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Detailed Example: 1st-Order Imprecise Differential Equation

A 1st-order imprecise differential equation, $a1*\frac {y(t)}{dt} + a0*y(t) = k$, has to be solved. Given the fuzzy values[*] for the ODE parameters: $a0=(m_{a0},\alpha_{a0},\beta_{a0})=(100,10,10)$, a1=(1,0.1,0.1), k=(10,1,1)and the initial fuzzy value (at t=0): y(0)=(0,0,0) (exactly known) The first order differential equation is represented by a chromosome as:
a0 a1 k y(0)
The maximum change parameter for the particular values at the $\ensuremath{\alpha} -cut-level \geq 0.0$ are:
$a0=[\alpha_{a0},\beta_{a0}]=[-10,10]$ a1=[-0.1,0.1] k=[-1,1] y(0)=[0,0]
The settings for the Interactive Evolutionary Algorithm for solving the ODE at $\ensuremath{\alpha} -cut-level \geq 0.0$ are:
Population size: 2+10 ES[*] Mutation Rate: 3
At each generation the parameter set of the chromosomes are evaluated using the Runge-Kutta-Method[*]. Runge-Kutta-Method settings:
Start Value: 0.000s
Stop Value: 0.600s
Step Size: 0.005s
The following tabular shows the best chromosome and the mutation strength per generation:
Generation Mutation Strength a0 a1 k y(0)
  (%)        
0 5 100.0 1.00 10.0 0
1 7 102.2 0.92 9.8 0
2 9 102.2 0.92 9.7 0
3 11 102.2 0.92 9.7 0
4 13 103.1 0.91 9.7 0
5 15 103.1 0.90 9.7 0
6 17 101.8 0.90 9.5 0
7 19 103.3 0.90 9.5 0
8 21 103.3 0.90 9.2 0
9 19 103.3 0.90 9.0 0
10 21 103.3 0.90 9.0 0
11 23 103.3 0.90 9.0 0
12 25 106.3 0.91 9.0 0
13 27 109.5 0.91 9.0 0
14 29 110.0 0.91 9.0 0
15 31 110.0 0.90 9.0 0
16 29 110.0 0.90 9.0 0
17 27 110.0 0.90 9.0 0
18 25 110.0 0.90 9.0 0
19 23 110.0 0.90 9.0 0
At each generation the Runge-Kutta-Method evaluates each chromosome and stores the generated y-t values in the database if they are better then the stored ones. Fig. 2 displays the solution space after the Interactive Evolutionary Algorithm for $\ensuremath{\alpha} -cut-level \geq 0.0$ and $\ensuremath{\alpha} -cut-level = 1.0$.
  
Figure: Simulation Result: $a1*\frac {y(t)}{dt} + a0*y(t) = k$
\begin{figure}\centering
\epsfig{file=LRseriesResult.ps,width=8cm}
\end{figure}


next up previous
Next: 2nd-Order Imprecise Differential Equation Up: Interactive Evolutionary Algorithm Approach Previous: Objective Function
Christoph Reich
1999-12-22