This article describes a laboratory component of a course in fractional calculus for undergraduates. It incorporates theoretical, experimental, and numerical analyses of the fractional harmonic oscillator. Three independent approaches were taken to obtain solutions to the fractional harmonic oscillator excited by a step function: 1) a power series expansion of the RiemannLiouville form, 2) a circuit using fractance devices, 3) a numerical integration using the Grünwald-Letnikov algorithm. The fractional harmonic oscillator was also subjected to steady state AC excitation. In both the transient and steady state cases, the Riemann-Liouville form proved to accurately model the system dynamics. The course demonstrated that undergraduates learned the fundamental concepts of fractional calculus quite readily.
Measurements of conductance fluctuations in undoped hydrogenated amorphous germanium (a-Ge:H) find power spectra that vary with inverse frequency (1/f) that are characterized by non-Gaussian statistics. The non-Gaussian aspect of the 1/f noise is reflected in (1) histograms of the noise power per octave that are described by lognormal distributions, (2) power-law second spectra, and (3) strong correlations of the noise power in frequency-space. In contrast, measurements of current fluctuations in polycrystalline germanium thin films find 1/f noise with Gaussian statistics. These results are discussed in terms of a model of filamentary conduction, where the filament structure and conductance in a-Ge:H are modulated by hydrogen motion.
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