Example Problem
Let us consider the SDE of the Langevin equation where p and о are positive constants.
The parameters for Equation 11.22 in terms of crisp and fuzzy are given in Table 11.3.
Case 1
In this case, only the initial condition is fuzzy. As mentioned earlier, EM and MM are used to handle the problem. The investigated results are depicted in Figures 11.11 through 11.14. FEMM solutions are given in Figures 11.11 and 11.12 for a = 0.5 and 0 (interval). Similarly, Figures 11.13 and 11.14 depict the FMM results for a = 0.5 and 0 (interval).
Case 2
Here, the involved parameters (except the initial condition) are considered only as fuzzy and the solutions for the FEMM and the FMM are shown in Figures 11.15 through 11.18. The values of alpha (a) are 0.5 and 0 as mentioned in Case 1.
Case 3
Finally, both the initial condition and parameters are taken as fuzzy and the problem is investigated using the FEMM and the MM. The solutions can be visualized from Figures 11.19 through 11.22 for the same values of a as in the earlier cases.
The obtained results in digital form are also incorporated in Table 11.4. The left, centre and right values of the TFNs are given for the different cases. Furthermore, the values of X at different time (t = 1, 2, 3, 4) denoted as X(1), X(2), X(3) and X(4) are presented in this table for both the FEMM and the FMM.
TABLE 11.3
Crisp and Fuzzy Values of the Used Parameters (for Section 11.2.2)
Parameters 
Crisp 
TFN 
X0 
1 
[0.5, 1, 1.5] 
P 
15 
[10, 15, 20] 
о 
1 
[0.5, 1, 1.5] 
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TABLE 11.4
Crisp 
Left 
Centre 
Right 

EMM 
MM 
FEMM 
FMM 
FEMM 
FMM 
FEMM 
FMM 

Case 1 
X(1) 
0.1003 
0.0847 
0.1337 
0.0847 
0.1003 
0.0847 
0.1003 
0.0847 
X(2) 
0.0732 
0.0708 
0.0523 
0.0708 
0.0732 
0.0708 
0.0732 
0.0708 

X(3) 
0.2367 
0.2250 
0.0319 
0.2250 
0.2367 
0.2250 
0.2367 
0.2250 

X(4) 
0.6246 
0.5883 
0.2565 
0.5883 
0.6246 
0.5883 
0.6246 
0.5883 

Case 2 
X(1) 
0.1003 
0.0847 
0.1337 
0.1316 
0.1003 
0.0847 
0.1029 
0.1525 
X(2) 
0.0732 
0.0708 
0.0523 
0.0519 
0.0732 
0.0708 
0.0770 
0.0700 

X(3) 
0.2367 
0.2250 
0.0319 
0.0399 
0.2367 
0.2250 
0.6432 
0.6647 

X(4) 
0.6246 
0.5883 
0.2565 
0.2507 
0.6246 
0.5883 
0.9620 
0.9499 

Case 3 
X(1) 
0.1003 
0.0847 
0.1003 
0.1316 
0.1003 
0.0847 
0.1029 
0.1525 
X(2) 
0.0732 
0.0708 
0.0732 
0.0519 
0.0732 
0.0708 
0.0770 
0.0700 

X(3) 
0.2367 
0.2250 
0.2367 
0.0399 
0.2367 
0.2250 
0.6432 
0.6647 

X(4) 
0.6246 
0.5883 
0.6246 
0.2507 
0.6246 
0.5883 
0.9620 
0.9499 
Fuzzy Solution of the Problem for Different Cases
Crisp Left Centre Right
The sensitiveness of the solution set is studied in terms of the uncertain widths. Here, we have considered the fuzzy solutions and widths of X at different time (t = 1, 2, 3, 4), which are encrypted in Table 11.5. The uncertain widths for both FEMM and FMM are given in Table 11.5, and it may be observed that the FMM gives less width.
Next, the uncertain solutions (X at t = 2) are plotted in terms of TFN for various cases in Figures 11.23 through 11.25.
FIGURE 11.11
Interval solutions using the FEMM at a = 0.5.
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FIGURE 11.12
Fuzzy solutions using the FEMM.
FIGURE 11.13
Interval solutions using the FMM at a = 0.5.
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Neutron Diffusion: Concepts and Uncertainty Analysis for Engineers and Scientists
FIGURE 11.14
Fuzzy solutions using the FMM.
FIGURE 11.15
Interval solutions using the FEMM at a = 0.5.
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FIGURE 11.16
Fuzzy solutions using the FEMM.
FIGURE 11.17
Interval solutions using the FMM at a = 0.5.
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Neutron Diffusion: Concepts and Uncertainty Analysis for Engineers and Scientists
FIGURE 11.18
Fuzzy solutions using the FMM.
FIGURE 11.19
Interval solutions using the FEMM at a = 0.5.
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FIGURE 11.20
Fuzzy solutions using the FEMM.
FIGURE 11.21
Interval solutions using the FMM at a = 0.5.
FIGURE 11.22
Fuzzy solutions using the FMM.
In Figure 11.23, it is seen that the solution obtained by using the FMM gives a TFN (where the left and right solutions are same), which is parallel to the membership function axis. Again, it has also been seen that the nonincreasing left continuous function of the obtained TFN by using the FFEM becomes parallel to the membership function axis, whereas in Figure 11.24, the left nondecreasing function values of the resultant TFNs are more approximate as compared to the left nonincreasing function values.
TABLE 11.5
Width of the Solutions at a = 0 Using the FEMM and FMM
Width 

FEMM 
FMM 

Case 1 
X(1) 
0.0334 
0 
X(2) 
0.0209 
0 

X(3) 
0.2686 
0 

X(4) 
0.3681 
0 

Case 2 
X(1) 
0.2366 
0.2841 
X(2) 
0.0247 
0.0189 

X(3) 
0.6751 
0.7046 

X(4) 
0.7055 
0.6992 

Case 3 
X(1) 
0.2032 
0.2841 
X(2) 
0.0038 
0.0189 

X(3) 
0.4065 
0.7046 

X(4) 
0.3374 
0.6992 
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FIGURE 11.23
Fuzzy plot for Case 1 at X(t = 2).
FIGURE 11.24
Fuzzy plot for Case 2 at X(t = 2).
FIGURE 11.25
Fuzzy plot for Case 3 at X(t = 2).
For Case 3, Figure 11.25 gives the straight line for the nonincreasing and nondecreasing left continuous function using FEMM and FEM, respectively, which are parallel to the membership function axis.