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# Conclusion

Solving differential equations is difficult even when there is no imprecision involved. Introducing imprecision makes it even harder to solve the problem correctly. The Interactive Evolutionary Algorithm Approach tries to find the overall maximal and minimal curves at each -cut level. It is based on the classical evolutionary algorithm using a protocol system that saves the minimum/maximum values during the process in a minimum/maximum database.

The Interactive Evolutionary Algorithm is looking for the minimum or maximum solution for the differential equation respectively. At the end of the evolutionary process the optimal chromosome should represent the minimum/maximum function determined by the Runge-Kutta Method. But it is impossible to tell whether another chromosome could have generated a better optimum for a particular y-t-value pair during the optimizing process or not. The overall minimum/maximum y-t-values are retrieved by protocolling them during the evolutionary algorithm process. The approach introduced can be seen as a mixture between the interacting and non interacting approach of Bonarini and Bontempi [1] (see Section 2).

An advantage is that the Interactive Evolutionary Algorithm is able to find the overall solution space. There is no spurious behaviour introduced but it might miss out some possible behaviour. Further it is sufficient to represent one problem set -- initial values and parameters of the differential equation -- in a chromosome. If there is more than one problem set necessary to represent the minimum/maximum curve the evolutionary process jumps between these solutions back and forth protocolling the minimum/maximum y-t-pairs.

A disadvantage is that at each generation a classical mathematical method for solving ODEs has to be performed for each chromosome which is computationally expensive.

Next: Bibliography Up: Simulation of Imprecise Ordinary Previous: 2nd-Order Imprecise Differential Equation
Christoph Reich
1999-12-22