Neumann Boundary Value Problem Research Articles

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

On optimal control problem for the Perona-Malik equation and its approximation

<p style='text-indent:20px;'>We discuss the existence of solutions to an optimal control problem for the Neumann boundary value problem for the Perona-Malik equations. The control variable <inline-formula><tex-math id="M5">\begin{document}$ v $\end{document}</tex-math></inline-formula> is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution <inline-formula><tex-math id="M6">\begin{document}$ u_d\in L^2(\Omega) $\end{document}</tex-math></inline-formula> and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and derive some optimality conditions for the approximating problems.</p>

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.

A new physically realisable internal 1:1 resonance in the coupled pendulum–slosh system

The problem of dynamic coupling between a rectangular container undergoing planar pendular oscillations and its interior potential fluid sloshing is studied. The Neumann boundary-value problem for the fluid motion inside the container is deduced from the Bateman–Luke variational principle. The governing integro-differential equation for the motion of the suspended container, from a single rigid pivoting rod, is the Euler–Lagrange equation for a forced pendulum. The fluid and rigid-body partial differential equations are linearised, and the characteristic equation for the natural and resonant frequencies of the coupled dynamical system are presented. It is found that internal 1:1 resonances exist for an experimentally realistic setup, which has important physical implications. In addition, a new instability is found in the linearised coupled problem whereby instability occurs when the rod length is shorter than a critical length, and an explicit formula is given.

Three boundary value problems of the Cauchy–Riemann equation in the Eclipse domain

In this article, by the parqueting-reflection technique, we obtain a covering of the complex plane from reflecting the Eclipse domain. Then, by the Cauchy–Pompeiu formula, we derive the generalized Cauchy–Pompeiu representation formula in the Eclipse domain. In addition, we consider the Schwarz boundary value problem (BVP) for the nonhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and the condition of solvability. Besides, by using the Schwarz BVP, we investigate the inhomogeneous Dirichlet and homogeneous Neumann BVPs.

Green's functions of the first and second boundary value problems for the Laplace equation in the nonclassical domain

The subject of study is the Green's functions of the first and second boundary value problems for the Laplace equation. The study constructs the Green's functions of the first and second boundary value problems for the Laplace equation in space with a spherical segment in analytical form, as well as numerical analysis of these functions. Research task: to formalize the problem of determining Green's functions for the specified domain; using methods of Fourier, pair summation equations and potential theory to reduce mixed boundary value problems for auxiliary harmonic functions to a system of equations that has an analytical solution; investigate the compatibility of the algebraic system for determining constants of integration; formulate and prove a theorem about the jump of the normal derivative of the potential of a simple layer on the surface of a segment, with the help of which to present the Green's function in the form of the potential of a simple layer; conduct a numerical experiment and identify algorithms and areas of changing the parameters of effective calculations; analyze the behavior of Green's functions. Scientific novelty: for the first time, Green's functions of Dirichlet and Neumann boundary value problems for the Laplace equation in three-dimensional space with a spherical segment were constructed in analytical form, the obtained results were substantiated, and a comprehensive numerical experiment was conducted to analyze the behavior of these functions. The obtained results: mixed boundary value problems in the interior and exterior of the spherical surface to which the segment belongs are set for the auxiliary harmonic functions; using the Fourier method, the problem is reduced to systems of paired equations in series by Legendre functions, the solutions of which are found using discontinuous Mehler-Dirichlet sums. The specified functions are obtained in an explicit view in two forms: series based on the basic harmonic functions in spherical coordinates and the potential of a simple layer on the surface of the segment. To substantiate the results, the lemma on the compatibility of the algebraic system for determining the constants of integration and the theorem on the jump of the normal derivative of the potential of a simple layer on a segment are proved. A numerical experiment was conducted to analyze the behavior of the constructed functions. Conclusions: the analysis of numerical values of Green's functions obtained by different algorithms showed that the highest accuracy of results outside the surface of the segment was obtained when using images of Green's functions in the form of series. On the basis of the calculations, the lines of the level of the Green's functions of two boundary value problems in the plane of the singular point, as well as the graphs of the potential density of the simple layer for the Dirichlet problem and the potential jump for the Neumann problem on the segment at different locations of the singular point were constructed. In the partial case of the location of a singular point at the origin of the coordinates, the potential of the electrostatic field of a point charge near a conductive grounded thin shell in the form of a spherical segment is found. The main characteristics of such a field are found in closed form.

Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂ t u − Δ u = λ | u | p , t > 0 , x ∈ R + n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R + n , − ∂ x u ( t , x ′ , 0 ) = γ | u | q ( t , x ′ , 0 ) , t > 0 , x ′ ∈ R n − 1 where p = 1 + 2 n , q = 1 + 1 n and ε > 0 is small enough. We investigate the life span of solutions for λ , γ > 0. Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ , γ < 0 and ∫ R + n u 0 ( x ) d x > 0.

