Fractional Hamiltonian analysis of higher order derivatives systems

The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski's formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part and a damped oscillator are analyzed. The classical results are obtained when fractional derivatives are replaced with the integer order derivatives.Comment: 13 page

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850

Complex order van der Pol oscillator

In this paper a complex-order van der Pol oscillator is considered. The complex derivative Dα±ȷβ , with α,β∈R + is a generalization of the concept of integer derivative, where α=1, β=0. By applying the concept of complex derivative, we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the two versions of the complex-order van der Pol oscillator are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed

Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of fractional distributed-order type for the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain case corresponding to stress relaxation. In solving system of differential and integro-differential equations we use the Laplace transformation in the time domain

Quadrature Phase-Domain ADPLL with Integrated On-line Amplitude Locked Loop Calibration for 5G Multi-band Applications

5th generation wireless systems (5G) have expanded frequency band coverage with the low-band 5G and mid-band 5G frequencies spanning 600 MHz to 4 GHz spectrum. This dissertation focuses on a microelectronic implementation of CMOS 65 nm design of an All-Digital Phase Lock Loop (ADPLL), which is a critical component for advanced 5G wireless transceivers. The ADPLL is designed to operate in the frequency bands of 600MHz-930MHz, 2.4GHz-2.8GHz and 3.4GHz-4.2GHz. Unique ADPLL sub-components include: 1) Digital Phase Frequency Detector, 2) Digital Loop Filter, 3) Channel Bank Select Circuit, and 4) Digital Control Oscillator. Integrated with the ADPLL is a 90-degree active RC-CR phase shifter with on-line amplitude locked loop (ALL) calibration to facilitate enhanced image rejection while mitigating the effects of fabrication process variations and component mismatch. A unique high-sensitivity high-speed dynamic voltage comparator is included as a key component of the active phase shifter/ALL calibration subsystem. 65nm CMOS technology circuit designs are included for the ADPLL and active phase shifter with simulation performance assessments. Phase noise results for 1 MHz offset with carrier frequencies of 600MHz, 2.4GHz, and 3.8GHz are -130, -122, and -116 dBc/Hz, respectively. Monte Carlo simulations to account for process variations/component mismatch show that the active phase shifter with ALL calibration maintains accurate quadrature phase outputs when operating within the frequency bands 600MHz-930MHz, 2.4GHz-2.8GHz and 3.4GHz-4.2GHz