Numerical Accuracy of the Predictor-Corrector Method to Solve Fuzzy Differential Equations Based on the Stochastic Arithmetic

Figures & data

Table 1. The optimal step size of Example 5.1.

θ	1/2			1/3
Method	FPC-3step	FIPC-3step	FMPC-4step	FPC-3step	FIPC-3step	FMPC-4step
$h_{opt}$	${\displaystyle \frac{1}{640}}$	${\displaystyle \frac{1}{160}}$	${\displaystyle \frac{1}{80}}$	${\displaystyle \frac{1}{2430}}$	${\displaystyle \frac{1}{810}}$	${\displaystyle \frac{1}{270}}$

Table 2. Numerical results of Example 5.1.

FPC-3step method, $\theta =1/3,t=0.1$
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
1	${\displaystyle \frac{0.1}{3}}$	0.0	0.9687483E+000	0.101378E+001
		0.1	0.9723573E+000	0.101288E+001
		0.2	0.9759662E+000	0.1011992E+001	–	–
		⋮	⋮	⋮
		1	0.1004837E+001	0.1004837E+001
2	${\displaystyle \frac{0.1}{9}}$	0.0	0.9689204E+000	0.1013609E+001
		0.1	0.972512E+000	0.1012732E+001
		0.2	0.9761037E+000	0.1011854E+001	0.276E−003	0.171E−003
		⋮	⋮	⋮
		1	0.100483E+001	0.100483E+001
⋮	⋮	⋮	⋮	⋮	⋮	⋮
4	${\displaystyle \frac{0.1}{81}}$	0.0	0.9686382E+000	0.101387E+001
		0.1	0.9722576E+000	0.1012069E+001
		0.2	0.976172E+000	0.101178E+001	0.36E−003	0.22E−004
		⋮	⋮	⋮
		1	0.100483E+001	0.100483E+001
5	${\displaystyle \frac{0.1}{243}}$	0.0	0.969014E+000	0.10135E+001
		0.1	0.972596E+000	0.101264E+001
		0.2	0.976178E+000	0.101177E+001	0.37E−003	@.0
		⋮	⋮	⋮
		1	0.100483E+001	0.100483E+001

Table 3. Numerical results of Example 5.1.

FIPC-3step method, $\theta =1/3,t=0.1$
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
1	${\displaystyle \frac{0.1}{3}}$	0.0	0.9686382E+000	0.101387E+001
		0.1	0.9722576E+000	0.1012974E+001
		0.2	0.9758761E+000	0.101116E+001	–	–
		⋮	⋮	⋮
		1	0.1004831E+001	0.100483E+001
2	${\displaystyle \frac{0.1}{9}}$	0.0	0.9686431E+000	0.101388E+001
		0.1	0.9722625E+000	0.101298E+001
		0.2	0.9758818E+000	0.101207E+001	@.0	0.5E−005
		⋮	⋮	⋮
		1	0.1004836E+001	0.1004836E+001
3	${\displaystyle \frac{0.1}{27}}$	0.0	0.9686438E+000	0.101388E+001
		0.1	0.972263E+000	0.101298E+001
		0.2	0.9758823E+000	0.101207E+001	@.0	0.7E−006
		⋮	⋮	⋮
		1	0.100483E+001	0.100483E+001
4	${\displaystyle \frac{0.1}{81}}$	0.0	0.968643E+000	0.10138E+001
		0.1	0.972263E+000	0.101298E+001
		0.2	0.975882E+000	0.101207E+001	@.0	@.0
		⋮	⋮	⋮
		1	0.100483E+001	0.100483E+001

Table 4. Numerical results of Example 5.1.

FMPC-4step method, $\theta =1/3,t=0.1$
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
1	${\displaystyle \frac{0.1}{3}}$	0.0	0.9686439E+000	0.1013885E+001
		0.1	0.9722631E+000	0.1012980E+001
		0.2	0.9758825E+000	0.1012075E+001	–	–
		⋮	⋮	⋮
		1	0.1004837E+001	0.100483E+001
2	${\displaystyle \frac{0.1}{9}}$	0.0	0.9686364E+000	0.1013893E+001
		0.1	0.9722564E+000	0.101298E+001
		0.2	0.9758765E+000	0.1012082E+001	0.75E−005	0.3E−005
		⋮	⋮	⋮
		1	0.1004837E+001	0.1004837E+001
3	${\displaystyle \frac{0.1}{27}}$	0.0	0.9686405E+000	0.101388E+001
		0.1	0.9722604E+000	0.101298E+001
		0.2	0.975879E+000	0.1012078E+001	@.0	@.0
		⋮	⋮	⋮
		1	0.100483E+001	0.100483E+001

Table 5. Numerical results of Example 5.1.

