A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.     

Isoperimetric Inequalities for a Wedge-Like Membrane

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber–Krahn. In this work, we introduce “relative torsional rigidity” for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya–Szeg?, Payne, Payne–Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.     

Coordination chemistry of tin: Part III. The crystal structure and thermal behavior of dipotassium dimethyl-tetrafluoro-stannate dihydrate, K2[(CH3)2SnF4]·2H2O

Reaction of (CH₃)₂SnF₂ with two equivalents of KF in aqueous medium leads to the formation of the complex salt K₂[(CH₃)₂SnF₄]·2H₂O (1). Its crystal structure was determined by single-crystal X-ray diffraction. Complex 1 crystallizes in the monoclinic space group C2 (No. 5) with the lattice parameters a=9.265(1), b=7.556(1), c=7.076(1) Å; β=98.21(1)° and Z=2. The structure is characterized by the anion [(CH₃)₂SnF₄]²⁻ in which the tin atom adopts a slightly distorted octahedral coordination, with the methyl groups in trans position. The potassium cations are pentacoordinated from three fluorine atoms and the oxygen atoms of two water molecules in the form of a distorted square pyramid. In addition, the thermal behavior of the compound was studied with the aid of TG/DSC-measurements coupled with MS, revealing that the dehydration of 1 takes place at 75 °C, with an enthalpy of 57.79 kJ mol⁻¹, and that it decomposes without melting in two further endothermic steps to undetermined phases in the system KF-SnF₂-SnF₄ and free carbon (∼0.1%).     

Comment on “Debye temperature and melting point of II‐VI and III‐V semiconductors” [Cryst. Res. Technol. 45, No. 9, 920–924 (2010)]

In the present work, we have readjusted some empirical parameters obtained by Kumar et al. in their work which contains some numerical errors.     

New routes to steroidal heterocyclic derivatives: Synthesis of biologically active pyrazolyl‐ and isoxazolylpregnene derivatives

Pyrazolyl‐ and isoxazolylpregnene derivatives were synthesized from the appropriate hydrazonoyl chlorides and hydroximoyl chlorides with enaminoprogesterone derivative. The newly synthesized compounds were elucidated by elemental analysis, spectral data and chemical transformation. Some products were tested towards some bacteria and some Fungal‐plant pathogens     

Absolute configuration of N-[(–)-2-(7-methoxy-1,2,3,4-tetra­hydro­-1-naphthyl)­ethyl]­cyclo­propyl­carbox­amide,a highly potent and selective melatonin analogue

The title compound, C₁₇H₂₃NO₂, a tetra­hydro­naphthalenic analogue of melatonin, crystallizes in the monoclinic space group P2₁ with one mol­ecule in the asymmetric unit. The crystal structure has been determined by X-ray analysis at room temperature. The absolute configuration of this compound was determined unambiguously as R at the chiral naphthalene C-1 position.     

Analysis of projectile motion in view of fractional calculus

The projectile motion is examined by means of the fractional calculus. The fractional differential equations of the projectile motion are introduced by generalizing Newton’s second law and Caputo’s fractional derivative is considered to use the physical initial conditions. In the absence of air resistance it is found that at certain conditions, the range and the maximum height of the projectile obtained by using the fractional calculus give the same results of the classical calculus. It is also found that, unlike the classical projectile motion, the launching angle that maximizes the horizontal range is a function of the arbitrary order of the fractional derivative α. Moreover, in a resistant medium, the parametric equations are expressed in terms of Mittag-Leffler function and the results agree with those of the classical projectile as α → 2. Moreover, the trajectories of the projectile are discussed in graphs and compared with those of the classical calculus. In order to explore the validity of modelling the projectile motion by the fractional approach, we compared our results with the experimental data of mortar.