10-6 | 4.90556×10-5 |
0.3 | 0.009000 | 0.009000 | 0.008995 | 0.009168 | 7.80626×10-17 | 4.76408×10-6 | 1.67919×10-4 |
0.4 | 0.021333 | 0.021333 | 0.021327 | 0.021671 | 1.00614×10-16 | 6.03412×10-6 | 3.37354×10-4 |
0.5 | 0.041667 | 0.041666 | 0.041659 | 0.042210 | 1.17961×10-16 | 7.38289×10-6 | 5.43291×10-4 |
0.6 | 0.072000 | 0.072000 | 0.071991 | 0.072768 | 1.52656×10-16 | 8.59316×10-6 | 7.67981×10-4 |
0.7 | 0.114333 | 0.114333 | 0.114324 | 0.115323 | 1.66533×10-16 | 9.40486×10-6 | 9.90150×10-4 |
0.8 | 0.170667 | 0.170667 | 0.170657 | 0.171852 | 1.94289×10-16 | 9.65586×10-6 | 1.18515×10-3 |
0.9 | 0.243000 | 0.243000 | 0.242991 | 0.244325 | 1.94289×10-16 | 9.42277×10-6 | 1.32513×10-3 |
1.0 | 0.333333 | 0.333333 | 0.333324 | 0.334712 | 2.22045×10-16 | 9.16170×10-6 | 1.37915×10-3 |
Cases | Algorithm | FV | |
|
|
|
Case I | DEA | 2.53316×10-29 | 1.00615×10-16 | 8.60394×10-17 | 1.000000 | 9.66226×10-32 |
BATA | 1.67350×10-9 | 9.72613×10-6 | 9.90934×10-6 | 1.000000 | 1.48160×10-8 | |
APSOA | 3.10631×10-6 | 2.64036×10-3 | 1.22854×10-3 | 0.999986 | 1.40408×10-5 | |
Case Ⅱ | DEA | 1.42109×10-14 | 2.82911×10-15 | 2.63426×10-15 | 1.000000 | 9.32853×10-28 |
BATA | 6.07290×10-9 | 2.38838×10-6 | 1.79948×10-6 | 1.000000 | 6.64846×10-10 | |
APSOA | 4.70803×10-5 | 4.43119×10-4 | 3.45271×10-4 | 0.999977 | 2.22885×10-5 | |
Case Ⅲ | DEA | 4.64252×10-14 | 4.06611×10-8 | 3.70035×10-8 | 1.000000 | 1.76247×10-14 |
BATA | 1.87730×10-9 | 1.19083×10-5 | 1.08990×10-5 | 1.000000 | 1.53321×10-9 | |
APSOA | 3.86227×10-5 | 1.37820×10-3 | 1.26506×10-3 | 0.999981 | 1.92306×10-5 | |
Case Ⅳ | DEA | 1.13794×10-16 | 1.45270×10-16 | 1.11022×10-16 | 1.000000 | 1.65656×10-30 |
BATA | 2.22120×10-9 | 7.48351×10-6 | 6.39056×10-6 | 1.000000 | 4.39610×10-9 | |
APSOA | 5.51747×10-5 | 8.40980×10-4 | 6.14017×10-4 | 0.999944 | 5.55517×10-5 |
Cases | |
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Case I | 1.23390×10-4 | 9.86141×10-5 | 1.000000 | 2.47705×10-10 |
Case Ⅱ | 1.62394×10-3 | 1.41170×10-3 | 1.000000 | 6.45741×10-8 |
Case Ⅲ | 1.36834×10-4 | 1.18751×10-4 | 1.000000 | 3.04623×10-10 |
Case Ⅳ | 1.64242×10-2 | 1.03170×10-2 | 9.999980 | 1.68620×10-6 |
Cases | k | MINF | MAXF | GMF | STDF | MDF |
Case I | 5 | 1.67350×10-9 | 2.60170×10-3 | 1.00588×10-4 | 3.71797×10-4 | 1.31929×10-4 |
6 | 7.42850×10-8 | 2.60550×10-4 | 1.56482×10-5 | 3.22757×10-5 | 1.59137×10-5 | |
7 | 3.01530×10-8 | 1.20840×10-3 | 9.44284×10-5 | 2.14560×10-4 | 1.24477×10-4 | |
Case Ⅱ | 5 | 6.07290×10-9 | 6.99340×10-2 | 8.57081×10-4 | 1.37830×10-4 | 9.04938×10-4 |
6 | 9.53340×10-8 | 3.57130×10-2 | 1.71610×10-3 | 4.16909×10-3 | 1.74473×10-3 | |
7 | 5.20710×10-8 | 7.52620×10-2 | 2.34578×10-3 | 9.42239×10-3 | 3.65243×10-3 | |
Case Ⅲ | 5 | 1.87730×10-9 | 3.62650×10-4 | 5.02668×10-5 | 6.38566×10-5 | 4.46301×10-5 |
