2nd-Order Imprecise Differential Equation Example

An electrical circuit (Fig.: 3) should be simulated.

  

Figure 3: RLC Series Circuit

This circuit can be modelled by a 2nd order differential equation, $\frac {d^2y}{dt^2}=-\frac {R*C}{L*C}*\frac {dy}{dt}-\frac {1}{L*C}*y$. Suppose the component values are imprecisely known (tolerances) and therefore defined with the fuzzy values R=(50,40,40), L=(1,0.1,0.1), C=(0.001,0.0001,0.0001), y(0)=(10,1,1), and dy(0)=(0,0,0). The Runge-Kutta method uses t_start=0 and t_end=1 with a step size of 0.02. Fig. 4 shows the result of the simulation. The solution of the simulation states the desired result. When several solutions have to be overlayed to get the overall solution space, the Interactive Evolutionary Algorithm Approach is jumping between these solutions back and forth saving the results in the minimum/maximum database. When only one solution of the differential equation represents the extreme boundaries, the evolutionary algorithm does not alternate between solutions and finds directly the optimum problem sets.

  

Figure: Result of: $\frac {d^2y}{dt^2}=-\frac {R*C}{L*C}*\frac {dy}{dt}-\frac {1}{L*C}*y$