Neumann Boundary Value Problem Research Articles

Hilbert problem with measurable data for semilinear equations of the Vekua type

We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type ∂z f (z ) = h (z )q(f (z )). The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form ΔU (z )=H (z )Q(U (z )) with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media.

Existence of solutions for a class of nonlinear Neumann boundary value problems in the presence of upper and lower solutions

In this article, we explore the monotone iterative technique (MI‐technique) to study the existence of solutions for a class of nonlinear Neumann four‐point, boundary value problems (BVPs) defined as, where and . The nonlinear term , where , is Lipschitz in and one‐sided Lipschitz in . Using lower solution and upper solutions , we develop MI‐technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti‐maximum principle. Then under certain assumptions, we prove that these sequences converge uniformly to the solution in the specific region, where or . To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi‐point BVPs. We don't require sign restriction on which is very common and strict condition.

Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller—Segel Model with Signal-Dependent Motilities for Local Sensing Chemotaxis

In this paper we consider the initial Neumann boundary value problem for a degenerate Keller—Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in the boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos—Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in [1] and the exponential stabilization is obtained for the first time.

Four Boundary Value Problems for a Nonlocal Biharmonic Equation in the Unit Ball

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green’s functions.

Neumann and Mix Boundary Value Problems on the Upper Half Plane

We give explicit representation of Neumann boundary value problem for Bitsadze equation on the upper half plane. We will also give solution of the inhomogeneous polyanalytic equation arising from Neumann and (n-1) Dirichlet boundary conditions on the upper half plane H.

On the Schwarz method for the eddy currents model

We investigate an iterative solver for the reduced eddy currents model set in open domains. Its formulation is the result of the coupling of a variational problem restricted to a truncated domain with a transparent condition at the fictitious boundary. The non-local boundary condition is derived using simple/double layer potentials. Applying a splitting technique yields an iterative algorithm that requires to solve, at each iteration, a local Neumann boundary value problem on the current potential. A re-interpretation of the algorithm as a Schwarz method enables a tractable convergence analysis which is the core subject of the paper. Using suitable variational tools, we establish an exponential convergence of the iterative solver. Some analytical and numerical examples are presented to check the theoretical findings and the relevance of our procedure.

Positive solutions of Neumann boundary value problems and applications to logistic type population models

Positive solutions of Neumann boundary value problems and applications to logistic type population models

Initial-boundary value problem for the time-fractional degenerate diffusion equation

In this paper the initial-boundary value problems for the one-dimensional linear time-fractional diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t and time-degenerate diffusive coefficients t β with β ≥ 1 − α are studied. The solutions of initialboundary value problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t, are shown. The second section present Dirichlet and Neumann boundary value problems, and in the third section has shown the solutions of the Dirichlet and Neumann boundary value problem for the one-dimensional linear time-fractional diffusion equation. The solutions of these fractional diffusive equations are presented using the Kilbas-Saigo function Eα,m,l(z). The solution of the problems is discovered by the method of separation of variables, through finding two problems with one variable. The existence and uniqueness to the solution of the problem are confirmed. In addition, the convergence of the solution has been proven using the estimate for the Kilbas-Saigo function Eα,m,l(z) from [13] and Parseval’s identity

Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem where k 1 and k 2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × ℝ+, ℝ), ℝ+ := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

On Solving Semilinear Singularly Perturbed Neumann Problems for Multiple Solutions

Based on the analysis of bifurcation points and Morse indices of trivial solutions at any perturbation value for a general semilinear singularly perturbed Neumann boundary value problem, in this paper, the exact critical perturbation value $\varepsilon_c$ which determines the existence or nonexistence of nontrivial positive solutions is obtained. As a result, the generating process of nontrivial positive solutions is studied and further used to guide algorithm design and numerical computation. An improved local minimax method is then proposed accordingly to compute both M-type and W-type saddle points by using an adaptive local refinement strategy and a Newton method to overcome singularity difficulty and accelerate local convergence. Extensive numerical results are reported to justify the critical perturbation value $\varepsilon_c$ and investigate some interesting solution properties of different types of problems.

Spectral characterization of the constant sign Green's functions for periodic and Neumann boundary value problems of even order

Spectral characterization of the constant sign Green's functions for periodic and Neumann boundary value problems of even order

Weak Formulation of the Laplacian on the Full Shift Space

We consider a Laplacian on the one-sided full shift space over a finite symbol set, which is constructed as a renormalized limit of finite difference operators. We propose a weak definition of this Laplacian, analogous to the one in calculus, by choosing test functions as those which have finite energy and vanish on various boundary sets. In the abstract setting of the shift space, the boundary sets are chosen to be the sets on which the finite difference operators are defined. We then define the Neumann derivative of functions on these boundary sets and establish a relation between three important concepts in analysis so far, namely, the Laplacian, the bilinear energy form and the Neumann derivative of a function. As a result, we obtain the Gauss-Green's formula analogous to the one in classical case. We conclude this paper by providing a sufficient condition for the Neumann boundary value problem on the shift space.

Complex boundary value problems in a lens

We apply the parqueting-reflection principle to solve the Dirichlet and Neumann boundary value problems for both the Cauchy–Riemann and Poisson equations on a lens. The Green and Neumann functions for the lens are constructed, and then the solvability of

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov—Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.

Existence results for damped regular equations under periodic or Neumann boundary conditions

We present a version of the averaging and a method to construct lower and upper solutions in the reversed order that are suitable for both the periodic and the Neumann boundary conditions. As an application, we obtain new results for a periodic problem related to the Liebau phenomenon and for a Neumann boundary value problem arising in fluid dynamics.

Nonlocal Neumann Boundary Value Problem for Fractional Symmetric Hahn Integrodifference Equations

In this article, we present a nonlocal Neumann boundary value problems for separate sequential fractional symmetric Hahn integrodifference equation. The problem contains five fractional symmetric Hahn difference operators and one fractional symmetric Hahn integral of different orders. We employ Banach fixed point theorem and Schauder’s fixed point theorem to study the existence results of the problem.

On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to H−1(Ω) or H˜−1(Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials, and hence the unique solvability of BDIEs. The equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Two‐operator boundary‐domain integral equation systems for variable‐coefficient mixed Boundary Value Problem in 2D

In this paper, the mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second‐order scalar elliptic differential equation with variable coefficients in a bounded two‐dimensional domain is considered. The two‐operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of two‐operator boundary‐domain integral equations (BDIEs). Although the theory of two‐operator BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix‐based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two‐operator BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the appropriate Sobolev spaces.

Diffusion processes associated with elliptic operators with measure-valued coefficients

In this paper, we give a survey on diffusion processes associated with elliptic operators with measure-valued coefficients. We first introduce the existence and uniqueness of Brownian motion and reflected Brownian with measure-valued drifts as well as the Gaussian type estimates of the transition functions. We then study the Dirichlet and Neumann boundary value problems for the associated generators of the diffusion processes and obtain the probabilistic representations for the solutions. we also present the small time asymptotics and large deviations for the Brownian motions with drifts.