Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H 1/2

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>¿. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.

From Differences to Derivatives

A relation showing that the Grünwald-Letnikov and generalized Cauchy derivatives are equal is deduced confirming the validity of a well known conjecture. Integral representations for both direct and reverse fractional differences are presented. From these the fractional derivative is readily obtained generalizing the Cauchy integral formula.

Fractional Calculus of the Generalized Wright Function

Mathematics Subject Classification: 26A33, 33C20.

Some Fractional Extensions of the Temperature Field Problem in Oil Strata

This survey is devoted to some fractional extensions of the incomplete lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for the fractional heat equation. By using Caputo’s differintegration operator and the Laplace transform, new integral forms of the solutions are obtained. In each of the different cases the integrands are expressed in terms of a convolution of two special functions of Wright’s type.

A Brief Story about the Operators of the Generalized Fractional Calculus

2000 Mathematics Subject Classification: 26A33, 33C60, 44A20

An Analog of the Tricomi Problem for a Mixed Type Equation with a Partial Fractional Derivative

Mathematics Subject Classification 2010: 35M10, 35R11, 26A33, 33C05, 33E12, 33C20.

Fractional Calculus of P-transforms

MSC 2010: 44A20, 33C60, 44A10, 26A33, 33C20, 85A99

Generalized Fractional Calculus, Special Functions and Integral Transforms: What is the Relation?

Виржиния С. Кирякова - В този обзор илюстрираме накратко наши приноси към обобщенията на дробното смятане (анализ) като теория на операторите за интегриране и диференциране от произволен (дробен) ред, на класическите специални функции и на интегралните трансформации от лапласов тип. Показано е, че тези три области на анализа са тясно свързани и взаимно индуцират своето възникване и по-нататъшно развитие. За конкретните твърдения, доказателства и примери, вж. Литературата.

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method.

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Fractional variational problems depending on indefinite integrals and with delay

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example, where we nd analytically the minimizer.

Design, Synthesis And Biological Evaluation Of Hybrid Bioisoster Derivatives Of N-acylhydrazone And Furoxan Groups With Potential And Selective Anti-trypanosoma Cruzi Activity.

Hybrid bioisoster derivatives from N-acylhydrazones and furoxan groups were designed with the objective of obtaining at least a dual mechanism of action: cruzain inhibition and nitric oxide (NO) releasing activity. Fifteen designed compounds were synthesized varying the substitution in N-acylhydrazone and in furoxan group as well. They had its anti-Trypanosoma cruzi activity in amastigotes forms, NO releasing potential and inhibitory cruzain activity evaluated. The two most active compounds (6, 14) both in the parasite amastigotes and in the enzyme contain the nitro group in para position of the aromatic ring. The permeability screening in Caco-2 cell and cytotoxicity assay in human cells were performed for those most active compounds and both showed to be less cytotoxic than the reference drug, benznidazole. Compound 6 was the most promising, since besides activity it showed good permeability and selectivity index, higher than the reference drug. Thereby the compound 6 was considered as a possible candidate for additional studies.

Synthesis And Antiproliferative Activity Of 8-hydroxyquinoline Derivatives Containing A 1,2,3-triazole Moiety.

Twelve novel 8-hydroxyquinoline derivatives were synthesized with good yields by performing copper-catalyzed Huisgen 1,3-dipolar cycloaddition (click reaction) between an 8-O-alkylated-quinoline containing a terminal alkyne and various aromatic or protected sugar azides. These compounds were evaluated in vitro for their antiproliferative activity on various cancer cell types. Protected sugar derivative 16 was the most active compound in the series, exhibiting potent antiproliferative activity and high selectivity toward ovarian cancer cells (OVCAR-03, GI50 < 0.25 μg mL(-1)); this derivative was more active than the reference drug doxorubicin (OVCAR-03, GI50 = 0.43 μg mL(-1)). In structure-activity relationship (SAR) studies, the physico-chemical parameters of the compounds were evaluated and docking calculations were performed for the α-glucosidase active site to predict the possible mechanism of action of this series of compounds.

Design And Synthesis Of N-acylated Aza-goniothalamin Derivatives And Evaluation Of Their In Vitro And In Vivo Antitumor Activity.

Herein we describe the synthesis of a focused library of compounds based on the structure of goniothalamin (1) and the evaluation of the potential antitumor activity of the compounds. N-Acylation of aza-goniothalamin (2) restored the in vitro antiproliferative activity of this family of compounds. 1-(E)-But-2-enoyl-6-styryl-5,6-dihydropyridin-2(1H)-one (18) displayed enhanced antiproliferative activity. Both goniothalamin (1) and derivative 18 led to reactive oxygen species generation in PC-3 cells, which was probably a signal for caspase-dependent apoptosis. Treatment with derivative 18 promoted Annexin V/7-aminoactinomycin D double staining, which indicated apoptosis, and also led to G2 /M cell-cycle arrest. In vivo studies in Ehrlich ascitic and solid tumor models confirmed the antitumor activity of goniothalamin (1), without signs of toxicity. However, derivative 18 exhibited an unexpectedly lower in vivo antitumor activity, despite the treatments being administered at the same site of inoculation. Contrary to its in vitro profile, aza-goniothalamin (2) inhibited Ehrlich tumor growth, both on the ascitic and solid forms. Our findings highlight the importance of in vivo studies in the search for new candidates for cancer treatment.

Esi(+)-ms And Gc-ms Study Of The Hydrolysis Of N-azobenzyl Derivatives Of Chitosan.

New N-p-chloro-, N-p-bromo-, and N-p-nitrophenylazobenzylchitosan derivatives, as well as the corresponding azophenyl and azophenyl-p-sulfonic acids, were synthesized by coupling N-benzylvchitosan with aryl diazonium salts. The synthesized molecules were analyzed by UV-Vis, FT-IR, 1H-NMR and 15N-NMR spectroscopy. The capacity of copper chelation by these materials was studied by AAS. Chitosan and the derivatives were subjected to hydrolysis and the products were analyzed by ESI(+)-MS and GC-MS, confirming the formation of N-benzyl chitosan. Furthermore, the MS results indicate that a nucleophilic aromatic substitution (SnAr) reaction occurs under hydrolysis conditions, yielding chloroaniline from N-p-bromo-, and N-p-nitrophenylazo-benzylchitosan as well as bromoaniline from N-p-chloro-, and N-p-nitrophenylazobenzyl-chitosan.