Discrete problems associated to elliptic equations

This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis

An algorithm for approximating solutions to 2nd-order linear differential equations with polynomial coefficients in B-polynomials (Bernstein polynomial basis) subject to Dirichlet conditions is introduced. The algorithm expands the desired solution in terms of B-polynomials over a closed interval [0, 1] and then makes use of the orthonormal relation of B-polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of B-polynomials, and the procedure is much simpler compared to orthogonal polynomials for solving differential equations. The current procedure is implemented to solve five linear equations and one first-order nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

On the suitable weak solutions to the Boussinesq equations in a bounded domain

In this paper we construct the suitable weak solution for the initial-boundary value problem of the Boussinesq equations and obtain some properties for these solutions. Also in the case of a two dimensional space the uniqueness of weak solution is proved.     

Three solutions of constant sign for a system of discrete equations

We consider the following system of discrete equations

An initial-boundary value problem for the Maxwell equations

In this paper, by using methods from complex analysis and quaternionic analysis, we investigate an initial-boundary value problem for the Maxwell equations and obtain the general solutions and solvable conditions of the problem respectively in different cases. In addition, by using a similar method, we also discuss an initial-boundary value problem for a hyperbolic complex system of first order equations in R³.     

A short note on solvability of systems of interval linear equations

In this note we formulate necessary and sufficient conditions for strong solvability and feasibility of systems of linear interval equations in terms of absolute value inequalities.     

Discrete nonlinear excitations on an inhomogeneous lattice in a spatially linear potential

Discrete nonlinear excitations on an inhomogeneous lattice in a spatially linear potential are investigated. Discrete breathers possessing multiple frequencies are always spatially localized and oscillatory in time, with oscillatory frequencies depending on an external potential. A number of phenomena associated with the translational symmetry of the system is addressed. The spatially localized, time periodic and quasi-periodic nonlinear coherent excitations comprising multiple-soliton components are presented with exact results for the corresponding energy, momentum and norm. The mean width of a pulse as a measure of localization is also studied. Project supported by the National Natural Science Foundation of China.     

Boundary value problems for higher order linear impulsive differential equations

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.     

Perturbation of semi-linear evolution equations under weak assumptions at initial time

We prove perturbation results for abstract semi-linear evolution equations in a Banach space. The main feature is that only very weak assumptions are needed at initial time. This allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi-linear parabolic and hyperbolic boundary value problems with various boundary conditions. The theory also implies the well known theory on parameter dependent equations.     

Gidas-Ni-Nirenberg results for finite difference equations: Estimates of approximate symmetry

Are positive solutions of finite difference boundary value problems Δhu=f(u) in Ωh, u=0 on ∂Ωh as symmetric as the domain? To answer this question we first show by examples that almost arbitrary non-symmetric solutions can be constructed. This is in striking difference to the continuous case, where by the famous Gidas-Ni-Nirenberg theorem [B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related problems via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243] positive solutions inherit the symmetry of the underlying domain. Then we prove approximate symmetry theorems for solutions on equidistantly meshed n-dimensional cubes: explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer. The quality of the estimates depends on whether or not f(0)<0. The one-dimensional case stands out in two ways: the proofs are elementary and the estimates for the defect of symmetry are O(h) compared to O(1/|log(h)|) in the higher-dimensional case.     

Fifth-order evolution equations describing pseudospherical surfaces

We consider differential equations which describe pseudospherical surfaces, with associated 1-forms , 1?i?3. We characterize all such equations of type ut=uxxxxx+G(u,ux,…,uxxxx) whose associated 1-forms satisfy fp1=μpf₁₁+ηp, μp,ηp∈R, 2?p?3, in addition to a generic technical assumption. We also classify all of these equations which are independent of any of the real parameters μp, ηp, obtaining as particular cases the fifth-order Korteweg-de Vries, the Sawada-Kotera and the Kaup-Kupershmidt equations. We determine huge classes of equations describing pseudospherical surfaces and their respective linear problems in which particular cases are obtained by merely specifying certain functions which depend on u and its derivatives with respect to x.     

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

一类非线性演化方程初值问题的幂级数解

研究了一类非线性演化方程初值问题.通过不变子空间方法,这类初值问题被约化为常微分方程组的初值问题.这类初值问题是适定的.本文给出了这类初值问题关于时间变量t的幂级数解.