Variational problems about maximum energy of different types of pulse generators with a capacitor modelled using conformable fractional derivative

In this study, the variational problems about the maximum energy of two different types of pulse generators containing capacitors (C_(α )) modeled with a conformable fractional derivative (CFD) are solved. In the first circuit discussed, the pulse generator circuit consists of a resistor and a CFD capacitor, namely R and C_(α ), connected in series. In the second circuit discussed, the pulse generator consists of a CFD capacitor (C_(α )), a resistor ( Rp ), which models its non-ideal insulator and connected in parallel with the CFD capacitor, and a resistor (R) connected in series to both of them. Conformable fractional order differential equations that describe both of the circuit topologies are derived. For each circuit, an optimization problem is obtained by using the Euler-Lagrange equation and the differential equation describing the circuit in order to maximize the energy of the pulse generator. In both problems, the voltage (V_(C_α ) (τ)) that the capacitor will reach as a result of charging and the time required for the realization of this charging, that is, the charging time (τ), are taken as constant for the clarity of the problem. It is assumed that the capacitor is initially uncharged, that is, the capacitor voltage is V_(C_α )(0) = 0 Volts. Then, these optimization equations are solved numerically by substituting the boundary values and using the finite difference method. The numerical solution of the CFD capacitor voltage of each pulse generator is done with Matlab.