OBLICZENIA NUMERYCZNE POCHODNEJ UŁAMKOWEGO RZĘDU W ZAGADNIENIACH POCZĄTKOWYCH, PRZYKŁADY W PROGRAMACH MATLAB I MATHEMATICA

The paper concerns a numerical method that deals with the computations of the fractional derivative in Caputo and Riemann-Liouville definitions. The method can be applied in time stepping processes of initial value problems. The name of the method is SubIval, which is an acronym of its previous name – the subinterval-based method. Its application in solving systems of fractional order state equations is presented. The method has been implemented into an ActiveX DLL. Exemplary MATLAB and Mathematica codes are given, which provide guidance on how the DLL can be used.Artykuł dotyczy numerycznej metody, którą wykorzystać można do obliczeń pochodnej ułamkowego rzędu w definicji Caputo i Riemanna-Liouville’a. Metoda ta może być wykorzystana przy rozwiązywaniu zagadnień początkowych. Metoda nosi nazwę SubIval, co jest akronimem jej poprzedniej, anglojęzycznej nazwy „subinterval-based method” (metoda podprzedziałów). Przedstawiono jej zastosowanie w rozwiązywaniu równań stanu ułamkowego rzędu. Metoda została zaimplementowana w bibliotece DLL z obsługą ActiveX. Przedstawiono przykładowe kody obliczeniowe (w oprogramowaniach MATLAB i Mathematica), które zawierają wskazówki dotyczące zastosowania biblioteki

Comparison of Step Response Characteristics of Simple Fractional Order Systems and Second Order Systems

The step response characteristics of first and second order systems are well known. On the other hand, the step response of fractional order systems (FOSs) with 2-term fractional denominator is like those of first and second order systems. But there are important differences between the two types of characteristics. Considering the step response, the behavior of simple FOS with a denominator polynomial having unity term and the other involves fractional power is investigated in this paper comparatively with 1st and 2nd order systems. The results bring light for the design of fractional order control systems (FOCSs)

Geophysics of Small Planetary Bodies

As a SETI Institute PI from 1996-1998, Erik Asphaug studied impact and tidal physics and other geophysical processes associated with small (low-gravity) planetary bodies. This work included: a numerical impact simulation linking basaltic achondrite meteorites to asteroid 4 Vesta (Asphaug 1997), which laid the groundwork for an ongoing study of Martian meteorite ejection; cratering and catastrophic evolution of small bodies (with implications for their internal structure; Asphaug et al. 1996); genesis of grooved and degraded terrains in response to impact; maturation of regolith (Asphaug et al. 1997a); and the variation of crater outcome with impact angle, speed, and target structure. Research of impacts into porous, layered and prefractured targets (Asphaug et al. 1997b, 1998a) showed how shape, rheology and structure dramatically affects sizes and velocities of ejecta, and the survivability and impact-modification of comets and asteroids (Asphaug et al. 1998a). As an affiliate of the Galileo SSI Team, the PI studied problems related to cratering, tectonics, and regolith evolution, including an estimate of the impactor flux around Jupiter and the effect of impact on local and regional tectonics (Asphaug et al. 1998b). Other research included tidal breakup modeling (Asphaug and Benz 1996; Schenk et al. 1996), which is leading to a general understanding of the role of tides in planetesimal evolution. As a Guest Computational Investigator for NASA's BPCC/ESS supercomputer testbed, helped graft SPH3D onto an existing tree code tuned for the massively parallel Cray T3E (Olson and Asphaug, in preparation), obtaining a factor xIO00 speedup in code execution time (on 512 cpus). Runs which once took months are now completed in hours