6 | 3.21290×10-8 | 3.50730×10-2 | 1.11512×10-3 | 4.56102×10-3 | 1.75846×10-3 | |
7 | 4.30290×10-8 | 1.39250×10-2 | 7.87759×10-4 | 2.10378×10-3 | 1.00833×10-3 | |
Case Ⅳ | 5 | 2.22120×10-9 | 6.18400×10-2 | 2.87805×10-3 | 1.13173×10-2 | 5.28277×10-3 |
6 | 5.75010×10-8 | 5.39330×10-2 | 2.61573×10-3 | 1.06274×10-2 | 4.69668×10-3 | |
7 | 5.86660×10-8 | 5.87900×10-2 | 6.89697×10-3 | 1.57383×10-2 | 1.12698×10-2 |
Cases | Case I | Case Ⅱ | Case Ⅲ | Case Ⅳ | ||||||||||||||
% Run with fitness ≤ | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | ||||||
10-3 | 100 | 100 | 100 | 100 | 97 | 95 | 100 | 96 | 99 | 94 | 95 | 83 | ||||||
10-4 | 97 | 100 | 98 | 100 | 85 | 81 | 100 | 90 | 86 | 93 | 92 | 81 | ||||||
10-5 | 86 | 97 | 81 | 75 | 56 | 37 | 84 | 59 | 43 | 80 | 72 | 67 | ||||||
10-6 | 34 | 61 | 43 | 30 | 15 | 14 | 30 | 19 | 13 | 38 | 28 | 34 | ||||||
10-7 | 10 | 13 | 9 | 8 | 6 | 5 | 10 | 5 | 6 | 14 | 8 | 8 |
BATA | APSOA | DEA | |||||
Parameter | Value | Parameter | Value | Parameter | Value | ||
|
0 | Maximum swarm size | 80 | Cross Probability | 0.5 | ||
|
2 | Maximum number of flights | 2000 | Initial Points | Default | ||
Maximum bat population size | 40 | Lower bound | -2 | Penelty Function | Default | ||
|
1 | Upper bound | 2 | Post Process | Default | ||
Loudness | 1 | Accelaration constant |
0.7 | Random Seed | 0 | ||
Pulse rate | 0.05 | Accelaration constant |
0.5 | Scaling Factor | 0.6 | ||
Maximum number of generations, |
2000 | Step size | 0.1 | Step size | 0.1 | ||
Step size | 0.1 | Search Points | Default | ||||
Tolerance | 0.001 |
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0.1 | 0.01 | 0.01 | 0.009988 | 0.012820 | 1.02349×10-16 | 1.17878×10-5 | 2.82030×10-3 |
0.2 | 0.04 | 0.04 | 0.039989 | 0.042810 | 7.63278×10-17 | 1.09495×10-5 | 2.81007×10-3 |
0.3 | 0.09 | 0.09 | 0.089999 | 0.092786 | 4.16334×10-17 | 1.00600×10-5 | 2.78584×10-3 |
0.4 | 0.16 | 0.16 | 0.159991 | 0.162745 | 5.55112×10-17 | 9.35407×10-6 | 2.74546×10-3 |
0.5 | 0.25 | 0.25 | 0.249991 | 0.252690 | 5.55112×10-17 | 8.93795×10-6 | 2.68994×10-3 |
0.6 | 0.36 | 0.36 | 0.359991 | 0.362623 | 5.55112×10-17 | 8.81746×10-6 | 2.62319×10-3 |
0.7 | 0.49 | 0.49 | 0.489991 | 0.492552 | 1.11022×10-16 | 8.92606×10-6 | 2.55189×10-3 |
0.8 | 0.64 | 0.64 | 0.639991 | 0.642485 | 1.11022×10-16 | 9.15278×10-6 | 2.48528×10-3 |
0.9 | 0.81 | 0.81 | 0.809991 | 0.812435 | 2.22045×10-16 | 9.37028×10-6 | 2.43497×10-3 |
1.0 | 1.00 | 1.00 | 0.999991 | 1.002410 | 0.000000 | 9.46283×10-6 | 2.41478×10-3 |
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0.1 | -0.09 | -0.09 | -0.089999 | -0.090088 | 7.49401×10-16 | 3.34267×10-7 | 8.76839×10-5 |
0.2 | -0.16 | -0.16 | -0.159999 | -0.160208 | 2.77556×10-15 | 8.73920×10-7 | 2.07961×10-4 |
0.3 | -0.21 | -0.21 | -0.209999 | -0.210359 | 4.41314×10-15 | 1.29197×10-6 | 3.58686×10-4 |
