Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems.

Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems

We investigate multiple solutions for the perturbation of a singular potential biharmonic problem with fixed energy. We get a theorem that shows the existence of at least one nontrivial weak solution under some conditions and fixed energy on which the corresponding functional of the equation satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.

In this paper, in order to develop a more efficient and more convenient Procedure for the strength design of a spur gear, substantial effort is devoted to improving the accuracy in the boundary integral equation analyses. First, corner problems associated with a gear tip are mathematically avoided by adopting Priwalow's formulae. Next, a systematic technique using higher-order shape functions is proposed for the numerical evaluation of singular integrals. Then, a Stieltjes integral is used to deal with a discontinuous change of the boundary direction a corners. The accuracy is investigated using some numerical examples and comparison is made with the results in previous reports.

In the conformal near-field acoustic holography (NAH) using the boundary element method (BEM), the transfer matrix relating the vibro-acoustic properties of source and field depends solely on the geometrical condition of the problem. This kind of NAH is known to be very powerful in dealing with the sources having irregular shaped boundaries. When the vibro-acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components. The conventional acoustic BEM based on the Kirchhoff-Helmholtz integral equation has the singularity problem in the close near field of the source surface. This problem stems from the singular kernel of the Green function of the boundary integral equation (BIE) and the singularity can influence the reconstruction accuracy greatly. In this paper, the nonsingular BIE is introduced to the NAH calculation and the holographic BIE is reformulated. The effectiveness of nonsingular BEM has been investigated for the reduction of reconstruction error. Through interior and exterior examples, it is shown that the resolution of predicted field pressure could be improved in the close near field by employing the nonsingular BIE. Because the BEM-based NAH inevitably requires the field pressure measured in the close proximity to the source surface, the present approach is recommended for improving the resolution of the reconstructed source field.

Novel regularized boundary integral equations for potential plane problems

Analysis and optimization of diffraction effects in nanolithography through multilayered media with a fast and accurate field-theoretical approach is presented. The scattered field through an arbitrary two-dimensional (2D) mask pattern in multilayered media illuminated by a TM-polarized incident wave is determined by using an electric field integral equation formulation. In this formulation the electric field is represented in terms of complex images Green's functions. The method of moments is then employed to solve the resulting integral equation. In this way an accurate and computationally efficient approximate method is achieved. The accuracy of the proposed method is vindicated through comparison with direct numerical integration results. Moreover, the comparison is made between the results obtained by the proposed method and those obtained by the full-wave finite-element method. The ray tracing method is combined with the proposed method to describe the imaging process in the lithography. The simulated annealing algorithm is then employed to solve the inverse problem, i.e., to design an optimized mask pattern to improve the resolution. Two binary mask patterns under normal incident coherent illumination are designed by this method, where it is shown that the subresolution features improve the critical dimension significantly.

: This document addresses 5 substantial problems in the theory and numerical analysis of ill-posed problems and integral equations: (1) Collocation, and Galerkin methods for Volterra and Abel equations of the first kind; (2) Galerkin and collocation methods for nonlinear Abel-Volterra integral equations on the half-line and on a finite interval; (3) a new approach to classification and regularization of ill-posed operator equations, and quantification of ill-posedness; (4) operator external problems in the theory of compensation and representation of control systems; (5) constrained least-squares solutions of linear inclusions and singular control problems in Hilbert space. New notions of bivariational and singular variational derivatives for functionals are also studies. They will be applied to extend the von Mises calculus for statistical functionals and its applications to robustness and approximation theorems. Keyboards: Multivalued linear mappings; Kernel functions.

An accurate method for the evaluation of the Cauchy principal value integrals arising in time-domain electromagnetic wave scattering is presented. This is applied to a boundary integral equation (BIE) method employing quadratic curvilinear surface elements where such singularities do not vanish (as they generally do when simpler but less accurate discretizations are employed). The technique involves weakening the singularity in the original kernel to a degree where conventional integration methods may be employed and transforming the strong singularity to a line integral in a form which allows cancellation of its singular components. The effectiveness of the method is demonstrated by scattering calculations on a variety of targets and the computational cost of the approach is trivial.

We study the existence of nontrivial weak solutions for a class of generalized $ p(x) $-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.

Solving of multi-connected curvilinear boundary value problems by the fast PIES

We consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudo-Newton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.

BackgroundBiomedical applications of high-throughput sequencing methods generate a vast amount of data in which numerous chromatin features are mapped along the genome. The results are frequently analysed by creating binary data sets that link the presence/absence of a given feature to specific genomic loci. However, the nucleosome occupancy or chromatin accessibility landscape is essentially continuous. It is currently a challenge in the field to cope with continuous distributions of deep sequencing chromatin readouts and to integrate the different types of discrete chromatin features to reveal linkages between them.ResultsHere we introduce the NucTools suite of Perl scripts as well as MATLAB- and R-based visualization programs for a nucleosome-centred downstream analysis of deep sequencing data. NucTools accounts for the continuous distribution of nucleosome occupancy. It allows calculations of nucleosome occupancy profiles averaged over several replicates, comparisons of nucleosome occupancy landscapes between different experimental conditions, and the estimation of the changes of integral chromatin properties such as the nucleosome repeat length. Furthermore, NucTools facilitates the annotation of nucleosome occupancy with other chromatin features like binding of transcription factors or architectural proteins, and epigenetic marks like histone modifications or DNA methylation. The applications of NucTools are demonstrated for the comparison of several datasets for nucleosome occupancy in mouse embryonic stem cells (ESCs) and mouse embryonic fibroblasts (MEFs).ConclusionsThe typical workflows of data processing and integrative analysis with NucTools reveal information on the interplay of nucleosome positioning with other features such as for example binding of a transcription factor CTCF, regions with stable and unstable nucleosomes, and domains of large organized chromatin K9me2 modifications (LOCKs). As potential limitations and problems we discuss how inter-replicate variability of MNase-seq experiments can be addressed.

For the approximate solution of the biharmonic Dirichlet problem we propose and study a boundary element method based on the integral equation of Muskhelishvili. Such an approach has a number of advantages, for instance, this equation does not have any critical geometry and in the case of smooth boundaries the method always converges. If the boundary has corner points, then the convergence of the method depends on the invertibility of some operators from a Toeplitz algebra.KeywordsGalerkin MethodBoundary Element MethodCorner PointLebesgue Measurable FunctionToeplitz AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.

Improved complex variable moving least squares approximation for three-dimensional problems using boundary integral equations