Approximate solutions of Fredholm integral equations of the second kind

This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail. (c) 2009 Elsevier Inc. All rights reserved.

A Fredholm-type theory for third-kind linear integral equations

The third-kind linear integral equation Image where g(t) vanishes at a finite number of points in (a, b), is considered. In general, the Fredholm Alternative theory [[5.]] does not hold good for this type of integral equation. However, imposing certain conditions on g(t) and K(t, t′), the above integral equation was shown [[1.], 49–57] to obey a Fredholm-type theory, except for a certain class of kernels for which the question was left open. In this note a theory is presented for the equation under consideration with some additional assumptions on such kernels.

Solution of a hypersingular integral equation of the second kind

A straightforward analysis involving the complex function-theoretic method is employed to determine the closed-form solution of a special hypersingular integral equation of the second kind, and its known solution is recovered.

Damage quantification under sliding and seizure condition using first-of-a-kind fretting machine

In a practical situation, it is difficult to model exact contact conditions clue to challenges involved in the estimation of contact forces, and relative displacements between the contacting bodies. Sliding and seizure conditions were simulated on first-of-a-kind displacement controlled system. Self-mated stainless steels have been investigated in detail. Categorization of contact conditions prevailing at the contact interface has been carried out based on the variation of coefficient of friction with number of cycles, and three-dimensional fretting loops. Surface and subsurface micro-cracks have been observed, and their characteristic shows strong dependence on loading conditions. Existence of shear bands in the subsurface region has been observed for high strain and low strain rate loading conditions. Studies also include the influence of initial surface roughness on the damage under two extreme contact conditions. (C) 2013 Elsevier B.V. All rights reserved.

A modified approach to numerical solution of Fredholm integral equations of the second kind

A modified approach to obtain approximate numerical solutions of Fredholin integral equations of the second kind is presented. The error bound is explained by the aid of several illustrative examples. In each example, the approximate solution is compared with the exact solution, wherever possible, and an excellent agreement is observed. In addition, the error bound in each example is compared with the one obtained by the Nystrom method. It is found that the error bound of the present method is smaller than the ones obtained by the Nystrom method. Further, the present method is successfully applied to derive the solution of an integral equation arising in a special Dirichlet problem. (C) 2015 Elsevier Inc. All rights reserved.

A C-0 Interior Penalty Method for a Fourth-order Variational Inequality of the Second Kind

In this article, we propose a C-0 interior penalty ((CIP)-I-0) method for the frictional plate contact problem and derive both a priori and a posteriori error estimates. We derive an abstract error estimate in the energy norm without additional regularity assumption on the exact solution. The a priori error estimate is of optimal order whenever the solution is regular. Further, we derive a reliable and efficient a posteriori error estimator. Numerical experiments are presented to illustrate the theoretical results. (c) 2015Wiley Periodicals, Inc.

Molecular Origin of the Intrinsic Bending Force for Helical Morphology Observed in Chiral Amphiphilic Assemblies: Concentration and Size Dependence

The formation of the helical morphology in monolayers and bilayers of chiral amphiphilic assemblies is believed to be driven at least partly by the interactions at the chiral centers of the amphiphiles. However, a detailed microscopic understanding of these interactions and their relation with the helix formation is still not clear. In this article a study of the molecular origin of the chirality-driven helix formation is presented by calculating, for the first time, the effective pair potential between a pair of chiral molecules. This effective potential depends on the relative sizes of the groups attached to the two chiral centers, on the orientation of the amphiphile molecules, and also on the distance between them. We find that for the mirror-image isomers (in the racemic modification) the minimum energy conformation is a nearly parallel alignment of the molecules. On the other hand, the same for a pair of molecules of one kind of enantiomer favors a tilt angle between them, thus leading to the formation of a helical morphology of the aggregate. The tilt angle is determined by the size of the groups attached to the chiral centers of the pair of molecules considered and in many cases predicted it to be close to 45 degrees. The present study, therefore, provides a molecular origin of the intrinsic bending force, suggested by Helfrich (J. Chem. Phys. 1986, 85, 1085-1087), to be responsible for the formation of helical structure. This effective potential may explain many of the existing experimental results, such as the size and the concentration dependence of the formation of helical morphology. It is further found that the elastic forces can significantly modify the pitch predicted by the chiral interactions alone and that the modified real pitch is close to the experimentally observed value. The present study is expected to provide a starting point for future microscopic studies.

Solution of singular integral equations with logarithmic and Cauchy kernels

A direct method of solution is presented for singular integral equations of the first kind, involving the combination of a logarithmic and a Cauchy type singularity. Two typical cages are considered, in one of which the range of integration is a Single finite interval and, in the other, the range of integration is a union of disjoint finite intervals. More such general equations associated with a finite number (greater than two) of finite, disjoint, intervals can also be handled by the technique employed here.

Multiple-quantum artifacts in single-quantum two-dimensional correlated NMR spectra of strongly coupled spins

Artifacts in the form of cross peaks have been observed along two- and three-quantum diagonals in single-quantum two-dimensional correlated (COSY) spectra of several peptides and oligonucleotides. These have been identified as due to the presence of a non-equilibrium state of kind I (a state describable by populations which differ from equilibrium) of strongly coupled spins carried over from one experiment to the next in the COSY algorithm.

On the axiomatic approach to the maximum entropy principle of inference

Recent axiomatic derivations of the maximum entropy principle from consistency conditions are critically examined. We show that proper application of consistency conditions alone allows a wider class of functionals, essentially of the form ∝ dx p(x)[p(x)/g(x)] s , for some real numbers, to be used for inductive inference and the commonly used form − ∝ dx p(x)ln[p(x)/g(x)] is only a particular case. The role of the prior densityg(x) is clarified. It is possible to regard it as a geometric factor, describing the coordinate system used and it does not represent information of the same kind as obtained by measurements on the system in the form of expectation values.

Brownian particles in stationary and moving traps: The mean and variance of the heat distribution function

A recent theoretical model developed by Imparato et al. Phys of the experimentally measured heat and work effects produced by the thermal fluctuations of single micron-sized polystyrene beads in stationary and moving optical traps has proved to be quite successful in rationalizing the observed experimental data. The model, based on the overdamped Brownian dynamics of a particle in a harmonic potential that moves at a constant speed under a time-dependent force, is used to obtain an approximate expression for the distribution of the heat dissipated by the particle at long times. In this paper, we generalize the above model to consider particle dynamics in the presence of colored noise, without passing to the overdamped limit, as a way of modeling experimental situations in which the fluctuations of the medium exhibit long-lived temporal correlations, of the kind characteristic of polymeric solutions, for instance, or of similar viscoelastic fluids. Although we have not been able to find an expression for the heat distribution itself, we do obtain exact expressions for its mean and variance, both for the static and for the moving trap cases. These moments are valid for arbitrary times and they also hold in the inertial regime, but they reduce exactly to the results of Imparato et al. in appropriate limits. DOI: 10.1103/PhysRevE.80.011118 PACS.

The harmonic torsional oscillations of a thin disk submerged in a fluid with a surfactant surface layer

A formulation in terms of a Fredholm integral equation of the first kind is given for the axisymmetric problem of a disk oscillating harmonically in a viscous fluid whose surface is contaminated with a surfactant film. The equation of the first kind is converted to a pair of coupled integral equations of the second kind, which are solved numerically. The resistive torque on the disk is evaluated and surface velocity profiles are computed for varying values of the ratio of the coefficient of surface shear viscosity to the coefficient of viscosity of the substrate fluid, and the depth of the disk below the surface.