An Integral Equation Method for a Problem with Mixed Boundary Conditions

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

When quantum field theory is formulated in terms of transition amplitudes between initial and final configurations, it is important to impose mathematically consistent boundary conditions on the fields. The choice of boundary conditions is also important when dealing with elliptic complexes and index theorems on bounded manifolds. Although Dirichlet and Neumann boundary conditions are adequate for many purposes, they are not always appropriate. However, a number of the remaining cases can be handled by imposing Dirichlet conditions on certain field components and Neumann conditions on the others. Quantum gravity and supergravity, spinor field theory, and various elliptic complexes all lend themselves to mixed boundary conditions of this sort. The mathematical significance of mixed boundary conditions is investigated, and a simple rule is obtained for determining which field components should obey which boundary conditions. It will be shown that gauge transformations on a bounded manifold give rise to intrinsic gauge transformations within the boundary. This result is used to define gauge-invariant and BRS-invariant quantum amplitudes for boundary configurations. The most general form for mixed boundary conditions on the Dirac operator and the de Rham complex will also be derived.

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

The processes of reflection of three-dimensional electromagnetic waves by locally irregular media interfaces are investigated. The boundary value problem for the system of Maxwell equations in an infinite half-space is reduced to the solution of two systems of singular integral equations. To construct a numerical algorithm for solving these systems of singular integral equations, the approximation and collocation method is used. Special attention is focused on calculation of the kernels of these equations. Solutions of systems of singular integral equations are used to calculated the far-field pattern for the scattered field in the far zone.

Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions

According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-${T}_{c}$)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-${T}_{c}$\ensuremath{\Vert}/${T}_{c}$\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

Based on consolidation equations proposed for unsaturated soil, an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed. The mixed boundary condition can be used for special applications, such as tests occur in laboratory. The analytical solution is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition. Then, the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition. Finally, using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution for the new boundary condition is obtained. The present method is applicable to various types of boundary conditions. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.

Let D be a non-empty open subset of ℝm,m≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb {R}^{m}, m\\ge 2$\\end{document}, with boundary ∂D, with finite Lebesgue measure |D|, and which satisfies a parabolic Harnack principle. Let K be a compact, non-polar subset of D. We obtain the leading asymptotic behaviour as ε↓ 0 of the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{\\infty }$\\end{document} norm of the torsion function with a Neumann boundary condition on ∂D, and a Dirichlet boundary condition on ∂(εK), in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that D ∖ K is a non-trap domain.

Boundary value problems (BVP) arise in many mathematical model of frontier science. In short, a BVP is a second order ordinary differential equation with boundary conditions prescribed at two distinct points. Usually, we prescribe three types of boundary conditions: Dirichlet, Neumann and Mixed. Among them mixed boundary conditions is little difficult to approach. One of the method to solve the BVP with mixed boundary conditions is the shooting method. In this method, we first convert the BVP into an initial value problems (IVP) and then solve it. We then check whether the solution of IVP is a solution of the BVP. There are standard ways of solving IVPs. Recently, some alternate methods are also used to solve them. For example, several authors use Haar wavelet. Frames are considered as an extension to wavelet. In this paper, we construct operational matrices from a frame and then apply them to solve the IVP which is obtained from the given BVP by shooting method. This method is simple and easy to use because we construct the operational matrix once, store it and then apply it any times we want. Moreover, we also show that the given operational matrix method converges quadratically.

A second-order sparse factorization method for Poisson's equation with mixed boundary conditions

Nonexistence of Positive Solutions for Singular Hyperbolic Differential Inequalities

Paper is devoted to determine the solution of a non-stationary heat equation in an axial symmetry cylindrical coordinates subject to a nonhomogeneous mixed discontinuous boundary conditions of the first and of the second kind inside the disk of an finite surface cylinder. The solution of the given mixed boundary value problem is obtained with the aid of a classical methods and based on the application of a new type of a dual series equations (DSE) with a Bessel function of the first kind of order zero as a kernel. The solution of obtained DSE which is discussed in this paper is introduced to a Fredholm integral equation of the first kind.

We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet-to-Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non-smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.