0.4 | -0.24 | -0.24 | -0.239998 | -0.240513 | 4.91274×10-15 | 1.60640×10-6 | 5.13425×10-4 |
0.5 | -0.25 | -0.25 | -0.249998 | -0.250633 | 4.27436×10-15 | 1.99844×10-6 | 6.33424×10-4 |
0.6 | -0.24 | -0.24 | -0.239997 | -0.240680 | 2.83107×10-15 | 2.63088×10-6 | 6.79560×10-4 |
0.7 | -0.21 | -0.21 | -0.209997 | -0.210624 | 1.33227×10-15 | 3.46638×10-6 | 6.24306×10-4 |
0.8 | -0.16 | -0.16 | -0.159996 | -0.160464 | 3.05311×10-16 | 4.08576×10-6 | 4.63690×10-4 |
0.9 | -0.09 | -0.09 | -0.089996 | -0.090229 | 9.71445×10-17 | 3.50629×10-6 | 2.29250×10-4 |
1.0 | 0.00 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
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0.1 | 1.01 | 1.01 | 1.009980 | 1.008052 | 6.48762×10-8 | 1.95158×10-5 | 1.94787×10-3 |
0.2 | 1.04 | 1.04 | 1.039982 | 1.038052 | 6.12838×10-8 | 1.81608×10-5 | 1.92087×10-3 |
0.3 | 1.09 | 1.09 | 1.089984 | 1.088150 | 5.55299×10-8 | 1.61406×10-5 | 1.84995×10-3 |
0.4 | 1.16 | 1.16 | 1.159986 | 1.158281 | 4.80703×10-8 | 1.37066×10-5 | 1.71900×10-3 |
0.5 | 1.25 | 1.25 | 1.249989 | 1.248481 | 3.95009×10-8 | 1.11155×10-5 | 1.51917×10-3 |
0.6 | 1.36 | 1.36 | 1.359991 | 1.358749 | 3.04728×10-8 | 8.58836×10-6 | 1.25134×10-3 |
0.7 | 1.49 | 1.49 | 1.489994 | 1.489071 | 2.16083×10-8 | 6.26947×10-6 | 9.28685×10-4 |
0.8 | 1.64 | 1.64 | 1.639996 | 1.639421 | 1.34155×10-8 | 4.18586×10-6 | 5.79129×10-4 |
0.9 | 1.81 | 1.81 | 1.809998 | 1.809752 | 6.20399×10-9 | 2.20620×10-6 | 2.47895×10-4 |
1.0 | 2.00 | 2.00 | 2.000000 | 2.000000 | 0.000000 | 2.22045×10-16 | 0.000000 |
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0.1 | 0.000333 | 0.000333 | 0.000331 | 0.000324 | 3.84892×10-17 | 2.27034×10-6 | 9.00070×10-6 |
0.2 | 0.002667 | 0.002666 | 0.002663 | 0.002716 | 5.98480×10-17 | 3.60640×10-6 | 4.90556×10-5 |
0.3 | 0.009000 | 0.009000 | 0.008995 | 0.009168 | 7.80626×10-17 | 4.76408×10-6 | 1.67919×10-4 |
0.4 | 0.021333 | 0.021333 | 0.021327 | 0.021671 | 1.00614×10-16 | 6.03412×10-6 | 3.37354×10-4 |
0.5 | 0.041667 | 0.041666 | 0.041659 | 0.042210 | 1.17961×10-16 | 7.38289×10-6 | 5.43291×10-4 |
0.6 | 0.072000 | 0.072000 | 0.071991 | 0.072768 | 1.52656×10-16 | 8.59316×10-6 | 7.67981×10-4 |
0.7 | 0.114333 | 0.114333 | 0.114324 | 0.115323 | 1.66533×10-16 | 9.40486×10-6 | 9.90150×10-4 |
0.8 | 0.170667 | 0.170667 | 0.170657 | 0.171852 | 1.94289×10-16 | 9.65586×10-6 | 1.18515×10-3 |
0.9 | 0.243000 | 0.243000 | 0.242991 | 0.244325 | 1.94289×10-16 | 9.42277×10-6 | 1.32513×10-3 |
1.0 | 0.333333 | 0.333333 | 0.333324 | 0.334712 | 2.22045×10-16 | 9.16170×10-6 | 1.37915×10-3 |
Cases | Algorithm | FV | |
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Case I | DEA | 2.53316×10-29 | 1.00615×10-16 | 8.60394×10-17 | 1.000000 | 9.66226×10-32 |
