The Hill's equation is the Schrodinger equation $$-y'+qy=la y$$ with a periodic one-dimensional potential function $q$ and coupled with periodic boundary conditions $y(0)=y(1)$, $y'(0)=y'(1)$ or anti-periodic boundary conditions $y(0)=-y(1)$, $y'(0)=-y'(1)$. We study the inverse nodal problem for Hill's equation, in particular the reconstruction problem. Namely, we want to reconstruct the potential function using only nodal data ( zeros of eigenfunctions ). In this thesis, we give a reconstruction formula for $q$ using the periodic nodal data or using anti-periodic nodal data We show that the convergence is pointwise for all $x in (0,1)$ where $q$ is continuous; and pointwise for $a.e.$ $x in (0,1)$ as well as $L^1$ convergence when $qin L^1(0,1)$. We do this by making a translation so that the problem becomes a Dirichlet problem. The idea comes from the work of Coskun and Harris.

Energy harvesting from the rotating system has been an influential topic for researchers over the past several years. Yet, most of these harvesters are linear resonance-based harvesters whose output power drops dramatically under random excitations. This poses a serious problem because a lot of vibrations in rotating systems are stochastic. In this dissertation, a novel energy harvesting strategy for rotating systems was proposed by taking advantage of stochastic resonance. Stochastic resonance is referred to as a physical phenomenon that is manifest in nonlinear bistable systems whereby a weak periodic signal can be significantly amplified with the aid of inherent noise or vice versa. Stochastic resonance can thus be used to amplify the noisy and weak vibration motion. Through mathematical modeling, this dissertation shows that stochastic resonance is particularly favorable to energy harvesting in rotating systems. The conditions for stochastic resonance are satisfied by adding a nonlinear bistable energy harvester to the rotating system because whirl noise and periodic signalㄴ already coexist in the rotating environment. Both numerical and experimental results show that stochastic resonance energy harvester has higher power and wider bandwidth than linear harvesters under a rotating environment. The dissertation also investigates how stochastic resonance changes for the various types of excitation that occur in real-world applications. Under the non-gaussian noise, the stochastic resonance frequency is shifted larger value. Furthermore, the co-existence of the vibrational and stochastic resonance is observed depending on the periodic signal to noise ratio. The dissertation finally proposed two real applications of stochastic resonance energy harvesting. First, stochastic resonance energy harvester for oil drilling applications is presented. In the oil drilling environment, the periodic force in rotating shafts is biased, which can lower the efficacy of stochastic resonance. To solve the problem, an external magnet was placed above the bi-stable energy harvester to compensate for the biased periodic signal. Energy harvester for smart tires is also proposed. The passively tuned system is implemented in a rotating tire via centrifugal force. An inward-oriented rotating beam is used to induce bistability via the centrifugal acceleration of the tire. The results show that larger power output and wider bandwidth can be obtained by applying the proposed harvesting strategy to the rotating system. / Doctor of Philosophy / In this dissertation, a novel energy harvesting strategy for rotating systems was proposed by taking advantage of stochastic resonance. Stochastic resonance is referred to as a physical phenomenon that is manifest in nonlinear bistable systems whereby a weak periodic signal can be significantly amplified with the aid of inherent noise or vice versa. Stochastic resonance can thus be used to amplify the noisy and weak vibration motion. Through mathematical modeling, this dissertation shows that stochastic resonance is particularly favorable to energy harvesting in rotating systems.Both numerical and experimental results show that stochastic resonance energy harvester has higher power and wider bandwidth than linear harvesters under a rotating environment. The dissertation also investigates how stochastic resonance changes for the various types of excitation that occur in real-world applications. The dissertation finally proposed two real applications of stochastic resonance energy harvesting. First, stochastic resonance energy harvester for oil drilling applications is presented. Energy harvester for smart tires is also proposed. The results show that larger power output and wider bandwidth can be obtained by applying the proposed harvesting strategy to the rotating system.

In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier functions may be used. However, when the solutions are non-periodical, the Legendre and Chebyshev polynomials are recommended, reported in many papers and books. There seems to exist few reports for the study of non-periodical solutions by spectral Fourier methods under the Dirichlet conditions and other boundary conditions. In this paper, we will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM) for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule. Furthermore, the error bounds are derived for both the CFM and the SFM. When there exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as accurate as the spectral method using Legendre and Chebyshev polynomials. However, once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the mesh length of uniform collocation grids, which are just equivalent to those by the linear elements and the finite difference method (FDM). The O(h^2) and even the superconvergence O(h4) are found numerically. The traditional condition number of the CFM is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods using the Legendre and Chebyshev polynomials. Also the effective condition number is only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems, to support the analysis made. The simplicity of algorithms and the promising numerical computation with O(h^4) may grant the CFM to be competent in application in numerical physics, chemistry, engineering, etc., see [7].