Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Fractional Calculus and Applied Analysis, Vol. 9, No 1, (2006), 33p-41p
We suggest a fractional differential equation that combines the simple
harmonic oscillations of an LC circuit with the discharging of an RC circuit.
A series solution is obtained for the suggested fractional differential
equation. When the fractional order α = 0, we get the solution for the RC
circuit, and when α = 1, we get the solution for the LC circuit. For arbitrary
α we get a general solution which shows how the oscillatory behavior
(LC circuit) go over to a decay behavior (RC circuit) as grows from 0 to
1, and vice versa. An explanation of the behavior is proposed based on the
idea of the evolution of a resistive property in the inductor giving a new
value to the inductance that affects the frequency of the oscillator.