
This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter σ in the initial data, we revealed a threshold value σ∗ such that spreading happens when σ>σ∗ and vanishing happens when σ≤σ∗. There exists a unique L∗>0 independent of the initial data such that σ∗=0 if and only if the length of initial occupying interval is no smaller than 2L∗. These theoretical results may have important implications for prediction and prevention of biological invasions.
Citation: Di Zhang, Ningkui Sun, Xuemei Han. Dynamical behavior of solutions of a free boundary problem[J]. Mathematical Modelling and Control, 2024, 4(1): 1-8. doi: 10.3934/mmc.2024001
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This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter σ in the initial data, we revealed a threshold value σ∗ such that spreading happens when σ>σ∗ and vanishing happens when σ≤σ∗. There exists a unique L∗>0 independent of the initial data such that σ∗=0 if and only if the length of initial occupying interval is no smaller than 2L∗. These theoretical results may have important implications for prediction and prevention of biological invasions.
The computationally relative simple interval type-2 fuzzy sets (IT2 FSs) are currently the most commonly used T2 FSs. Therefore, interval type-2 fuzzy logic systems (IT2 FLSs [1,2,57]) based on IT2 FSs have superior ability to cope with uncertainties environments like power systems [3], permanent magnetic drive [4,5,6], intelligent controllers [7], medical systems [8], pattern recognition systems [9], database and information systems [10] and so on. The footprint of uncertainty (FOU) of an IT2 FS [11] make it own more design degrees of freedom compared with a T1 FS. Generally speaking, a T2 FLS (see Figure 1) is composed of fuzzifier, rules, inference, type-reducer and defuzzifier. Among which, the block of type-reduction plays the central role of transforming the T2 to the T1 FS. Finally the defuzzification changes the T1 FS to the crisp output.
In the past decades, many types of type-reduction (TR) were proposed gradually. Among which, the most famous one is the Karnik-Mendel (KM) algorithms [13]. This type of algorithms has the advantages of preserving the uncertainties flow between the upper and lower membership functions (MFs) of T2 FLSs. Then the continuous KM (CKM) algorithms [14] were put forward, in addition, the monotoncity and super convergence property of them were proved. For the sake of improving the calculation efficiency, Wu and Mendel proposed the enhanced KM (EKM) algorithms [15]. Extensive simulation experiments show that the EKM algorithms can save two iterations on average compared with the KM algorithms. Then Liu et al. gave the theoretical explanations for the initialization of EKM algorithms and extended the EKM algorithms to three different forms of weighted-based EKM (WEKM) algorithms [16,17] to calculate the more accurate centroids of type-2 fuzzy sets.
However, the iterative natures of KM types of algorithms make the applications of corresponding IT2 FLSs more challenge. Therefore, other types of noniterative algorithms [18] are proposed gradually, and they are Nagar-Bardini (NB) algorithms [19], Nie-Tan (NT) algorithms [20], Begian-Melek-Mendel (BMM) algorithms [21,22] and so on. Among which, IT2 FLSs based on the NB algorithms are proved to have superior performances to respond to the affect of uncertainties in systems’ parameters to the IT2 FLSs according to other algorithms like EKM, BMM, Greenfield-Chiclana Collapsing Defuzzifier (GCCD [24]) and Wu-Mendel Uncertainty Bound (UB [24]). Moreover, Nie-Tan (NT) algorithms have the simple closed form. Recent studies prove that the continuous NT (CNT) algorithms [20] to be accurate algorithms to calculate the centroid of IT2 FSs. In addition, BMM algorithms [18] are proved to be more generalized forms of NB and NT algorithms. All these works have laid theoretical foundations for studying the TR of T2 FLSs.
In order to obtain more accurate centroid TR of IT2 FLSs, this paper extends the NB, NT, and BMM algorithms to the corresponding WNB, WNT, and WBMM algorithms according to the Newton-Cotes quadrature formulas. The rest of this paper is organized as follows. Section 2 introduces the background of IT2 FLSs. Section 3 provides the Newton-Cotes formulas, the weighted-based noniterative algorithms, and how to adopt them to perform the centroid TR of IT2 FLSs. Section 4 gives four computer simulation examples to illustrate the performances of weighted-based noniterative algorithms. Finally Section 5 is the conclusions and expectations.