FIPC-3step method, $\theta =1/2,t=0.5$
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
1	${\displaystyle \frac{0.5}{2}}$	0.0	0.1082269E+001	0.1112596E+001
		0.1	0.1084695E+001	0.1111989E+001
		0.2	0.1087121E+001	0.1111382E+001	–	–
		⋮	⋮	⋮
		1	0.1106530E+001	0.1106530E+001
2	${\displaystyle \frac{0.5}{4}}$	0.0	0.1082064E+001	0.1112380E+001
		0.1	0.108448E+001	0.1111773E+001
		0.2	0.108691E+001	0.111116E+001	0.213E-003	0.213E−003
		⋮	⋮	⋮
		1	0.1106317E+001	0.1106317E+001
3	${\displaystyle \frac{0.5}{8}}$	0.0	0.1082195E+001	0.111251E+001
		0.1	0.1084621E+001	0.1111912E+001
		0.2	0.108704E+001	0.1111305E+001	0.13E−003	0.76E−004
		⋮	⋮	⋮
		1	0.110645E+001	0.1106454E+001
⋮	⋮	⋮	⋮	⋮	⋮	⋮
6	${\displaystyle \frac{0.5}{64}}$	0.0	0.108226E+001	0.111259E+001
		0.1	0.108469E+001	0.111198E+001
		0.2	0.108711E+001	0.111138E+001	0.4E−005	@.0
		⋮	⋮	⋮
		1	0.110652E+001	0.110652E+001
7	${\displaystyle \frac{0.1}{128}}$	0.0	0.108226E+001	0.111259E+001
		0.1	0.108469E+001	0.111198E+001
		0.2	0.108712E+001	0.111138E+001	@.0	@.0
		⋮	⋮	⋮
		1	0.11065E+001	0.110653E+001

Table 6. Comparison of the common significant digits $\left(C\right)$ in the FMPC-4step method, $h_{opt}=\frac{0.1}{27}$ for Example 5.1.

r	$C_{v_i^{-}(h;r),v_i^{-}(\theta h;r)}$	$C_{v_i^{-}(h;r),x_i^{-}\left(0.1\right)}$	$C_{v_i^{+}(h;r),v_i^{+}(\theta h;r)}$	$C_{v_i^{+}(h;r),x_i^{+}\left(0.1\right)}$
0.0	0.12E+002	0.12E+002	0.12E+002	0.12E+002
0.1	0.12E+002	0.12E+002	0.12E+002	0.123E+002
0.2	0.12E+002	0.12E+002	0.12E+002	0.12E+002
0.3	0.12E+002	0.12E+002	0.121E+002	0.13E+002
0.4	0.128E+002	0.13E+002	0.122E+002	0.13E+002
0.5	0.13E+002	0.122E+002	0.1E+002	0.1E+002
0.6	0.13E+002	0.123E+002	0.1E+002	0.1E+002
0.7	0.13E+002	0.13E+002	0.1E+002	0.1E+002
0.8	0.14E+002	0.134E+002	0.133E+002	0.14E+002
0.9	0.1E+002	0.1E+002	0.1E+002	0.1E+002
1.0	@.0	@.0	@.0	@.0

Table 7. The optimal step size of Example 5.2.

θ	1/2			1/3
Method	FPC-3step	FIPC-3step	FMPC-4step	FPC-3step	FIPC-3step	FMPC-4step
$h_{opt}$	${\displaystyle \frac{0.1}{2048}}$	${\displaystyle \frac{0.1}{512}}$	${\displaystyle \frac{0.1}{512}}$	${\displaystyle \frac{0.1}{6561}}$	${\displaystyle \frac{0.1}{729}}$	${\displaystyle \frac{0.1}{729}}$

Table 8. Numerical results of Example 5.2.

