Fractional Fourier Series with Separation of Variables Technique and Its Application on Fractional differential Equations

When using some classical methods, such us separation of variables; it is impossible to find a general solution for some differential equations. Therefore, we suggest adding conformable fractional Fourier series to get a new technique to solve fractional Benjamin Bana Mahony and Heat Equations. Furtheremore, we give new numerical approximation for functions using mathematica coding called conformable fractional Fourier series approximation

Fractional Spectral Graph Wavelets and Their Applications

One of the key challenges in the area of signal processing on graphs is to design transforms and dictionary methods to identify and exploit structure in signals on weighted graphs. In this paper, we first generalize graph Fourier transform (GFT) to spectral graph fractional Fourier transform (SGFRFT), which is then used to define a novel transform named spectral graph fractional wavelet transform (SGFRWT), which is a generalized and extended version of spectral graph wavelet transform (SGWT). A fast algorithm for SGFRWT is also derived and implemented based on Fourier series approximation. Some potential applications of SGFRWT are also presented.

Convergence of a series associated with the convexification method for coefficient inverse problems

This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L^{2} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H^{1}-norm for a sequence of L^{2}-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

Optimum inelastic tuned mass dampers without using viscous elements for minimizing steady-state vibration of base-excited systems

A special type of a tuned mass damper, which consists of a mass and an elasto-plastic spring without using any viscous damper, is used to reduce the steady-state response of structures to base excitation. Previous work of the authors showed that the elasto-plastic tuned mass damper (P-TMD) could help reduce the seismic responses, and a method based on energy equalization was proposed to design it. In this study, the effectiveness of the P-TMD is investigated under harmonic support motion, and a direct approach is developed to find its optimum parameters. To estimate the nonlinear steady-state response of P-TMD-controlled systems, an analytical framework is established using the Fourier series approximation, which is validated by direct numerical integration of the equations of motion. The obtained results for the optimum P-TMD are discussed and compared with those of the optimum elastic tuned mass damper.

Adaptive Repetitive Control of A Linear Oscillating Motor under Periodic Hydraulic Step Load

A linear oscillating motor has a direct and efficient linear motion output and is widely used in linear actuation systems. The motor is often applied to compact hybrid electrohydraulic actuators to drive a linear pump. However, the periodic switch of the rectification valve in the pump brings the hydraulic step load to the linear motor, which causes periodic oscillation waveform distortions. The distortion results in the reduction of pumping capacity. The conventional feedback proportional-integral-derivative control is applied to the pump, however, this solution cannot handle the step load as well as resolving nonlinear properties and uncertainties. In this paper, we introduce a nonlinear model to identify periodic hydraulic load. Then, the loads are broken up into a set of simple terms by Fourier series approximation. The uncertain terms and other modeling uncertainties are estimated and compensated by the practical adaptive controller. A robust control term is also developed to handle uncertain nonlinearities. The controller overcame drawbacks of conventional repetitive controllers, such as heavy memory requirements and noise sensitivity. The controller can achieve a prescribed final tracking accuracy under periodic hydraulic load via Lyapunov analysis. Finally, experimental results on the linear oscillating motor-pump are provided for validation of the effectiveness of the scheme.