Definition 1. A T2 FS
˜A={(x,u),μ˜A(x,u)|∀x∈X,∀u∈[0,1]} | (1) |
where the primary variable
˜A=∫x∈X∫u∈[0,1]μ˜A(x,u)/(x,u) | (2) |
Definition 2. The secondary MF of
μ˜A(x=x′,u)≡μ˜A(x′)=∫∀u∈[0,1]fx′(u)/u. | (3) |
Definition 3. The two dimensional support of
FOU(˜A)=∪x∈XJx={(x,u)∈X×[0,1]|μ˜A(x,u)>0} | (4) |
The upper and lower bounds of
UMF(˜A)=¯μ˜A(x)=¯FOU(˜A),LMF(˜A)=μ_˜A(x)=FOU(˜A)_. | (5) |
Because the secondary membership grades of IT2 FSs are all uniformly equal to 1, i.e.,
˜A=∫∀x∈Xμ˜A(x)/x=∫∀x∈X˜A(x)/x | (6) |
Here
Definition 4. An embedded T1 FS
Ae={(x,u(x)|∀x∈X,u∈Jx}. | (7) |
Definition 5. An IT2 FS can be considered as the union of all its embedded T2 FSs
˜A=m∑j=1˜Aje | (8) |
where
From the aspect of inference structure, IT2 FLSs can usually be divided into two categories: Mamdani type [3,6,12,18,25] and Takagi-Sugeno-Kang type [4,25,26,27]. Without loss of generality, here we consider a Mamdani IT2 FLS with
˜Rs:Ifx1is˜Fs1and⋯andxnis˜Fsn, thenyis ˜Gs(s=1,2,⋯,M) | (9) |
in which
For simplicity, here we use the singleton fuzzifier, i.e., the input measurements are modeled as crisp sets (type-0 FSs). As
Fs:{Fs(x′)≡[f_s(x′),¯fs(x′)],f_s(x′)≡Tni=1μ_˜Fsi(x′i),¯fs(x′)≡Tni=1¯μ˜Fsi(x′i) | (10) |
where
As for the centroid TR, we combine the firing interval of each rule with its consequent IT2 FS to obtain the fired-rule output FS
˜Bs:{FOU(˜Bs)=[μ_˜Bs(y|x′),¯μ˜Bs(y|x′)],μ_˜Bs(y|x′)=f_s(x′)∗μ_˜Gs(y),¯μ˜Bs(y|x′)=¯fs(x′)∗¯μ˜Gs(y) | (11) |
where
Then the output IT2 FS
˜B:{FOU(˜B)=[μ_˜B(y|x′),¯μ˜B(y|x′)],μ_˜B(y|x′)=μ_˜B1(y|x′)∨μ_˜B2(y|x′)∨⋯∨μ_˜BM(y|x′),¯μ˜B(y|x′)=¯μ˜B1(y|x′)∨¯μ˜B2(y|x′)∨⋯∨¯μ˜BM(y|x′) | (12) |
where
Finally the type-reduced set
YC(x′)=1/[l˜B(x′),r˜B(x′)] | (13) |
where the two end points
Before introducing the weighted-based noniterative algorithms, we first give the preliminary knowledge: Newton-Cotes quadrature formulas [16,30].
Generally speaking, the numerical integration is an approach that approximates the definite integral
Definition 6. (Quadrature formula [16,17,46,47]) Let the discrete points satisfy that
Q(f)=N∑i=0wif(xi)=w0f(x0)+⋯+wNf(xN) | (14) |
then Eq (14) is referred to as the numerical integration or quadrature formula, where
Next, the composite trapezoidal rule, composite Simpson rule, and composite Simpson 3/8 rule to adopted to approximate
Theorem 1. (Composite trapezoidal rule [16,17,46,47]) Let
∫baf(x)dx=h2[f(a)+f(b)+2N−1∑i=1f(xi)]+ET(f,h) | (15) |
Suppose that
Theorem 2. (Composite Simpson rule [16,17,46,47]) Let
∫baf(x)dx=h3[f(a)+f(b)+2N−1∑i=1f(x2i)+4N−1∑i=0f(x2i+1)]+ES(f,h) | (16) |
Suppose that
Theorem 3. (Composite Simpson 3/8 rule [16,17,46,47]) Let
∫baf(x)dx=3h8[f(a)+f(b)+N∑i=12f(x3i)+N∑i=13f(x3i−2)+N∑i=13f(x3i−1)]+ESC(f,h) | (17) |
Suppose that
Here all the integrals are measured in the Lebesgue sense.
