Neumann Boundary Value Problem Research Articles

On the homogeneous torsion problem for heterogeneous and orthotropic cross-sections: theoretical and numerical aspects

For many years, torsion of arbitrary cross-sections has been a subject of numerous investigations from theoretical and numerical points of view. As it is well known, the resulting boundary value problem (BVP) governing such phenomenon happens to be a pure Neumann BVP and, therefore, its solutions are determined up to a constant. Among a large plethora of finite element method (FEM) techniques that can be used in this context, most of FEM practitioners resolve this uniqueness issue by fixing the candidate solution to a node of the domain. Although such popular and pinpointing technique is widely spread and works well for practical purposes, it does not have a continuous counterpart and therefore its justification remains a matter of debate. Hence, this self-contained work aims to address the modeling of arbitrary heterogeneous and orthotropic cross-sections as well as the theoretical and numerical aspects of their solutions. In particular, we discuss the existence of weak solutions, well-posedness, regularity of solutions, and convergence of Galerkin’s method for different variational settings (with special focus on a regularized variational approach). Moreover, we establish a connection, at a discrete level, between the convergence of solutions of well-posed variational settings and those solutions coming from the usual practice of fixing a datum at a node. Finally, we discuss some numerical aspects of all the FEM discrete formulations proposed here by performing convergence analysis in L2 and H1 norms. The section of numerical results is closed by presenting a series of study cases ranging from a square cross-section composed of two different materials to an isotropic bridge cross-section for which no analytical solution exists.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Polyanalytic Neumann-n problem in the unit disk

An iterated Neumann boundary value problem for the polyanalytic operator is investigated in the unit disc of the complex plane. Although this problem can be treated for domains in the complex plane having a harmonic Green function, the unit disk is particularly interesting as it is the origin of complex boundary value problems and the prototype of simply connected domains. Moreover, the resulting formulas for solutions and for solvability conditions are expressed by integral formulas with explicit kernels. By the way, the result verifies the findings for domains with harmonic Green function.

Asymptotic analysis of the narrow escape problem in general shaped domain with several absorbing necks

This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several absorbing narrow necks. The narrow escape problem is to compute the mean first passage time (MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet–Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann–Robin type boundary value problem. By solving the new model, we obtain the three-term asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for narrow escape problem (NEP) in a general shaped domain with several absorbing neck windows. This work is motivated by Li (2014 J. Phys. A: Math. Theor. 47 505202), where the Neumann–Robin model was proposed to solve the NEP in a domain with a single absorbing neck.

On a new p(x)-Kirchhoff type problems with p(x)-Laplacian-like operators and Neumann boundary conditions

Abstract In this paper we study a Neumann boundary value problem of a new p(x)-Kirchhoff type problems driven by p(x)-Laplacian-like operators. Using the theory of variable exponent Sobolev spaces and the method of the topological degree for a class of demicontinuous operators of generalized (S+) type,weprove theexistenceofaweak solutionsof this problem. Our results are a natural generalisation of some existing ones in the context of p(x)-Kirchhoff type problems.

Interaction of Boundary Singular Points in an Elliptic Boundary Value Problem

The paper continues the construction of the Lp-theory of elliptic Dirichlet and Neumann boundary value problems with discontinuous piecewise constant coefficients in divergent form for an unbounded domain R2 with a piecewisesmooth noncompact Lipschitz boundary and C1 smooth discontinuity lines of the coefficients. An earlier constructed Lp-theory is generalized to the case of different smallest eigenvalues corresponding to a finite and an infinite singular point, and the effect of their interaction is further studied in the class of functions with first derivatives from Lp( ) in the entire range of the exponent p (1, )..

Hilbert and Poincaré problems for semi-linear equations in rectifiable domains

The study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in \cite{R7}, we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.

Existence and approximate solutions of a nonlinear model for the Antarctic circumpolar current

We consider a nonlinear Neumann boundary value problem which is derived for the Antarctic Circumpolar Current. By the theory of topological degree, we prove the existence results for the problem with semilinear oceanic vorticity term. We also construct the approximate solutions for such a nonlinear model.

SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER

The physical model of many electrical objects is a hollow cylinder of finite length. As a basis for constructing analytical models of electrical machines, linear equations of mathematical physics are used. They are the solution of the Laplace three-dimensional partial differential equation, which is widely used in analytical calculations.
 
 The purpose of the study is to solve the Laplace three-dimensional differential equation for a hollow cylinder of finite length, which can be adapted for the electromagnetic calculation of electromechanical devices with cylindrical active parts.
 
 Materials and methods. To solve the Laplace equation in a cylindrical coordinate system, the Fourier variable separation method was used. To obtain a non-trivial solution of the equation, eigenfunctions of the Sturm–Liouville problem were used. Dirichlet, Neumann boundary value problems of the mixed type for hollow cylinders are adapted to the electromagnetic calculation of electromechanical devices having cylindrical active parts.
 
 The results of the study. The Laplace equation given in a cylindrical coordinate system is considered, on the basis of which the Sturm–Liouville equation with zero initial values is compiled to find eigenfunctions. The complete solution of the Lapalace equation with given boundary conditions is obtained as the sum of the solutions of two separate Dirichlet problems with different boundary conditions.
 
 Findings. The obtained analytical expression can be used as a mathematical basis for constructing three-dimensional analytical models of electrical machines with cylindrical active parts and carrying out electromagnetic calculations of the corresponding electromechanical devices.

Dirichlet and Neumann Boundary Value Problems for Dunkl Polyharmonic Equations

Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl–Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green’s function.

Neumann boundary value problems for elliptic operators with measure-valued coefficients

In this paper, we prove that there exists a unique weak solution to the Neumann boundary value problem for second order elliptic operators whose coefficients are signed measures. We will give a probabilistic representation of the solution. The heat kernel estimates and reflected diffusion processes play important roles.

Guaranteed Lower Eigenvalue Bounds for Steklov Operators Using Conforming Finite Element Methods

Abstract For the eigenvalue problem of the Steklov differential operator, an algorithm based on the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed lower eigenvalue bounds utilize the a priori error estimation for FEM solutions to non-homogeneous Neumann boundary value problems, which is obtained by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples demonstrate the efficiency of our proposed method.

Existence result for a Neumann boundary value problem governed by a class of p ( x )-Laplacian-like equation

In this article, we consider a Neumann boundary value problem driven by p ( x )-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: − Δ p ( x ) l u + δ | u | ζ ( x ) − 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) in  Ω , ∂ u ∂ η = 0 on  ∂ Ω , where Δ p ( x ) l u is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , δ, μ and λ are three real parameters, p ( x ) , ζ ( x ) ∈ C + ( Ω ‾ ), η is the outer unit normal to ∂ Ω and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.

The study of approximate solution of one class of hypersingular integral equations of first kind

This work presents the method for calculating the approximate solution to the integral equation of the first kind of the Neumann boundary value problems for the Helmholtz equation in two-dimensional space

Global multiplicity for parametric anisotropic Neumann (p,q)-equations

We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.

Lower and Upper Solution Method for Semilinear, Quasi-Linear and Quadratic Singularly Perturbed Neumann Boundary Value Problems

In this paper, by using the method of lower and upper solutions and notion of (Iq)-stability, we established sufficient conditions for the uniform convergence of the solutions of singularly perturbed Neumann boundary value problems for second-order differential equations to the solution of their reduced problems.

"Far-field interaction" of concentrated masses in two-dimensional Neumann and Dirichlet problems

We study the eigenvalues of the Neumann and Dirichlet boundary-value problems in a two-dimensional domain containing several small, of diameter $O(\varepsilon)$, inclusions of large "density" $O(\varepsilon^{-\gamma})$, $\gamma\geqslant2$, that is, the "mass" $O(\varepsilon^{2-\gamma})$ of each of them is comparable ($\gamma=2$) or much bigger ($\gamma>2$) than that of the embracing material. We construct a model of such spectral problems on concentrated masses which (the model) provides an asymptotic expansions of the eigenvalues with remainders of power-law smallness order $O(\varepsilon^{\vartheta})$ as $\varepsilon\to+0$ and $\vartheta\in(0,1)$. Besides, the correction terms are real analytic functions of the parameter $|{\ln \varepsilon}|^{-1}$. A "far-field interaction" of the inclusions is observed at the levels $|{\ln \varepsilon}|^{-1}$ or $|{\ln \varepsilon}|^{-2}$. The results are obtained with the help of the machinery of weighted spaces with detached asymptotics and also by using weighted estimates of solutions to limit problems in a bounded punctured domain and in the intact plane.