FPC-3step method, $\theta =1/2$, $t=0.1$
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
2	${\displaystyle \frac{0.1}{4}}$	0.0	0.803239E−001	0.1016841E+000
		0.1	0.814453E−001	0.100669E+000
		0.2	0.825666E−001	0.996547E−001	0.16882E−001	0.1016841E+000
		⋮	⋮	⋮
		1	0.915343E−001	0.915342E−001
3	${\displaystyle \frac{0.1}{8}}$	0.0	0.768867E−001	0.8756085E−001
		0.1	0.774680E−001	0.870753E−001
		0.2	0.780496E−001	0.865899E−001	0.27592E−002	0.141232E−001
		⋮	⋮	⋮
		1	0.8271046E−001	0.827104E−001
4	${\displaystyle \frac{0.1}{16}}$	0.0	0.783973E−001	0.849148E−001
		0.1	0.787701E−001	0.846366E−001
		0.2	0.791432E−001	0.843586E−001	0.1132E−003	0.26460E−002
		⋮	⋮	⋮
		1	0.8213854E−001	0.821385E−001
⋮	⋮	⋮	⋮	⋮	⋮	⋮
10	${\displaystyle \frac{0.1}{1024}}$	0.0	0.793893E−001	0.83816E−001
		0.1	0.796574E−001	0.83642E−001
		0.2	0.799261E−001	0.834693E−001	0.984E−003	0.1E−004
		⋮	⋮	⋮
		1	0.82084E−001	0.82084E−001
11	${\displaystyle \frac{0.1}{2048}}$	0.0	0.79394E−001	0.83811E−001
		0.1	0.79662E−001	0.83638E−001
		0.2	0.79930E−001	0.83465E−001	0.69E−003	@.0
		⋮	⋮	⋮
		1	0.82085E−001	0.820851E−001

Table 9. Numerical results of Example 5.2.

FIPC-3step method, $\theta =1/2$, $t=0.1$
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
2	${\displaystyle \frac{0.1}{4}}$	0.0	0.741218E−001	0.801331E−001
		0.1	0.744630E−001	0.798738E−001
		0.2	0.748044E−001	0.7961454E−001	0.46683E−002	0.801331E−001
		⋮	⋮	⋮
		1	0.775463E−001	0.775463E−001
3	${\displaystyle \frac{0.1}{8}}$	0.0	0.780348E−001	0.843083E−001
		0.1	0.783911E−001	0.840377E−001
		0.2	0.787476E−001	0.837674E−001	0.4932E−003	0.41751E−002
		⋮	⋮	⋮
		1	0.816095E−001	0.816095E−001
4	${\displaystyle \frac{0.1}{16}}$	0.0	0.783540E−001	0.84656E−001
		0.1	0.787119E−001	0.84384E−001
		0.2	0.790702E−001	0.841130E−001	0.143E−003	0.345E−003
		⋮	⋮	⋮
		1	0.819451E−001	0.81945E−001
⋮	⋮	⋮	⋮	⋮	⋮	⋮
8	${\displaystyle \frac{0.1}{256}}$	0.0	0.78485E−001	0.84799E−001
		0.1	0.78843E−001	0.84527E−001
		0.2	0.79203E−001	0.842550E−001	@.0	0.4E−005
		⋮	⋮	⋮
		1	0.82083E−001	0.82083E−001
9	${\displaystyle \frac{0.1}{512}}$	0.0	0.784861E−001	0.84801E−001
		0.1	0.78845E−001	0.84529E−001
		0.2	0.79203E−001	0.84255E−001	@.0	@.0
		⋮	⋮	⋮
		1	0.82084E−001	0.82084E−001

Table 10. Numerical results of Example 5.2.