IT2 FLSs based on the closed form of Nagar-Bardini (NB) algorithms [19,46,47] can make distinctly improvement on coping with uncertainties. For the centroid output IT2 FS
l˜B=N∑i=1yiμ_˜B(yi)N∑i=1μ_˜B(yi), and r˜B=N∑i=1yi¯μ˜B(yi)N∑i=1¯μ˜B(yi). | (18) |
Then the defuzzified output can be obtained as:
yNB=l˜B+r˜B2. | (19) |
Similar to the continuous KM types of algorithms [13,14,15,16,17,31], the continuous NB (CNB) algorithms can be adopted for studying the theoretical property of centroid TR and defuzzification of IT2 FLSs.
Let
l˜B=∫bayμ_˜B(y)dy∫baμ_˜B(y)dy, and r˜B=∫bay¯μ˜B(y)dy∫ba¯μ˜B(y)dy. | (20) |
Here the output centroid two end points can be computed without iterations. Furthermore, the output is a linear combination of the output of two T1 FLSs: one constructed from the LMFs, and the other constructed from the UMFs.
In this section, we propose a type of weighted NB (WNB) algorithms, i.e.,
l˜B=N∑i=1wiyiμ_˜B(yi)N∑i=1wiμ_˜B(yi), and r˜B=N∑i=1wiyi¯μ˜B(yi)N∑i=1wi¯μ˜B(yi). | (21) |
WNB algorithms can be considered as the numerical implementation of CNB algorithms. Comparing Eqs (18) and (20), it is found that the CNB and NB algorithms are very similar, i.e., the sum operations in the discrete version are transformed into the definite integral operations in the continuous version, i.e., the sum operations for the sampling points
According to the quadrature formula (see Eq (14)), the corresponding weights
Algorithms | Integration rule | Weights |
NB, NT, BMM | ______________ | |
TWNB, TWNT, TWBMM | Composite Trapezoidal rule | |
SWNB, SWNT, SWBMM | Composite Simpson rule | |
S3/8WNB, S3/8WNT, S3/8WBMM | Composite Simpson 3/8 rule |
The closed form of discrete Nie-Tan (NT) algorithms can compute the centroid output of
yNT=N∑i=1yi[μ_˜B(yi)+¯μ˜B(yi)]N∑i=1[μ_˜B(yi)+¯μ˜B(yi)]. | (22) |
Most recent studies prove the continuous NT (CNT) algorithms [20] to be an accurate method for computing the centroids of IT2 FSs, i.e.,
yCNT=∫bay[μ_˜B(y)+¯μ˜B(y)]dy∫ba[μ_˜B(y)+¯μ˜B(y)]dy. | (23) |
This section proposes a type of weighted NT (WNT) algorithms, i.e.,
yWNT=N∑i=1wiyi[μ_˜B(yi)+¯μ˜B(yi)]N∑i=1wi[μ_˜B(yi)+¯μ˜B(yi)]. | (24) |
WNT algorithms can be viewed as the numerical implementation of CNT algorithms. The sum operations in the discrete NT algorithms are transformed into the definite integral operations in the continuous NT (CNT) algorithms, i.e., the sum operations for the sampling points
Begian-Melek-Mendel (BMM) algorithms can also obtain the output of IT2 FLSs directly, i.e.,
yBMM=αN∑i=1yiμ_˜B(yi)N∑i=1μ_˜B(yi)+βN∑i=1yi¯μ˜B(yi)N∑i=1¯μ˜B(yi) | (25) |
where
IT2 FLSs based on BMM algorithms [13,21,22] are superior to T1 FLSs counterparts on both robustness and stability. In addition, the BMM algorithms are more generalized form of NB and NT algorithms. Observing the Eqs (19) and (25), it can be found that BMM and NB algorithms are exactly the same while