РЕШЕНИЕ ЗАДАЧИ РИКЬЕ–НЕЙМАНА ДЛЯ ПОЛИГАРМОНИЧЕСКОГО УРАВНЕНИЯ В ШАРЕ

In this paper, an elementary solution for polyharmonic equations is determined and its properties are given. This elementary solution coincides with previously known elementary solutions of biharmonic and triharmonic equations. Using the elementary solution, an integral representation of the solutions of a non-homogeneous polyharmonic equation in a bounded domain with a smooth boundary is found. Based on the integral representation, the solvability of the Riquier–Neumann problem is investigated. First, the concept of the Green's function of the Riquier–Neumann problem is defined, and then the Green's function is proved. Using the integral representation of the solutions of the polyharmonic equation and the Green's function of the Riquier–Neumann problem, the integral representation of the solution of the Riquier–Neumann problem in a unit ball is found. An example of the solution of the Neumann problem for the Poisson equation with the simplest right-hand side is given, which is necessary in what follows. On the basis of the Green's function of the Riquier–Neumann problem, a theorem on the integral representation of the solution of the Riquier–Neumann boundary value problem with boundary data, the integral of which over the unit sphere vanishes, is proved. In conclusion, on the basis of the theorem, an example of calculating the solution of the Riquier–Neumann problem with boundary functions coinciding with the traces of homogeneous harmonic polynomials on a unit sphere is given.

Neumann boundary value problem for the Beltrami equation in a ring domain

Neumann boundary value problem for the Beltrami equation in a ring domain

Sound radiation by a vibrating circular plate located at the bottom of a non-rigid flanged circular cylindrical tube

The sound transmission through a non-rigid circular tube has been investigated in this study. For the same, the Neumann–Robin boundary value problem has been solved. The problem deals with sound radiation by considering a complex system consisting of a fluid filled in the circular cylindrical tube having a finite impedance and the half-space. The outlet of the tube is embedded into a rigid infinite flat screen. The fluid inside the tube is disturbed by a vibrating circular plate embedded at the bottom of the tube. A coupled system of partial differential equations governing the vibration of the plate and the fluid has been solved. For this purpose, the superposition of the two solutions has been applied: one is the Neumann boundary value problem in the form of the Dini series and the other is the correction for the impedance boundary in the form of a Fourier series. The continuity conditions at the tube outlet have been expressed in terms of the modal coupling acoustic impedance coefficients. The use of the radial polynomials together with Cerjan’s expansion greatly improve the numerical analysis. Application of the additional Fourier series helps to avoid finding the complex eigenvalues. Truncation conditions in the Dini series and in the Fourier series have been applied to assure valid and accurate numerical results for frequencies up to 8kHz, a tube radius from 30 to 70 mm, and a tube length from 3 to 7 times the tube radius with an arbitrary excitation of the plate. The selected tube materials range from being acoustically soft, such as mineral wool, to acoustically hard, such as a porous material. The other advantage of using the Dini and the Fourier series is that they require only real eigenvalues, and thus it is not necessary to find the cumbersome complex eigenvalues of the tube. One more advantage of the rigorous solution is that it automatically satisfies the radiation condition in the half-space, whereas the finite elements method always satisfies it approximately. The findings of this study can be summarized as follows: The main computational cost is associated with a relatively significant number of the Fourier expansion terms. As a result, for frequencies smaller than 1.2kHz, the first result can be obtained rapidly using the finite elements method. In addition, it has been shown that using the radial polynomials to calculate the coupling integrals enables faster calculations by at least 120 times.