Neumann Boundary Value Conditions Research Articles

Synchronization of fractional-order delayed coupled networks with reaction–diffusion terms and Neumann boundary value conditions

This paper investigates the adaptive pinning strategies for achieving synchronization of fractional-order delayed coupled networks with reaction–diffusion terms and a digraph topology by incorporating Neumann boundary value conditions. By employing the inf–sup method, a novel fractional-order inequality is proved. The classical Poincaré inequality is also extended by utilizing the Hölder inequality. Two types of control laws are developed to achieve synchronization: one with control gains dependent solely on time, and another with control gains dependent on both space and time. For each case, adaptive control laws and synchronization criteria based on matrix inequalities are proposed. Finally, the effectiveness of the synchronization results is demonstrated through two numerical examples.

The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes.

Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

Limit Behavior of Complex Special Lagrangian Equations With Neumann Boundary-Value Conditions

Abstract We consider the following complex special Lagrangian equations with Neumann boundary conditions on a domain $\Omega \subset \mathbb{C}^{n} $: $$\begin{align*}& \left\{\begin{array}{ll} {\sum\limits_{i=1}^n \arctan k \lambda_{i} (H(u_{k,\varepsilon} ))=\Theta_k(z)} & {\textrm{ in } \Omega} \\{D_{\nu} {u_{k,\varepsilon }}=-\varepsilon{u_{k,\varepsilon }}+\varphi(z)} & {\textrm{ on } \partial \Omega}\end{array}\right.. \end{align*}$$We prove uniform global estimates for these equations. As $k\to +\infty $, we prove that $u_{ k,\varepsilon }$ converges to the solution of $J$-type equation with the same Neumann boundary condition on $\Omega $.

A gradient maximum principle of solutions for a quasilinear parabolic equation

This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017).

On Neumann problem for the degenerate Monge\u2013Amp\xe8re type equations

In this paper, we study the global C^{1, 1} regularity for viscosity solution of the degenerate Monge–Ampère type equation det [D^{2}u-A(x, Du)]=B(x, u, Du) with the Neumann boundary value condition D_{nu }u=varphi (x), where the matrix A is under the regular condition and some structure conditions, and the right-hand term B is nonnegative.

The degenerate Monge-Ampère equations with the Neumann condition

In this paper, we study a priori derivative estimates (up to the second order) of solutions for the Monge-Ampere equation \begin{document}$ \det D^{2}u = f(x) $\end{document} with the Neumann boundary value condition, which are independent of \begin{document}$ \inf f $\end{document} . Based on these uniform estimates, the existence and uniqueness of the global \begin{document}$ C^{1,1} $\end{document} solution to the Neumann problem of the degenerate Monge-Ampere equation are established under the assumption \begin{document}$ f^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}) $\end{document} .

A finite element iterative solver for a PNP ion channel model with Neumann boundary condition and membrane surface charge

In this paper, an effective finite element iterative algorithm is presented for solving a Poisson-Nernst-Planck ion channel (PNPic) model with Neumann boundary value condition and a membrane surface charge density. It is constructed by a solution decomposition scheme to avoid singularity problems caused by atomic charges, an alternating block iterative scheme to sharply reduce computation complexity and computer memory requirement, and a Slotboom variable transformation scheme to significantly enhance numerical stability, as well as a modified Newton iterative scheme to efficiently solve each related nonlinear finite element equation. This PNPic finite element solver is then implemented as a software package that works for an ion channel protein with a crystallographic structure in a mixture solution of multiple ionic species. Furthermore, a finite element scheme is presented to compute a volume integral of a potential/concentration function over a block of a solvent region. This work can greatly improve the accuracy of a visualization tool for depicting the distribution pattern of a three-dimensional potential/concentration function across membrane in a simple two-dimensional curve. Numerical results for a mouse voltage-dependent anion-channel isoform (mVDAC1) in a solution of up to four ionic species are reported. They demonstrate the convergence of the PNPic iterative solver, the performance of the software package, and the valuable usage of the visualization tool in the comparison study of different potential and concentration functions. They also validate that this PNPic model can well retain the anion selectivity property of mVDAC1.

Existence of two solutions for a fourth-order difference problem with p(k) exponent

The existence of nontrivial solutions for a fourth-order discrete anisotropic boundary value problem involving the p(k)-Laplacian operator with the Dirichlet and the Neumann boundary value conditions is investigated. Variational approach based on a new critical point theorem is applied. An example is inserted to illustrate main results.

Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity

This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity ulog |u|-fint_{Omega } ulog |u|,dx in a bounded domain, subject to homogeneous Neumann boundary value condition. By using the logarithmic Sobolev inequality and energy estimate methods, we get the results under appropriate conditions on blow-up and non-extinction of the solutions, which extend some recent results.

Steklov approximations of harmonic boundary value problems on planar regions

Error estimates for approximations of solutions of Laplace’s equation with Dirichlet, Robin or Neumann boundary value conditions are described. The solutions are represented by orthogonal series using the harmonic Steklov eigenfunctions. Error bounds for partial sums involving the lowest eigenfunctions are found. When the region is a rectangle, explicit formulae for the Steklov eigenfunctions and eigenvalues are known. These were used to find approximations for problems with known explicit solutions. Results about the accuracy of these solutions, as a function of the number of eigenfunctions used, are given.

Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip

A time-delayed reaction-diffusion system of mistletoes and birds with nonlocal effect in a two-dimensional strip is considered in this paper. By the background of model deriving, the bird diffuses with a Neumann boundary value condition, and the mistletoes does not diffuse and thus without boundary value condition. Making use of the theory of monotone semiflows and Kuratowski measure of non-compactness, we discuss the existence of spreading speed $c^\ast$. The value of $c^*$ is evaluated by using two auxiliary linear systems accompanied with approximate process.

Analytical Solution for the Time-Fractional Pennes Bioheat Transfer Equation on Skin Tissue

This study focuses on analytical solution of a fractional Pennes bioheat transfer equation on skin tissue. The method of separating variables, finite Fourier sine transformation, Laplace transformation and their corresponding inverse transforms are used to solve this equation with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin boundary value conditions. The exact solutions are discussed and derived in the form of generalized Mittag-Leffler function. In addition, numerical results are presented graphically for various values of order factional derivative.

On the prescribed variable exponent mean curvature impulsive system boundary value problems

Abstract This paper investigates the existence of solutions for prescribed variable exponent mean curvature impulsive system, with periodic-like, Dirichlet, and Neumann boundary value conditions, respectively. The proof of our main result is based upon the Leray-Schauder degree. The sufficient conditions for the existence of solutions are given. MSC: 34B37, 34B15.

Bifurcation and stability of solutions to a logistic equation with harvesting

In this paper, we study the global structure of the positive solutions to a logistic equation with constant yield harvesting under Neumann boundary value conditions. Moreover, we show that the logistic equation with the variable coefficients has exactly either zero, or one, or two solutions depending on the harvesting parameter. Copyright © 2014 John Wiley & Sons, Ltd.

Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ2−α+h4) in L2 norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.

Exact multiplicity and stability of solutions of second-order Neumann boundary value problem

The purpose of this paper is to establish the exact multiplicity and stability of solutions of the equation u″+g(x,u)=f(x) with the Neumann boundary value conditions u′(0)=u′(1)=0. Exactly three ordered solutions are obtained by taking advantage of the anti-maximum principle combined with the methods of upper and lower solutions. Moreover, we obtain that one of three solutions is negative, while the other two are positive, the middle solution is unstable, and the remaining two are stable.

Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions

This paper studies a fast diffusive p-Laplace equation with the nonlocal source |u|q−⨍Ω|u|qdx in a bounded domain, subject to homogeneous Neumann boundary value condition. A critical criterion is determined that the changing sign solutions blow up in finite time with q>1 and non-positive initial energy associated, and must be global for any initial energy if q⩽1. In particular, the conditions are obtained under which the changing sign solutions vanish in finite time.

MONOTONICITY METHOD FOR A KIND OF GENERALIZED CAPILLARITY EQUATIONS

Based on some results on the existence of solutions for variational inequalities,the existence and uniqueness of solution for a class of generalized capillarity equations with Dirichlet boundary value conditions are proved.By using the relationship between the capillarity equations with Neumann boudary value conditions and the capillarity equations with Dirichlet boundary value conditions and by employing a perturbation result on the ranges for maximal monotone operators,one sufficient condition of the existence of solution of the generalized capillarity equations with Neumann boundary value conditions is obtained.Some new methods are proposed in this paper,which extend and complement some existing ones.

Multiple solutions for asymptotically linear Duffing equations with Neumann boundary value conditions (II)

We study multiple solutions of asymptotically linear Duffing equations satisfying Neumann boundary value conditions with resonance by using index theory and Lusternik–Schnirelmann theory, and obtain some new results.