yNT=N∑i=1μ_˜B(yi)N∑i=1[μ_˜B(yi)+¯μ˜B(yi)]×N∑i=1yiμ_˜B(yi)N∑i=1μ_˜B(yi)+N∑i=1¯μ˜B(yi)N∑i=1[μ_˜B(yi)+¯μ˜B(yi)]×N∑i=1yi¯μ˜B(yi)N∑i=1¯μ˜B(yi) =αNTN∑i=1yiμ_˜B(yi)N∑i=1μ_˜B(yi)+βNTN∑i=1yi¯μ˜B(yi)N∑i=1¯μ˜B(yi) | (26) |
in which
αNT=N∑i=1μ_˜B(yi)N∑i=1[μ_˜B(yi)+¯μ˜B(yi)], and βNT=N∑i=1¯μ˜B(yi)N∑i=1[μ_˜B(yi)+¯μ˜B(yi)]. | (27) |
Comparing the Eqs (25) and (26), it can be found that, as
Continuous BMM (CBMM) algorithms can also be used for studying the theoretical properties of TR of IT2 FLSs, i.e.,
yCBMM=α∫bayμ_˜B(y)dy∫baμ_˜B(y)dy+β∫bay¯μ˜B(y)dy∫ba¯μ˜B(y)dy. | (28) |
Based on the quadrature formula, this section gives a type of weighted BMM (WBMM) algorithms, i.e.,
yWBMM=αN∑i=1wiyiμ_˜B(yi)N∑i=1wiμ_˜B(yi)+βN∑i=1wiyi¯μ˜B(yi)N∑i=1wi¯μ˜B(yi). | (29) |
Here WBMM algorithms can be considered as the numerical implementation of CBMM algorithms. Comparing Eqs (25) and (28), it is found that the CBMM and BMM algorithms are very similar, i.e., the sum operations in the discrete version are transformed into the definite integral operations in the continuous version, i.e., the sum operations for the sampling points
For the above three types of weighed-based noniterative algorithms (see Table 1), suppose that the primary variable be the letter
Next we can make conclusions about assigning weights for three types of weighted-based noniterative algorithms as:
1) Substitute
2) The coefficients
3) In Tables 1–3, for the TWNB, TWNT, TWBMM, and SWNB, SWNT, SWBMM algorithms, the weights are assigned as
Num | Expression |
1 | |
2 | |
3 | |
4 |
Num | Approach | ||
CNB | CNT | CBMM | |
1 | 4.3050 | 4.3208 | 4.3041 |
2 | 3.7774 | 3.7141 | 3.7747 |
3 | 4.3702 | 4.3953 | 4.3690 |
4 | 5.0000 | 5.0000 | 5.0000 |
4) In Tables 1–3, for the SWNB, SWNT, and SWBMM, and S3/8WNB, S3/8WNT, and S3/8WBMM algorithms, the number of sampling
Finally the inner relations between weighted-based noniterative algorithms and continuous noniterative algorithms for performing the centroid TR of IT2 FLSs can be made as:
1) Weighted-based noniterative algorithms calculate the type-reduced set
2) As the number of sampling increases, weighted-based noniterative algorithms may obtain more accurate computational results.
3) Weighted-based noniterative algorithms perform the numerical calculations according to the sum operation, whereas the continuous noniterative algorithms perform the calculations symbolically by means of the integral operations. On the whole, weighted-based noniterative algorithms can be viewed as the numerical implementation of continuous noniterative algorithms according to the numerical integration approaches.
