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## Detailed Example: 1st-Order Imprecise Differential Equation

A 1st-order imprecise differential equation, , has to be solved. Given the fuzzy values for the ODE parameters: , 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 are:
a1=[-0.1,0.1] k=[-1,1] y(0)=[0,0]
The settings for the Interactive Evolutionary Algorithm for solving the ODE at 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 and .

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