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Introduction

Differential equation problems occur in many different technical disciplines. Many mathematical models derived in the study of physical systems involve instantaneous rates of change that are given mathematically by the derivatives of one variable with respect to another. Nonetheless engineers are used to modeling their dynamic nonlinear physical systems by differential equations. N-th order differential equations, are defined as:
$a_n*\frac{d^n y}{dt^n} + a_{n-1}*\frac{d^{n-1}y}{dt^{n-1}}+... + a_2*\frac{d^2 y}{dt^2} + a_1*\frac{dy}{dt} + a_0*y=F[t,y,\frac{dy}{dt},...,\frac{d^n y}{dt^n}]$
with the initial conditions (values) $\frac{d^n y}{dt^n}(t_0)$, ..., $\frac{d^2 y}{dt^2}(t_0)$, $\frac{dy}{dt}(t_0)$, y(t0). These differential equations belong to the category of problems called initial value problems, which usually model time-dependent phenomena. There are several well-known mathematical methods for obtaining the numerical solution of a differential equation, e.g. Euler Method, Modified Euler, Runge-Kutta [2], [9].

Suppose the initial conditions (y(t0), $\frac{dy}{dt}(t_0)$, ..., $\frac{d^n y}{dt^n}$) and/or the differential equation parameters (a0, a1, ..., an) are not known exactly. Fuzzy logic allows to model such imprecision by fuzzy values based on fuzzy sets [12] in [11].

The proposed Interactive Evolutionary Algorithm Approach is able to solve ordinary differential equations (ODE) whose initial conditions and/or differential equation parameters are represented by fuzzy values.


Christoph Reich
1999-12-22