Instability results for reaction diffusion equations with Neumann boundary conditions

In this paper linear stochastic integral evolution equations are studied. They are associated with formal stochastic partial differential equations as well as stochastic delay differential equations. The existence and uniqueness of a solution is established for systems with disturbances depending on the state, both current and past, using semigroups or more generally evolution operators and known properties of such operators.     

Stabilized finite element method for Navier-Stokes equations with physical boundary conditions

This paper deals with the numerical approximation of the 2D and 3D Navier-Stokes equations, satisfying nonstandard boundary conditions. This lays on the finite element discretisation of the corresponding Stokes problem, which is achieved through a three-fields stabilized mixed formulation. A priori and a posteriori error bounds are established for the nonlinear problem, ascertaining the convergence of the method. Finally, numerical tests are presented, including mesh refinement via error indicators.

     

Some results for non-linear elliptic problems with mixed boundary conditions

We prove existence and uniqueness results for non-linear elliptic equations with lower order terms, L¹ data, and mixed boundary conditions that include as particular cases the Dirichlet and the Neumann problems. Mathematics Subject Classification (2000) 35J25, 35D05, 35J70, 35J60     

New results for the second order impulsive integro-differential equations with nonlinear boundary conditions

This paper considers the second order integro-differential equations with impulses. Some sufficient conditions for the existence of solutions are proposed by using monotone iterative method and Schauder fixed point theorem. Moreover, new concepts of lower and upper solutions are introduced for nonlinear boundary value problems.     

Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions

In this paper, we study existence and uniqueness of fractional integrodifferential equations with boundary value conditions. A new generalized singular type Gronwall inequality is given to obtain an important a priori bounds. Existence and uniqueness results of solutions are established by virtue of fractional calculus and fixed point method under some weak conditions. An example is given to illustrate the results.     

Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions

This paper deals with the existence of solutions for a class of p(x)-biharmonic equations with Navier boundary conditions. The approach is based on variational methods and critical point theory. Indeed, we investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of the third solution for the problem.     

Some existence and uniqueness results for boundary value problems for difference equations

Existence and uniqueness results for bvp problems for difference equations are discussed. The weighted norm technique and the Banach contraction mapping principle are employed     

Attractors for parabolic equations with dynamic boundary conditions

In this paper we prove existence of global solutions and (L²(Ω)×L²(Γ),(H¹(Ω)∩L^p(Ω))×L^p(Γ))$(L^2\left(\Omega \right)\times L^2\left(\Gamma \right),(H^1\left(\Omega \right)\cap L^p\left(\Omega \right))\times L^p\left(\Gamma \right))$-global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary, where there is no other restriction on p(≥2)$p(\ge 2)$.     

Evolution equations with white-noise boundary conditions

The paper is devoted to nonlinear evolution equations with nonhomogenous boundary conditions of white noise type. Necessary and sufficient conditions for the existence of solutions in the linear case are given. It is also shown that if the nonlinearity satisfies appropriate dissipativity conditions the nonlinear equation has a solution as well. The results are applied to equations with polynomial nonlinearities     

Asymptotics for wave equations with Wentzell boundary conditions and boundary damping

We are concerned with linear wave equations with Wentzell boundary conditions of dynamical type, where only one velocity feedback force acts on the Wentzell boundary. By using the theory of strongly continuous semigroups of linear operators, we prove that the energies of the solutions are strongly stable. Moreover, we show in the one dimensional case that there are solutions decaying at arbitrarily slow rates.     

Unique existence results and numerical solutions for fourth-order impulsive differential equations with nonlinear boundary conditions

The work is concerned with three kinds of fourth-order impulsive differential equations with nonlinear boundary conditions. We at first focused on studying the existence and uniqueness of positive solutions for these kinds of problems. By converting the problem to an equivalent integral equation, then applying the new class of fixed point theorems for the sum operator on cone, we obtain the sufficient conditions which not only guarantee the existence of a unique positive solution, but also be applied to construct two iterative sequences for approximating it. Further, we present the numerical methods for solving the fourth-order differential equations. At last, some examples are given with numerical verifications to illustrate the main results.     

Absorbing boundary conditions for diffusion equations

Summary. We construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients, curved artifical boundary, and arbitrary convection. It relies on the geometric identification of the Dirichlet to Neumann map and rational interpolation of in the complex plane. The boundary conditions are stable, accurate, and practical for computations. Received December 12, 1992 / Revised version received July 4, 1994     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Absorbing boundary conditions for reaction-diffusion equations

We treat here of the question of absorbing boundary conditionsfor nonlinear diffusion equations. We use the conditions designedfor the linear equation, we prove them to be well posed forthe nonlinear problem, and through numerical experiments thatthey are well suited for reaction–diffusion equations.     

Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

Existence results of fractional differential equations with irregular boundary conditions and p-Laplacian operator

In this paper, we discuss the existence of solutions for irregular boundary value problems of nonlinear fractional differential equations with p-Laplacian operator $$\left \{ \begin{array}{l} {\phi}_p(^cD_{0+}^{\alpha}u(t))=f(t,u(t),u'(t)), \quad 0< t<1, \ 1< \alpha \leq2, \\ u(0)+(-1)^{\theta}u'(0)+bu(1)=\lambda, \qquad u(1)+(-1)^{\theta}u'(1)=\int_0^1g(s,u(s))ds,\\ \quad \theta=0,1, \ b \neq \pm1, \end{array} \right . $$ where \(^{c}D_{0+}^{\alpha}\) is the Caputo fractional derivative, ? _p (s)=|s| ^p?2 s, p>1, \({\phi}_{p}^{-1}={\phi}_{q}\) , \(\frac {1}{p}+\frac{1}{q}=1\) and \(f: [0,1] \times\mathbb{R} \times\mathbb {R} \longrightarrow\mathbb{R}\) . Our results are based on the Schauder and Banach fixed point theorems. Furthermore, two examples are also given to illustrate the results.