Four computer simulation examples are provided in this section. Here we suppose that the centroid output IT2 FS [16,18,29,32] has been obtained by merging or weighting fuzzy rules under the guidance the inference before the TR and defuzzification. For the first example, the FOU is bounded by the piece-wise linear functions. For the second example, the FOU is bounded by both the piece-wise linear functions and Gaussian functions. For the third example, the FOU is bounded by the Gaussian functions. For the last example, the FOU is defined as the symmetric Gaussian primary MF with uncertainty derivations. Then Figure 2 and Table 2 show the defined FOUs for four examples. In examples 1, 3 and 4, here we choose the primary variable
Firstly, the CNB, CNT, and CBMM algorithms are considered the benchmarks to compute the centroid defuzzified values for four examples. And they are provided in Table 3. Here the adjustable coefficient
Next, we study the performances of proposed three types of weighted-based noniteraitve algorithms, respectively. Here the number of sampling is chosen as
Next, let’s quantitatively measure the calculation accuracies of proposed weighted-based noniterative algorithms. For the number of sampling
Algorithm | NB | TWNB | SWNB | S3/8WNB |
Example 1 | 0.000029 | 0.000029 | 0.000572 | 0.000067 |
Example 2 | 0.038400 | 0.001690 | 0.001450 | 0.001390 |
Example 3 | 0.009800 | 0.000094 | 0.000122 | 0.001908 |
Example 4 | 0.000110 | 0.000110 | 0.000110 | 0.000731 |
Total average | 0.016110 | 0.000640 | 0.000750 | 0.001370 |
Algorithm | NT | TWNT | SWNT | S3/8WNT |
Example 1 | 0.000043 | 0.000043 | 0.000519 | 0.00096 |
Example 2 | 0.067610 | 0.002170 | 0.002430 | 0.002280 |
Example 3 | 0.013730 | 0.000050 | 0.000050 | 0.002770 |
Example 4 | 0.000028 | 0.000028 | 0.000028 | 0.000458 |
Total average | 0.027140 | 0.000760 | 0.001010 | 0.001870 |
Algorithm | BMM | TWBMM | SWBMM | S3/8WBMM |
Example 1 | 0.000028 | 0.000028 | 0.000576 | 0.000056 |
Example 2 | 0.039650 | 0.001590 | 0.001320 | 0.001290 |
Example 3 | 0.009598 | 0.000096 | 0.000125 | 0.001866 |
Example 4 | 0.000112 | 0.000112 | 0.000112 | 0.000738 |
Total average | 0.016460 | 0.000610 | 0.000710 | 0.001320 |
Then we study the specific computation times of weighted-based noniterative algorithms for better applications. The number of sampling is still chosen as
Considering the Figures 3–8, Tables 4–6, and Figures 9–11 comprehensively, the following conclusions can be obtained:
1) In these four examples, the absolute errors of three types of weighted-based noniterative algorithms all converge as the number of sampling increases. In example 1, NB, TWNB, NT, TWNT, and BMM, TWBMM algorithms can obtain the smallest absolute errors and errors amplitudes of variation, while SNB, SNT, and SBMM algorithms get the largest absolute errors and errors amplitude of variation. In both examples 2 and 3, the proposed weighted-based noniterative can obtain the values of absolute errors and errors amplitudes of variation that are obviously less than their corresponding original noniterative algorithms. In the last example, the first three types of weighted-based noniterative algorithms can get almost the same absolute errors and errors amplitudes of variation, while the last type of weighted-based noniterative algorithms obtain the larger ones.
2) For the NB algorithms, the largest average of relative error is 3.84%. While the largest average of relative error of proposed WNB algorithms is only 0.169%. For the NT algorithms, the largest average of relative error is 6.761%. The largest average of relative error of proposed WNT algorithms is only 0.243%. For the BMM algorithms, the largest average of relative error is 3.965%. The largest average of relative error of proposed WBMM algorithms is only 0.159%.
3) The total mean of average of relative error of NB algorithms is 1.611%. While the largest total mean of average of relative error of proposed WNB algorithms is only 0.137%. The total mean of average of relative error of NT algorithms is 2.714%. While the largest total mean of average of relative error of proposed WNT algorithms is only 0.187%. The total mean of average of relative error of BMM algorithms is 1.646%. While the largest total mean of average of relative error of proposed WBMM algorithms is only 0.132%.
