A New Predictor-Corrector Method for Solving Nonlinear Fractional Differential Equations with Graded Meshes

In this dissertation we consider the numerical methods for solving non-linear fractional differential equations. We first review the predictor-corrector methods for solving the nonlinear fractional differential equation with uniform meshes and discussed in detail how to prove the error estimates. The convergence orders of the predictorcorrector methods for solving nonlinear fractional differential equations available in the literature are only O(h1+α ), where α ∈ (0, 1) denotes the fractional order and h is the step size. It will take a long time to obtain the good approximate solutions by using such method. Therefore it is necessary to construct some higher order numerical methods to solve the nonlinear fractional differential equations. We construct a higher order numerical method with the convergence order O(h1+2α) by approximating the Riemann-Liouville fractional integral with the quadratic interpolation polynomials. The graded meshes can be used in the numerical methods to capture the singularity of the problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results

Incidence of digital papillomatosis ("hairy warts") in a dairy herd. Response to surgery and autogenous vaccination

Digital papillomatosis (hairy warts) was diagnosed in a dairy herd with a high level of lameness (20%). Warts ranged from mild to moderate to severe, with severity increasing with length of lactations. All milking cows (100%) had at least mild lesions. Cows with severe lesions were more likely to be lame. Severity of lesions had no influence on 305-day ME milk production, days open, or somatic cell counts. Cows in milk more than 150 days and lame produced 3 kg less milk per day than cows that were not lame. Almost all warts were in the interdigital cleft near the heel of the rear feet. A few cows had lesions in the front of the interdigital cleft or on the front feet. No viral particles were observed or isolated. A new, Gram-negative, motile, facultatively anaerobic, spiral-shaped bacteria was isolated from one lesion. The cellular fatty acid profile of this bacterium had no match to any other known bacteria in any of three computer databases examined. Cows with severe lesions were assigned randomly to one of four groups: Group 1: surgical removal and autogenous vaccination; Group 2: surgical removal only; Group 3: autogenous vaccination only, and Group 4: control. Neither surgical removal nor autogenous vaccination had a significant effect on wart severity, lameness, or milk production when cows were inspected 10 wk later. Contemporary evaluation of 249 herdmates revealed a substantial number of severely affected cows naturally improved. Of 25 severely affected herd contemporaries, only 8 were severely affected 10 wk later. Evidently, natural improvement of lesions is a common phenomenon with “hairy warts.

Transverse and longitudinal characterization of electron beams using interaction with optical near-fields

We demonstrate an experimental technique for both transverse and longitudinal characterization of bunched femtosecond free electron beams. The operation principle is based on monitoring of the current of electrons that obtained an energy gain during the interaction with the synchronized optical near-field wave excited by femtosecond laser pulses. The synchronous accelerating/decelerating fields confined to the surface of a silicon nanostructure are characterized using a highly focused sub-relativistic electron beam. Here the transverse spatial resolution of 450 nm and femtosecond temporal resolution achievable by this technique are demonstrated

Subrelativistic Alternating Phase Focusing Dielectric Laser Accelerators

We demonstrate a silicon-based electron accelerator that uses laser optical near fields to both accelerate and confine electrons over extended distances. Two dielectric laser accelerator (DLA) designs were tested, each consisting of two arrays of silicon pillars pumped symmetrically by pulse front tilted laser beams, designed for average acceleration gradients 35 and 50 MeV/m respectively. The DLAs are designed to act as alternating phase focusing (APF) lattices, where electrons, depending on the electron-laser interaction phase, will alternate between opposing longitudinal and transverse focusing and defocusing forces. By incorporating fractional period drift sections that alter the synchronous phase between $\pm 60^\circ$ off crest, electrons captured in the designed acceleration bucket experience half the peak gradient as average gradient while also experiencing strong confinement forces that enable long interaction lengths. We demonstrate APF accelerators with interaction lengths up to 708 ${\mu}$m and energy gains up to 23.7 $\pm$ 1.07 keV FWHM, a 25$\%$ increase from starting energy, demonstrating the ability to achieve substantial energy gains with subrelativistic DLA.Comment: 16 page

Energy efficiency studies for dual-grating dielectric laser-driven accelerators

Dielectric laser-driven accelerators (DLAs) can provide high accelerating gradients in the GV/m range due to their having higher breakdown thresholds than metals, which opens the way for the miniaturization of the next generation of particle accelerator facilities. Two kinds of scheme, the addition of a Bragg reflector and the use of pulse-front-tilted (PFT) laser illumination, have been studied separately to improve the energy efficiency for dual-grating DLAs. The Bragg reflector enhances the accelerating gradient of the structure, while the PFT increases the effective interaction length. In this paper, we investigate numerically the advantages of using the two schemes in conjunction. Our calculations show that, for a 100-period structure with a period of 2 micrometer, such a design effectively increases the energy gain by more than 100 % when compared to employing the Bragg reflector with a normal laser, and by about 50 % when using standard structures with a PFT laser. A total energy gain of as much as 2.6 MeV can be obtained for a PFT laser beam when illuminating a 2000-period dual-grating structure with a Bragg reflector.Comment: 7 pages, 4 Figures, Proceedings of EAAC2017 Conferenc