Fractional Integration and Fractional Differentiation of the M-Series

Mathematics Subject Classification: 26A33, 33C60, 44A15

Fractional Calculus: Quo Vadimus? (Where Are We Going?)

Discussions under this title were held during a special session in frames of the International Conference “Fractional Differentiation and Applications” (ICFDA ’14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.

Design of Digital Fractional-Order Integrators and Differentiators by Least Squares

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Short-Circuit Impedance of Power Transformers: the Fractional Order Calculus Approach

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45

Calculus of Variations with Classical and Fractional Derivatives

MSC 2010: 49K05, 26A33

A new least-squares approach to differintegration modeling

Signal Processing, Vol. 86, nº 10

Определение мощности и фракционной дифференциации селевых наносов в пойме р. Дуруджи по геофизическим данным

სტატიაში მოცემულია დურუჯის აუზში სელური ნალექების სიმძლავრის, ფრაქციული დიფერენციაციის, გაწყლიანების ხარისხის, ძირითადი ქანების ლითოლოგიური არაერთგვაროვნების, მიკროტექტონიკური სურათის დაზუსტება და მიწისქვეშა ფილტრაციული ნაკადის გაბატონებული მიმართულების დადგენა ძიების გეოფიზიკური (გეოელექტრული და სეისმომეტრიული) მეთოდებით.

Chemical composition of rocks and minerals from the Karymsky volcanic center, Eastern Kamchatka

The powerful eruption in the Akademii Nauk caldera on January 2, 1996 marked a new activity phase of the Karymsky volcano and became a noticeable event in the history of modern volcanism in Kamchatka. The paper reports data obtained by studying more than 200 glassy melt inclusions in phenocrysts of olivine (Fo82-72), plagioclase (An92-73), and clinopyroxene (Mg# 83-70) in basalts of the 1996 eruption. The data were used to estimate composition of the parental melt and physicochemical parameters of the magma evolution. According to our data, the parental melt corresponded to low magnesium, high aluminum basalt (SiO2 = 50.2%, MgO = 5.6%, Al2O3 = 17%) of the mildly potassium type (K2O = 0.56%) and contained much dissolved volatile components (H2O = 2.8%, S = 0.17%, and Cl = 0.11%). Melt inclusions in the minerals are similar in chemical composition, a fact testifying that the minerals crystallized simultaneously with one another. Their crystallization started at pressure ~1.5 kbar, proceeded within a narrow temperature range of 1040+/-20°C, and continued until near-surface pressure ~100 bar was reached. Degree of crystallization of the parental melt during its eruption was close to 55%. Massive crystallization was triggered by H2O degassing under pressure <1 kbar. Magma degassing in an open system resulted in escape of 82% H2O, 93% S, and 24% Cl (of their initial contents in the parental melt) to the fluid phase. Release of volatile compounds to the atmosphere during the eruption that lasted for 18 h was estimated as 1.7x10**6 t H2O, 1.4x10**5 t S, and 1.5x10**4 t Cl. Concentrations of most incompatible trace elements in the melt inclusions are close to those in the rocks and to the expected fractional differentiation trend. Melt inclusions in plagioclase were found to be selectively enriched in Li. The Li-enriched plagioclase with melt inclusions thought to originate from cumulate layers in the feeding system beneath Karymsky volcano, in which plagioclase interacted with Li-rich melts/brines and was subsequently entrapped and entrained by the magma during the 1996 eruption.

Calculation of fractional derivatives of noisy data with genetic algorithms

This paper addresses the calculation of derivatives of fractional order for non-smooth data. The noise is avoided by adopting an optimization formulation using genetic algorithms (GA). Given the flexibility of the evolutionary schemes, a hierarchical GA composed by a series of two GAs, each one with a distinct fitness function, is established.

On the fractional order control of heat systems

The differentiation of non-integer order has its origin in the seventeenth century, but only in the last two decades appeared the first applications in the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated namely the fractional PID and the Smith predictor. Extensive simulations are presented assessing the performance of the proposed fractional-order algorithms.

Fractional control of heat diffusion systems

The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared. Simulations are presented assessing the performance of the proposed fractional-order algorithms.

Enriched basaltic andesites from mid-crustal fractional crystallization, recharge, and assimilation (Pilavo Volcano, Western Cordillera of Ecuador)