4) In general, see Figures 9–11, the computational speeds of original noniteraive algorithms are faster than their corresponding weighted-based noniterative algorithms. However, the computational speeds first two types on weighted-based noniterative algorithms are almost completely the same. As the number of sampling is fixed, the size relation of computation times is as: S3/8WNoiteraive > SWNoiteraive > TWNoniterative > Noniterative. It may just because the weights of proposed weighted-based noniterative algorithms are more complex than the noniterative algorithms. In other words, the convergence speeds of proposed weighted-based noniteraive algorithms are faster than the original noniterative algorithms.
5) From the above analysis, it can be found that, by choosing the proposed weighted-based noniteraive algorithms appropriately, which can improve both the calculation accuracies and convergence speeds.
The proposed weighted-based noniteraive algorithms can be used to investigate the TR and defuzzification of IT2 FLSs. If only the computational accuracy were considered, the proposed three types of weighted-based noniteraive algorithms outperformed the original noniteraive algorithms, in which the second type of weighed-based noniteraive algorithms were the best. Moreover, the computation time of proposed weighted-based noniteraive algorithms were not much different from the original noniteraive algorithms. Considering the above analysis comprehensively, we advise to use the second or third types of weighted-based noniterative algorithms for the TR and defuzzification of IT2 FLSs with the combination of linear functions and nonlinear functions as in examples 1 and 4, and adopt the first or second types of weighted-based noniteraive algorithms for the TR and defuzzification of IT2 FLSs with the combination of linear and nonlinear functions and nonlinear functions as in examples 2 and 3.
Finally, it is important to point out that, we only focus on the experimental performances of weighted-based noniteraive algorithms. It can be obtained from the simulation examples that, compare with the noniteraive algorithms, the proposed weighted-based noniteraive algorithms can improve the computational accuracies. However, if the requirements of computational accuracy were not high, the weighted-based noniteraive algorithms can not show their advantages, as the simplest noniteraive algorithms could attain well results.
This paper compares the operations between three types of discrete noniterative algorithms with their corresponding continuous versions. According to the Newton-Cotes quadrature formulas in the numerical integration technique, three types of noniterative algorithms are extended to the weighted-based noniterative algorithms. The continuous noniterative algorithms are considered as the benchmarks for performing the centroid TR and defuzzification of IT2 FLSs. Four simulation examples illustrate and analyze the computational accuracies and computation times of the proposed algorithms. Compared with the original noniterative algorithms, the proposed weighted-based noniteraive algorithms can obtain both higher calculation accuracies and faster convergence speeds.
In the future work, we will concentrate on designing the centroid TR of T2 FLSs [13,14,15,16,17,18,19,20,21,22,23,24,29,31,33,56] with weighted-based reasonable initialization enhanced Karnik-Mendel algorithms, the center-of-sets TR [12,13,34,45,46,47,48,49] of T2 FLSs, and seeking for global optimization algorithms [3,4,5,6,25,26,27,35,36,37,38,39,40,41,42,43,44,52,53] for designing and applying IT2 or GT2 FLSs in real world problems like forecasting, control [50,51,54,55] and so on.
The paper is supported by the National Natural Science Foundation of China (No. 61973146, No. 61773188, No. 61903167, No. 61803189), the Liaoning Province Natural Science Foundation Guidance Project (No. 20180550056), and Talent Fund Project of Liaoning University of Technology (No. xr2020002). The author is very thankful to Professor Jerry Mendel, who has given the author some important advices.
The authors declare that they have no conflict of interest.
