A Fractional LC − RC Circuit

Mathematics Subject Classification: 26A33, 30B10, 33B15, 44A10, 47N70, 94C05

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.