The origin of andesite is an important issue in petrology because andesite is the main eruptive product at convergent margins, corresponds to the average crustal composition and is often associated with major Cu-Au mineralization. In this study we present petrographic, mineralogical, geochemical and isotopic data for basaltic andesites of the latest Pleistocene Pilavo volcano, one of the most frontal volcanoes of the Ecuadorian Quaternary arc, situated upon thick (30-50 km) mafic crust composed of accreted Cretaceous oceanic plateau rocks and overlying mafic to intermediate Late Cretaceous-Late Tertiary magmatic arcs. The Pilavo rocks are basaltic andesites (54-57 center dot 5 wt % SiO(2)) with a tholeiitic affinity as opposed to the typical calc-alkaline high-silica andesites and dacites (SiO(2) 59-66 wt %) of other frontal arc volcanoes of Ecuador (e.g. Pichincha, Pululahua). They have much higher incompatible element contents (e.g. Sr 650-1350 ppm, Ba 650-1800 ppm, Zr 100-225 ppm, Th 5-25 ppm, La 15-65 ppm) and Th/La ratios (0 center dot 28-0 center dot 36) than Pichincha and Pululahua, and more primitive Sr ((87)Sr/(86)Sr similar to 0 center dot 7038-0 center dot 7039) and Nd (epsilon(Nd) similar to +5 center dot 5 to +6 center dot 1) isotopic signatures. Pilavo andesites have geochemical affinities with modern and recent high-MgO andesites (e.g. low-silica adakites, Setouchi sanukites) and, especially, with Archean sanukitoids, for both of which incompatible element enrichments are believed to result from interactions of slab melts with peridotitic mantle. Petrographic, mineral chemistry, bulk-rock geochemical and isotopic data indicate that the Pilavo magmatic rocks have evolved through three main stages: (1) generation of a basaltic magma in the mantle wedge region by flux melting induced by slab-derived fluids (aqueous, supercritical or melts); (2) high-pressure differentiation of the basaltic melt (at the mantle-crust boundary or at lower crustal levels) through sustained fractionation of olivine and clinopyroxene, leading to hydrous, high-alumina basaltic andesite melts with a tholeiitic affinity, enriched in incompatible elements and strongly impoverished in Ni and Cr; (3) establishment of one or more mid-crustal magma storage reservoirs in which the magmas evolved through dominant amphibole and clinopyroxene (but no plagioclase) fractionation accompanied by assimilation of the modified plutonic roots of the arc and recharge by incoming batches of more primitive magma from depth. The latter process has resulted in strongly increasing incompatible element concentrations in the Pilavo basaltic andesites, coupled with slightly increasing crustal isotopic signatures and a shift towards a more calc-alkaline affinity. Our data show that, although ultimately originating from the slab, incompatible element abundances in arc andesites with primitive isotopic signatures can be significantly enhanced by intra-crustal processes within a thick juvenile mafic crust, thus providing an additional process for the generation of enriched andesites.

Igneous layering, fractional crystallization and growth of granitic plutons: The Dolbel batholith in SW Niger

This study reassesses the development of compositional layering during the growth of granitic plutons, with emphasis on fractional crystallization and its interaction with both injection and inflation-related deformation. The Dolbel batholith (SW Niger) consists of 14, kilometre-sized plutons emplaced by pulsed magma inputs. Each pluton has a coarse-grained core and a peripheral layered series. Rocks consist of albite (An(<= 11)), K-feldspar (Or(96 99), Ab(1) (4)), quartz, edenite (X(Mg)=0337-0.55), augite (X(Mg)=0.65-0.72) and accessories (apatite, titanite and Fe-Ti-oxides). Whole-rock compositions are metaluminous, sodic (K(2)O/Na(2)O=0.49-0.62) and iron-rich [FeO(tot)/(FeO(tot)+MgO)=0.65-0.82]. The layering is present as size-graded and modally graded, sub-vertical, rhythmic units. Each unit is composed of three layers, which are, towards the interior: edenite +/- plagioclase (C(a/p)), edenite+plagioclase+augite+quartz (C(q)), and edenite+plagioclase+augite+quartz+K-feldspar (C(k)). All phases except quartz show zoned microstructures consisting of external intercumulus overgrowths, a central section showing oscillatory zoning and, in the case of amphibole and titanite, complexly zoned cores. Ba and Sr contents of feldspars decrease towards the rims. Plagioclase crystal size distributions are similar in all units, suggesting that each unit experienced a similar thermal history. Edenite, characteristic of the basal C(a/p) layer, is the earliest phase to crystallize. Microtextures and phase diagrams suggest that edenite cores may have been brought up with magma batches at the site of emplacement and mechanically segregated along the crystallized wall, whereas outer zones of the same crystals formed in situ. The subsequent C(q) layers correspond to cotectic compositions in the Qz-Ab-Or phase diagram at P(H2O)=5 kbar. Each rhythmic unit may therefore correspond to a magma batch and their repetition to crystallization of recurrent magma recharges. Microtextures and chemical variations in major phases allow four main crystallization stages to be distinguished: (1) open-system crystallization in a stirred magma during magma emplacement, involving dissolution and overgrowth (core of edenite and titanite crystals); (2) in situ fractional crystallization in boundary layers (C(a/p) and C(q) layers); (3) equilibrium `en masse' eutectic crystallization (C(k) layers); (4) compaction and crystallization of the interstitial liquid in a highly crystallized mush (e. g. feldspar intercumulus overgrowths). It is concluded that the formation of the layered series in the Dolbel plutons corresponds principally to in situ differentiation of successive magma batches. The variable thickness of the Ck layers and the microtextures show that crystallization of a rhythmic unit stops and it is compacted when a new magma batch is injected into the chamber. Therefore, assembly of pulsed magma injections and fractional crystallization are independent, but complementary, processes during pluton construction.

Magma differentiation fractionates Mo isotope ratios: Evidence from the Kos Plateau Tuff (Aegean Arc)

We investigated high temperature Mo isotope fractionation in a hydrous supra-subduction volcano-plutonic system (Kos, Aegean Arc, Greece) in order to address the debate on the δ98/95Mo variability of the continental crust. In this igneous system, where differentiation is interpreted to be dominated by fractional crystallization, bulk rock data from olivine basalt to dacite show δ98/95Mo ratios increasing from +0.3 to +0.6‰ along with Mo concentrations increasing from 0.8 to 4.1 μg g−1. Data for hornblende and biotite mineral separates reveal the extraction of light Mo into crystallizing silicates, with minimum partition coefficients between hornblende- silicate melt and biotite-silicate melt of 0.6 and 0.4 δ98/95Mo, respectively. Our data document significant Mo isotope fractionation at magmatic temperatures, hence, the igneous contribution to continental runoff is variable, besides probable source-related variability. Based on these results and published data an average continental δ98/95Mo of +0.3 to +0.4‰ can be derived. This signature corresponds more closely to the average of published data of dissolved Mo loads of large rivers than previous estimates and is consistent with an upper limit of δ98/95Mo = 0.4‰ of the Earth's upper crust as derived from the analysis of molybdenites.