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1. | Chuang Liu, Jinxia Wu, Weidong Yang, Robust H∞ output feedback finite-time control for interval type-2 fuzzy systems with actuator saturation, 2022, 7, 2473-6988, 4614, 10.3934/math.2022257 | |
2. | Xiaoyu Peng, Xiaodong Pan, Interval type-2 fuzzy systems on the basis of vague partitions and their approximation properties, 2024, 43, 2238-3603, 10.1007/s40314-024-02629-2 |
Algorithms | Integration rule | Weights |
NB, NT, BMM | ______________ | |
TWNB, TWNT, TWBMM | Composite Trapezoidal rule | |
SWNB, SWNT, SWBMM | Composite Simpson rule | |
S3/8WNB, S3/8WNT, S3/8WBMM | Composite Simpson 3/8 rule |
Num | Expression |
1 | |
2 | |
3 | |
4 |
Num | Approach | ||
CNB | CNT | CBMM | |
1 | 4.3050 | 4.3208 | 4.3041 |
2 | 3.7774 | 3.7141 | 3.7747 |
3 | 4.3702 | 4.3953 | 4.3690 |
4 | 5.0000 | 5.0000 | 5.0000 |
Algorithm | NB | TWNB | SWNB | S3/8WNB |
Example 1 | 0.000029 | 0.000029 | 0.000572 | 0.000067 |
Example 2 | 0.038400 | 0.001690 | 0.001450 | 0.001390 |
Example 3 | 0.009800 | 0.000094 | 0.000122 | 0.001908 |
Example 4 | 0.000110 | 0.000110 | 0.000110 | 0.000731 |
Total average | 0.016110 | 0.000640 | 0.000750 | 0.001370 |
Algorithm | NT | TWNT | SWNT | S3/8WNT |
Example 1 | 0.000043 | 0.000043 | 0.000519 | 0.00096 |
Example 2 | 0.067610 | 0.002170 | 0.002430 | 0.002280 |
Example 3 | 0.013730 | 0.000050 | 0.000050 | 0.002770 |
Example 4 | 0.000028 | 0.000028 | 0.000028 | 0.000458 |
Total average | 0.027140 | 0.000760 | 0.001010 | 0.001870 |
Algorithm | BMM | TWBMM | SWBMM | S3/8WBMM |
Example 1 | 0.000028 | 0.000028 | 0.000576 | 0.000056 |
Example 2 | 0.039650 | 0.001590 | 0.001320 | 0.001290 |
Example 3 | 0.009598 | 0.000096 | 0.000125 | 0.001866 |
Example 4 | 0.000112 | 0.000112 | 0.000112 | 0.000738 |
Total average | 0.016460 | 0.000610 | 0.000710 | 0.001320 |
Algorithms | Integration rule | Weights |
NB, NT, BMM | ______________ | |
TWNB, TWNT, TWBMM | Composite Trapezoidal rule | |
SWNB, SWNT, SWBMM | Composite Simpson rule | |
S3/8WNB, S3/8WNT, S3/8WBMM | Composite Simpson 3/8 rule |
Num | Expression |
1 | |
2 | |
3 | |
4 |
Num | Approach | ||
CNB | CNT | CBMM | |
1 | 4.3050 | 4.3208 | 4.3041 |
2 | 3.7774 | 3.7141 | 3.7747 |
3 | 4.3702 | 4.3953 | 4.3690 |
4 | 5.0000 | 5.0000 | 5.0000 |
Algorithm | NB | TWNB | SWNB | S3/8WNB |
Example 1 | 0.000029 | 0.000029 | 0.000572 | 0.000067 |
Example 2 | 0.038400 | 0.001690 | 0.001450 | 0.001390 |
Example 3 | 0.009800 | 0.000094 | 0.000122 | 0.001908 |
Example 4 | 0.000110 | 0.000110 | 0.000110 | 0.000731 |
Total average | 0.016110 | 0.000640 | 0.000750 | 0.001370 |
Algorithm | NT | TWNT | SWNT | S3/8WNT |
Example 1 | 0.000043 | 0.000043 | 0.000519 | 0.00096 |
Example 2 | 0.067610 | 0.002170 | 0.002430 | 0.002280 |
Example 3 | 0.013730 | 0.000050 | 0.000050 | 0.002770 |
Example 4 | 0.000028 | 0.000028 | 0.000028 | 0.000458 |
Total average | 0.027140 | 0.000760 | 0.001010 | 0.001870 |
Algorithm | BMM | TWBMM | SWBMM | S3/8WBMM |
Example 1 | 0.000028 | 0.000028 | 0.000576 | 0.000056 |
Example 2 | 0.039650 | 0.001590 | 0.001320 | 0.001290 |
Example 3 | 0.009598 | 0.000096 | 0.000125 | 0.001866 |
Example 4 | 0.000112 | 0.000112 | 0.000112 | 0.000738 |
Total average | 0.016460 | 0.000610 | 0.000710 | 0.001320 |