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: $a1*\frac {y(t)}{dt} + a0*y(t) = k$