FMPC-4step method, $\theta =1/2$, t = 0.1
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
2	${\displaystyle \frac{0.1}{4}}$	0.0	0.729963E−001	0.8622912E−001
		0.1	0.737009E−001	0.856108E−001
		0.2	0.744055E−001	0.849927E−001	0.54906E−002	0.8622913E−001
		⋮	⋮	⋮
		1	0.8004966E−001	0.800496E−001
3	${\displaystyle \frac{0.1}{8}}$	0.0	0.762335E−001	0.869008E−001
		0.1	0.7681009E−001	0.864111E−001
		0.2	0.773869E−001	0.859216E−001	0.2253E−002	0.32372E−002
		⋮	⋮	⋮
		1	0.820095E−001	0.820095E−001
4	${\displaystyle \frac{0.1}{16}}$	0.0	0.762335E−001	0.869008E−001
		0.1	0.7681009E−001	0.864111E−001
		0.2	0.773869E−001	0.859216E−001	0.2253E−002	0.32372E−002
		⋮	⋮	⋮
		1	0.820095E−001	0.820095E−001
⋮	⋮	⋮	⋮	⋮	⋮	⋮
8	${\displaystyle \frac{0.1}{256}}$	0.0	0.784538E−001	0.84834E−001
		0.1	0.78815E−001	0.84559E−001
		0.2	0.79177E−001	0.84283E−001	0.30E−004	0.37E−004
		⋮	⋮	⋮
		1	0.820848E−001	0.82084E−001
9	${\displaystyle \frac{0.1}{512}}$	0.0	0.78471E−001	0.84817E−001
		0.1	0.78831E−001	0.84543E−001
		0.2	0.79191E−001	0.84270E−001	@.0	@.0
		⋮	⋮	⋮
		1	0.82085E−001	0.82085E−001

Table 11. Numerical results of Example 5.2.

FPC-3step method, $\theta =1/2$, t = 0.5
i	h	r	$v_i^{-}(h;r)$	$v_i^{+}(h;r)$	$D_1$	$D_2$
6	${\displaystyle \frac{0.1}{64}}$	0.0	0.330108E−005	0.414505E−005
		0.1	0.33461E−005	0.41060E−005
		0.2	0.339136E−005	0.406714E−005	0.25995E−004	0.24859E−006
		⋮	⋮	⋮
		1	0.37569E−005	0.37569E−005
7	${\displaystyle \frac{0.1}{128}}$	0.0	0.358700E−005	0.380149E−005
		0.1	0.360097E−005	0.379414E−005
		0.2	0.361500E−005	0.378683E−005	0.3435E−006	0.97753E−007
		⋮	⋮	⋮
		1	0.372933E−005	0.37293E−005
⋮	⋮	⋮	⋮	⋮	⋮	⋮
12	${\displaystyle \frac{0.1}{4096}}$	0.0	0.36379E−005	0.37442E−005
		0.1	0.36466E−005	0.37423E−005
		0.2	0.36553E−005	0.374051E−005	0.4E−009	0.1522E−008
		⋮	⋮	⋮
		1	0.37266E−005	0.37266E−005
13	${\displaystyle \frac{0.1}{8192}}$	0.0	0.363839E−005	0.37438E−005
		0.1	0.36470E−005	0.37419E−005
		0.2	0.365568E−005	0.37401E−005	@.0	0.13E−009
		⋮	⋮	⋮
		1	0.37266E−005	0.37266E−005

Table 12. Comparison of the common significant digits $\left(C\right)$ in the FIPC-3step method, $h_{opt}=\frac{0.1}{512}$ for Example 5.2.

r	$C_{v_i^{-}(h;r),v_i^{-}(\theta h;r)}$	$C_{v_i^{-}(h;r),x_i^{-}\left(0.1\right)}$	$C_{v_i^{+}(h;r),v_i^{+}(\theta h;r)}$	$C_{v_i^{+}(h;r),x_i^{+}\left(0.1\right)}$
0.0	0.99E+001	0.1E+002	0.1E+002	0.1E+002
0.1	0.93E+001	0.1E+002	0.98E+001	0.1E+002
0.2	0.93E+001	0.1E+002	0.99E+001	0.1E+002
0.3	0.98E+001	0.1E+002	0.99E+001	0.1E+002
0.4	0.94E+001	0.1E+002	0.97E+001	0.1E+002
0.5	0.93E+001	0.1E+002	0.98E+001	0.1E+002
0.6	0.96E+001	0.1E+002	0.99E+001	0.1E+002
0.7	0.99E+001	0.1E+002	0.98E+001	0.1E+002
0.8	0.98E+001	0.1E+002	0.99E+001	0.1E+002
0.9	0.99E+001	0.1E+002	0.98E+001	0.1E+002
1.0	@.0	@.0	@.0	@.0