A New and Elegant Approach for Solving $$\varvec{n \times n}$$-Order Linear Fractional Differential Equations

The application of fractional differential equations (FDEs) in the fields of science and engineering are gradually increasing day by day during the last two decades. The solutions of linear systems of FDEs are of great importance. Several investigations are carried out on such systems using eigenvalue analysis or Laplace transform method. But both the methods have limitations, and as of now there are no methods for solving \(n \times n\)-order linear FDEs. In the present investigation, the issues of such difficulties are addressed, and the exact solutions of linear \(2 \times 2\)-order linear FDEs are presented by Laplace transform. We are unable to provide the exact solutions of such system of order \(n \times n\) by Laplace transform. To overcome this, we provide a new and elegant approach to find the approximate solutions of \(n \times n\)-order linear FDEs with the help of residual power series (RPS) method. The results thus obtained are verified by providing numerous examples.     

Spectroscopic detection of a phytochrome-like photoreceptor in the myxomycete Physarum polycephalum and the kinetic mechanism for the photocontrol of sporulation by Pfr.

Sporulation of the true slime mold Physarum polycephalum (Myxomycetales) can be triggered by the far-red/red reversible Physarum phytochrome. Physarum plasmodia were analyzed with a purpose-built dual-wavelength photometer that is designed for phytochrome measurements. A photoreversible absorbance change at 670 nm was monitored after actinic red (R) and far-red (FR) irradiation of starved plasmodia, confirming the occurrence of a phytochrome-like photoreceptor in Physarum spectroscopically. These signals were not found in growing plasmodia, suggesting the Physarum phytochrome to be synthesized during starvation, which makes the cells competent for the photoinduction of sporulation. The photoconversion rates by R and FR light were similar in the phytochromes of Physarum and etiolated oat shoots. In dark-grown Physarum plasmodia that had not been preexposed to any light only R induced a detectable absorbance change while FR did not. This indicates that most (at least 90%) of the photoreversible pigment occurs in the red-absorbing form. Since the effectiveness of FR in triggering sporulation was enhanced by preirradiation with R, it is concluded that at least part of the Pr can be photoconverted to the active Pfr photoreceptor species. We propose a kinetic mechanism for the photocontrol of sporulation by photoconversion of Pfr, which may also hold for the high-irradiance response to FR in Arabidopsis and Cuscuta.     

Dynamism of two-mode’s parameters on the field function for third-order dispersive Fisher: application for fibre optics

Two-mode equations model the spread of two different wave modes simultaneously. A new two-mode dispersive Fisher of third order (TMDF) is proposed and different types of solutions are obtained. We studied the effect of TMDF’s factors (dispersive, nonlinearity and the phase velocity) on the behavior of the obtained field function. We believe that our findings can be useful for investigating further analysis in fibre optics and engineering applications.     

Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series

Transport in Porous Media - Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to...     

A proper generalized decomposition approach for optical flow estimation

This paper introduces the use of the proper generalized decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, ie, optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularization term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (a) the weak convergence based on the Brézis-Lieb decomposition and (b) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely, low image resolution and/or low signal-to-noise ratio (SNR).     

Bromination of Enamines from Tertiary Amides Using the Petasis Reagent: A Convenient One-Pot Regioselective Route to Bromomethyl Ketones

An original one-pot synthesis of bromomethyl ketones is achived using the Petasis reagent (dimethyltitanocene) as a key for enamine generation. Several amides were used to test the limits of the procedure by changing either the alkyl chain R or the amino portion of the starting materials. The enamines generated in situ were allowed to react with bromine at low temperature followed by hydrolysis to yield bromomethyl ketones in excellent yields (85 to 95%). Mechanistic details and optimum conditions for the reaction are briefly discussed. The present approach offers several advantages such as regioselectivity in enamine formation, good yields, mild reaction conditions, and ease of experimentation.

[Supplementary materials are available for this article. Go to the publisher's online edition of Synthetic Communications® for the following free supplemental resource: Full experimental and spectral details.]     

An exact Block–Newton algorithm for solving fluid–structure interaction problems

In this Note, we introduce a partitioned Newton based method for solving nonlinear coupled systems arising in the numerical approximation of fluid–structure interaction problems. The originality of this Schur–Newton algorithm lies in the exact Jacobians evaluation involving the fluid–structure linearized subsystems which are here fully developed. To cite this article: M.Á. Fernández, M. Moubachir, C. R. Acad. Sci. Paris, Ser. I 336 (2003).