A representation theorem approach to probabilistic interpretation for viscosity solutions of Isaacs equations

The main objective of this paper is to show a certain representation theorem for generators of generalized backward stochastic differential equations (GBSDEs) and its application to probabilistic interpretation for viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary value problems. In the representation theorem a time change method is adopted to treat the difficulty brought from the random measure contained in GBSDEs. By means of the representation theorem the viscosity solution of Isaacs equations with nonlinear Neumann problems is proved to be the value function of a two-player zero-sum stochastic differential game problem with state constraints and recursive cost functionals.

Elastic Solutions to 2D Plane Strain Problems: Nonlinear Contact and Settlement Analysis for Shallow Foundations

The classical Neumann boundary value problem of an isotropic, homogeneous elastic half-plane under plane strain conditions is readdressed as the limiting case of the fully three-dimensional problem. Analytical solutions of the stress and strain tensors are obtained by taking the limit from known three-dimensional solutions. It is shown that the displacement fields for the plane strain problem are not well defined. A small number of simple expressions are developed, which provide a general solution for linearly-varying traction over arbitrary regions on the boundary. A simple, efficient, and rapidly convergent algorithm is developed which uses these solutions as analytic elements and provides a solution approach to the general boundary value problem. The method is verified against known solutions for Hertzian contact between parallel cylinders. Two numerical examples are presented for the analysis of shallow foundation systems. In the first, the boundary conditions are informed by analytical elastoplastic calculations and a strain influence analysis is performed and compared with the Schmertmann method. Subsequently, empirical laboratory contact traction distributions measured by Bauer et al., in both the normal and tangential directions are employed as boundary conditions for an analysis of the underlying stress field.

Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem

In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems εy″+ky=f(t), y′(a)=0, y′(b)=0, k>0, with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem.

The boundary value problem for the mean field equation on a compact Riemann surface

Let (Σ, g) be a compact Riemann surface with smooth boundary ∂E, ∆g be the Laplace-Beltrami operator, and h be a positive smooth function. Using a min-max scheme introduced by Djadli and Malchiodi (2008) and Djadli (2008), we prove that if Σ is non-contractible, then for any ρ Σ (8kπ, 8(k +1)π) with k Σ ℕ*, the mean field equation $$\left\{ {\matrix{{{\Delta _g}u = \rho {{h{{\rm{e}}^u}} \over {\int_\Sigma {h{{\rm{e}}^u}d{v_g}} }}} \hfill & {{\rm{in}}\,\,\Sigma ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\,\partial \Sigma } \hfill \cr } } \right.$$ has a solution. This generalizes earlier existence results of Ding et al. (Ann Inst H Poincaré Anal Non Linéaire, 1999) and Chen and Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If h is a positive smooth function, then for any ρ ∈ (4kπ, 4(k + 1)π) with k ∈ ℕ*, the mean field equation $$\left\{ {\matrix{{{\Delta _g}u = \rho \left( {{{h{{\rm{e}}^u}} \over {\int_\Sigma {h{{\rm{e}}^u}d{v_g}} }} - {1 \over {\left| \Sigma \right|}}} \right)} \hfill & {{\rm{in}}\,\,\Sigma ,} \hfill \cr {\partial u/\partial v = 0} \hfill & {{\rm{on}}\,\,\partial \Sigma } \hfill \cr } } \right.$$ has a solution, where v denotes the unit normal outward vector on ∂Σ. Note that in this case we do not require the surface to be non-contractible.

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

AbstractIn this paper, the fourth‐order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth‐order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ2 + h2) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.

Weak solution of a Neumann boundary value problem with 𝑝(𝑥)-Laplacian-like operator

Abstract In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the p ⁢ ( x ) p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized ( S + ) (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.

Boundary Value Problems for Liénard-Type Equations with Quadratic Dependence on the “Velocity”

The estimates were obtained for the number of solutions for the Neumann and Dirichlet boundary value problems associated with the Liénard equation with a quadratic dependence on the “velocity.” Sabatini’s transformation is used to reduce this equation to a conservative one, which does not contain the derivative of an unknown function. Despite the one-to-one correspondence between the equilibria, the topological structure of the phase portraits of both equations can differ significantly.

Multiplicity of solutions for a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x )-Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

Rigorous enclosures of solutions of Neumann boundary value problems

This paper is dedicated to the problem of isolating and validating zeros of non-linear two point boundary value problems. We present a method for such purpose based on the Newton-Kantorovich Theorem to rigorously enclose isolated zeros of two point boundary value problem with Neumann boundary conditions.