Fitted modifications of classical Runge‐Kutta pairs of orders 5(4)

Runge‐Kutta pairs sharing orders 5 and 4 are among the most celebrated methods for solving initial value problems. Here, after considering the phase lag as a function of frequency v, we derive a modification of a particular pair. Namely, we present a zero dissipative pair with vanished phase error and its first derivative. After extended numerical tests in various oscillatory problems, it seems that this modified pair outperforms existing methods in the literature.     

Trigonometric fitted,eighth‐order explicit Numerov‐type methods

We consider the integration of the special second‐order initial value problem of the form . A recently introduced family of 7 stages, eighth‐order methods, sharing constant coefficients, is used as base. This family is properly modified to derive phase fitted and zero dissipative methods (ie, trigonometric fitted) that are best suited for integrating oscillatory problems. Numerical tests over a set of problems shows enhanced performance when the purely linear part of the problems is rather large in comparison with the rest of nonlinear parts. An appendix implementing a MATLAB listing with the coefficients of the new method is also given.     

Onsager consistency checks for multicomponent diffusion models

The Onsager reciprocity relations are applied to several recently proposed multicomponent diffusion models in an attempt to gauge their validity and ascertain their applicability. Each of these friction‐based diffusion models stems from the more general Bearman formalism through various assumptions regarding the individual friction coefficients. By assessing the compliance of the Bearman model with respect to the Onsager relations, we ascertain the validity of the simplifications introduced to each diffusion model and suggest which postulates lead to results consistent with the Onsager relations. Although some models are not consistent with the Onsager relations, each model predicts the multicomponent drying of polymer films reasonably well. The necessity for consistency with the Onsager development is, therefore, revisited. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1496–1504, 2001     

Wake structures and vortex-induced vibrations of a long flexible cylinder—Part 2: Drag coefficients and vortex modes

Drag coefficients and vortex structures in the wake of a vertical long flexible cylinder vibrating at low mode numbers are presented in this paper. A model with an external diameter of 16 mm and a total length of 1.5 m giving an aspect ratio of about 94 was used to perform more than 100 runs in which Reynolds numbers ranged between 1200 and 12 000. Only the lower 40% of its length was exposed to the water current in the flume and applied top tensions varied from 15 to 110 N giving fundamental natural frequencies in the range from 3 to 7.1 Hz. Reduced velocities based on the fundamental natural frequency up to 16 were reached. The mass ratio was 1.8 and the combined mass-damping parameter about 0.05. The largest drag coefficients were found related to the largest x–y synchronised motions. Digital particle image velocimetry was used to investigate the vortex structures in the wake of the cylinder model. Two modes of vortex shedding were observed, depending on the response branch and the position along the length of the model at which the interrogations were performed.     

An NMR investigation on the phase structure and molecular mobility of the novel exfoliated polyethylene/palygorskite nanocomposites

Novel exfoliated polyethylene (PE)/palygorskite nanocomposites prepared by in situ polymerization are characterized by solid‐state nuclear magnetic resonance (NMR). The phase structure and molecular mobility are investigated by a combination of proton and carbon NMR. The results showed that incorporation of small amounts of palygorskite had great influence on the phase structure and molecular mobility. The incorporated palygorskite hindered the crystallization process and introduced motion‐hindered chains in the NMR crystalline and amorphous phase. ¹³C cross‐polarization and magic‐angle spinning NMR revealed two orthorhombic crystalline phase with different line‐width. The chain mobility of orthorhombic crystalline phase with broad resonance line is obviously hindered compared with the phase with narrow resonance line when the filler is introduced. Additionally, the results of pulsed field gradient NMR technique show those the tortuosities in the nanocomposites are much higher than that in the bulk PE. The self‐diffusion process of probe molecules is also influenced by the palygorksite load. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1363–1371, 2010     

Synthesis and nonlinear optical properties of side chain liquid crystalline polymers containing azobenzene push–pull chromophores

Azobenzene monomeric precursors bearing piperazine as donor moiety with different withdrawing groups and derived side chain polymethacrylates have been prepared and characterized. Monomers having terminal cyano or nitro groups, and the corresponding polymers, exhibited smectic A phases. Linear and nonlinear optical properties of every monomer and thin films of the cyano polymer ( pol‐PZ‐CN ) have been also studied. UV‐vis spectroscopy revealed out‐of‐plane orientation in the as prepared films, as confirmed by waveguide refractive index measurements. Moreover, absorption spectra indicated the presence of azo aggregates in these films. The initial molecular arrangement has been modified by applying thermal annealing within the mesophase range and UV‐blue irradiation. Although thermal annealing resulted in a significant amplification of the out‐of‐plane optical anisotropy due to thermotropic self‐organization of side chain azo moieties, irradiation with 440 nm light induced some disruption of aggregates. The nonlinear optical response of Corona poled films has been studied by second harmonic generation measurements, and the influence of the molecular arrangement on the nonlinear d_ij coefficients has been analyzed. The more efficient poling corresponded to preirradiated films. In any case, a noticeable degree of polar order (70% of the initial d₃₃ value) remained for several months after the poling in films kept at RT. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 232–242, 2010     

Hardy inequalities for fractional integrals on general domains

We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.     

Accurate Evaluation of the Cubic Lattice Green Functions Using Binomial Expansion Theorems

A new, simple, and efficient technique for the accurate evaluation of the lattice Green functions is presented. Using binomial expansion theorems, these functions are expressed through the binomial coefficients and basic integrals. The extensive test calculations show that the proposed algorithm in this work is the most efficient method in practical computations. Finally, in order to show the practical use of analytical expressions found some computation examples and comparisons with literature are made.     

A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u _t + _i=1 ^d V _i(x, t)f _i(u) _{x i} = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.     

Double roots of power series and related matters

For a given collection of distinct arguments , multiplicities and a real interval containing zero, we are interested in determining the smallest for which there is a power series with coefficients in , and roots of order respectively. We denote this by . We describe the usual form of the extremal series (we give a sufficient condition which is also necessary when the extremal series possesses at least non-dependent coefficients strictly inside , where is 1 or 2 as is real or complex). We focus particularly on , the size of the smallest double root of a power series lying on a given ray (of interest in connection with the complex analogue of work of Boris Solomyak on the distribution of the random series ). We computed the value of for the rationals in of denominator less than fifty. The smallest value we encountered was . For the one-sided intervals and the corresponding smallest values were and .