Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

THE BLOW-UP PROPERLIES OF SOLUTIONS TO SEMILINEAR HEAT EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.

An extension result for the Landau-Lifshitz equation with Neumann boundary condition

We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimension space. The partial regularity is proved up to the boundary and this result is an important supplement to those for the Dirichlet problem or the homogeneous Neumann problem.

Boundary Control for Cooperative Elliptic Systems under Conjugation Conditions

The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed. The set of equations and inequalities that characterizes this boundary control is found by theory of Lions, Sergienko and Deineka. The problem for cooperative Neumann elliptic systems under conjugation conditions is also considered. Finally, the problem for n × n cooperative elliptic systems under conjugation conditions is established.

Finding the Time-dependent Term in 2D Heat Equation from Nonlocal Integral Conditions

The aim of this paper is to find the time-dependent term numerically in a two-dimensional heat equation using initial and Neumann boundary conditions and nonlocal integrals as over-determination conditions.This is a very interesting and challenging nonlinear inverse coefficient problem with important applications in various fields ranging from radioactive decay,melting or cooling processes,electronic chips,acoustics and geophysics to medicine.Unique solvability theo-rems of these inverse problems are supplied.However,since the problems are still ill-posed(a small modification in the input data can lead to bigger impact on the ultimate result in the output solution)the solution needs to be regularized.Therefore,in order to obtain a stable solution,a regularized objective function is minimized in order to retrieve the unknown coefficient.The two-dimensional inverse problem is discretized using the forward time central space(FTCS)finite-difference method(FDM),which is conditionally stable and recast as a non-linear least-squares minimization of the Tikhonov regularization function.Numerically,this is effectively solved using the MATLAB subroutine lsqnonlin.Both exact and noisy data are inverted.Numerical results for a few benchmark test examples are presented,discussed and assessed with respect to the FTCS-FDM mesh size discretisation,the level of noise with which the input data is contaminated,and the choice of the regularization parameter is discussed based on the trial and error technique.

Solving Different Types of Differential Equations Using Modified and New Modified Adomian Decomposition Methods

The Modified Adomian Decomposition Method (MADM) is presented. A number of problems are solved to show the efficiency of the method. Further, a new solution scheme for solving boundary value problems with Neumann conditions is proposed. The scheme is based on the modified Adomian decomposition method and the inverse linear operator theorem. Several differential equations with Neumann boundary conditions are solved to demonstrate the high accuracy and efficiency of the proposed scheme.

Lipschitz Continuity and Explicit Form of Solution in a Class of Free Boundary Problem with Neumann Boundary Condition

We consider a class of free boundary problems with Neumann boundary conditions.We would like to give certain results with regularity of solutions(mainly the local interior and boundary Lipschitz continuity).We will also show an explicit form of solution under well-specified conditions.

Boundary Control Problems for 2 ×2 Cooperative Hyperbolic Systems with Infinite Order Operators

In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.

LOWER BOUNDS ESTIMATE FOR THE BLOW-UP TIME OF A NONLINEAR NONLOCAL POROUS MEDIUM EQUATION

The lower bounds for the blow-up time of blow-up solutions to the nonlinear nolocal porous equation ut=△u^m＋u^p∫Ωu^qdxwith either null Dirichlet boundary condition or homogeneous Neumann boundary condi- tion is given in this article by using a differential inequality technique.

The Blow-up Rate for a System of Heat Equations with Neumann Boundary Conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u_t=Δ_u,v_t=Δv in B_R×(0,T) with the Neumann boundary conditions u/η=e^v,v/η=e^u on S_R×[0,T).The exact blow-up rates are established.It is also proved that the blow-up will occur only on the boundary.

NONLINEAR COMPLEX DYNAMIC PHENOMENA OF THE PERTURBED METALLIC BAR CONSIDERING DISSIPATING EFFECT

Considering Peierls-Nabarro effect, one-dimensional finite metallic bar subjected with periodic field was researched under Neumann boundary condition. Dynamics of this system was described with displacement by perturbed sine-Gordon type equation. Finite difference scheme with fourth-order central differences in space and second-order central differences in time was used to simulate dynamic responses of this system. For the metallic bar with specified sizes and physical features, effect of amplitude of external driving on dynamic behavior of the bar was investigated under initial “breather” condition. Four kinds of typical dynamic behaviors are shown: x-independent simple harmonic motion; harmonic motion with single wave; quasi-periodic motion with single wave; temporal chaotic motion with single spatial mode. Poincaré map and power spectrum are used to determine dynamic features.