BATA | 1.67350×10-9 | 9.72613×10-6 | 9.90934×10-6 | 1.000000 | 1.48160×10-8 | |
APSOA | 3.10631×10-6 | 2.64036×10-3 | 1.22854×10-3 | 0.999986 | 1.40408×10-5 | |
Case Ⅱ | DEA | 1.42109×10-14 | 2.82911×10-15 | 2.63426×10-15 | 1.000000 | 9.32853×10-28 |
BATA | 6.07290×10-9 | 2.38838×10-6 | 1.79948×10-6 | 1.000000 | 6.64846×10-10 | |
APSOA | 4.70803×10-5 | 4.43119×10-4 | 3.45271×10-4 | 0.999977 | 2.22885×10-5 | |
Case Ⅲ | DEA | 4.64252×10-14 | 4.06611×10-8 | 3.70035×10-8 | 1.000000 | 1.76247×10-14 |
BATA | 1.87730×10-9 | 1.19083×10-5 | 1.08990×10-5 | 1.000000 | 1.53321×10-9 | |
APSOA | 3.86227×10-5 | 1.37820×10-3 | 1.26506×10-3 | 0.999981 | 1.92306×10-5 | |
Case Ⅳ | DEA | 1.13794×10-16 | 1.45270×10-16 | 1.11022×10-16 | 1.000000 | 1.65656×10-30 |
BATA | 2.22120×10-9 | 7.48351×10-6 | 6.39056×10-6 | 1.000000 | 4.39610×10-9 | |
APSOA | 5.51747×10-5 | 8.40980×10-4 | 6.14017×10-4 | 0.999944 | 5.55517×10-5 |
Cases | |
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|
Case I | 1.23390×10-4 | 9.86141×10-5 | 1.000000 | 2.47705×10-10 |
Case Ⅱ | 1.62394×10-3 | 1.41170×10-3 | 1.000000 | 6.45741×10-8 |
Case Ⅲ | 1.36834×10-4 | 1.18751×10-4 | 1.000000 | 3.04623×10-10 |
Case Ⅳ | 1.64242×10-2 | 1.03170×10-2 | 9.999980 | 1.68620×10-6 |
Cases | k | MINF | MAXF | GMF | STDF | MDF |
Case I | 5 | 1.67350×10-9 | 2.60170×10-3 | 1.00588×10-4 | 3.71797×10-4 | 1.31929×10-4 |
6 | 7.42850×10-8 | 2.60550×10-4 | 1.56482×10-5 | 3.22757×10-5 | 1.59137×10-5 | |
7 | 3.01530×10-8 | 1.20840×10-3 | 9.44284×10-5 | 2.14560×10-4 | 1.24477×10-4 | |
Case Ⅱ | 5 | 6.07290×10-9 | 6.99340×10-2 | 8.57081×10-4 | 1.37830×10-4 | 9.04938×10-4 |
6 | 9.53340×10-8 | 3.57130×10-2 | 1.71610×10-3 | 4.16909×10-3 | 1.74473×10-3 | |
7 | 5.20710×10-8 | 7.52620×10-2 | 2.34578×10-3 | 9.42239×10-3 | 3.65243×10-3 | |
Case Ⅲ | 5 | 1.87730×10-9 | 3.62650×10-4 | 5.02668×10-5 | 6.38566×10-5 | 4.46301×10-5 |
6 | 3.21290×10-8 | 3.50730×10-2 | 1.11512×10-3 | 4.56102×10-3 | 1.75846×10-3 | |
7 | 4.30290×10-8 | 1.39250×10-2 | 7.87759×10-4 | 2.10378×10-3 | 1.00833×10-3 | |
Case Ⅳ | 5 | 2.22120×10-9 | 6.18400×10-2 | 2.87805×10-3 | 1.13173×10-2 | 5.28277×10-3 |
6 | 5.75010×10-8 | 5.39330×10-2 | 2.61573×10-3 | 1.06274×10-2 | 4.69668×10-3 | |
7 | 5.86660×10-8 | 5.87900×10-2 | 6.89697×10-3 | 1.57383×10-2 | 1.12698×10-2 |
Cases | Case I | Case Ⅱ | Case Ⅲ | Case Ⅳ | ||||||||||||||
% Run with fitness ≤ | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | k=5 | k=6 | k=7 | ||||||
10-3 | 100 | 100 | 100 | 100 | 97 | 95 | 100 | 96 | 99 | 94 | 95 | 83 | ||||||
10-4 | 97 | 100 | 98 | 100 | 85 | 81 | 100 | 90 | 86 | 93 | 92 | 81 | ||||||
10-5 | 86 | 97 | 81 | 75 | 56 | 37 | 84 | 59 | 43 | 80 | 72 | 67 | ||||||
10-6 | 34 | 61 | 43 | 30 | 15 | 14 | 30 | 19 | 13 | 38 | 28 | 34 | ||||||
10-7 | 10 | 13 | 9 | 8 | 6 | 5 | 10 | 5 | 6 | 14 | 8 | 8 |