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

In this paper,we establish the lower bounds estimate of the blow up time for solutions to the nonlocal cross-coupled porous medium equations with nonlocal source terms under Dirichlet and Neumann boundary conditions.The results are obtained by using some differential inequality technique.

A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions

In this paper,we study a lattice Boltzmann method for the advectiondiffusion equation with Neumann boundary conditions on general boundaries.A novel mass conservative scheme is introduced for implementing such boundary conditions,and is analyzed both theoretically and numerically.Second order convergence is predicted by the theoretical analysis,and numerical investigations show that the convergence is at or close to the predicted rate.The numerical investigations include time-dependent problems and a steady-state diffusion problem for computation of effective diffusion coefficients.

ON THE SINGULAR ONE DIMENSIONAL P-LAPLACIAN-LIKE EQUATION WITH NEUMANN BOUNDARY CONDITIONS

In this paper, the solvability of singular one dimensional p-Laplacian-like equation with Neumann boundary conditions is considered. Under certain conditions on the operator and the nonlinear term which allows singularity, the sufficient and necessary condition for the existence of the solution to the problem is obtained. The main method is based on a-priori estimate for the lower bound of solutions, truncation arguments and the Schauder's fixed point theorem.

The Neumann Problem for Special Lagrangian Equations with Critical Phase

In this paper,we consider the Neumann problem for special Lagrangian equations with critical phase.The global gradient and Hessian estimates are obtained.Using the method of continuity,we prove the existence of solutions to this problem.

The Asymptotic Behavior of Chern-Simons Higgs Model on a Compact Riemann Surface with Boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter A 〉 0 has at least two solutions （uλ^-, uλ^2） for A sufficiently large, which satisfy that uλ^1 - -u0 almost everywhere as λ →∞, and that uλ^2 →-∞ almost everywhere as λ→∞, where u0 is a （negative） Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ →∞, and prove that uλ^2 - uλ^2- converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary OM is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.

Spectral Elliptic Solvers in a Finite Cylinder

New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented.A purely variational(no collocation)formulation of the problem is adopted,based on Fourier series expansion of the angular dependence and Legendre polynomials for the axial dependence.A new Jacobi basis is proposed for the radial direction overcoming the main disadvantages of previously developed bases for the Dirichlet problem.Nonhomogeneous Dirichlet boundary conditions are enforced by a discrete lifting and the vector problem is solved by means of a classical uncoupling technique.In the considered formulation,boundary conditions on the axis of the cylindrical domain are never mentioned,by construction.The solution algorithms for the scalar equations are based on double diagonalization along the radial and axial directions.The spectral accuracy of the proposed algorithms is verified by numerical tests.

From geometry to non-geometry via T-duality

Reconsideration of the T-duality of the open string allows us to introduce some geometric features in non-geometric theories.First,we have found what symmetry is T-dual to the local gauge transformations.It includes transformations of background fields but does not include transformations of the coordinates.According to this we have introduced a new,up to now missing term,with additional gauge field Ai^D（D denotes components with Dirichlet boundary conditions）.It compensates non-fulfilment of the invariance under such transformations on the end-points of an open string,and the standard gauge field AaN（N denotes components with Neumann boundary conditions）compensates non-fulfilment of the gauge invariance.Using a generalized procedure we will perform T-duality of vector fields linear in coordinates.We show that gauge fields AaNand AiDare T-dual to ADaand AN^irespectively.We introduce the field strength of T-dual non-geometric theories as derivatives of T-dual gauge fields along both T-dual variable yμand its double？yμ.This definition allows us to obtain gauge transformation of non-geometric theories which leaves the T-dual field strength invariant.Therefore,we introduce some new features of non-geometric theories where field strength has both antisymmetric and symmetric parts.This allows us to define new kinds of truly non-geometric theories.

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u^2) = 0 in a smooth bounded domain Ω R^3, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.

Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method.Part II:Higher-Order Schemes

Compact higher-order(HO)schemes for a new finite difference method,referred to as the Cartesian cut-stencil FD method,for the numerical solution of the convection-diffusion equation in complex shaped domains have been addressed in this paper.The Cartesian cut-stencil FD method,which employs 1-D quadratic transformation functions to map a non-uniform(uncut or cut)physical stencil to a uniform computational stencil,can be combined with compact HO Pad´e-Hermitian formulations to produce HO cut-stencil schemes.The modified partial differential equation technique is used to develop formulas for the local truncation error for the cut-stencil HO formulations.The effect of various HO approximations for Neumann boundary conditions on the solution accuracy and global order of convergence are discussed.The numerical results for second-order and compact HO formulations of the Cartesian cut-stencil FD method have been compared for test problems using the method of manufactured solutions.