On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem

     

Method for solving the equations describing the interaction of a three-dimensional boundary layer with an outer inviscid flow

A new method is developed for solving the three-dimensional time-independent equations describing the interaction of a laminar boundary layer with an outer inviscid flow. The method also applies to the interaction of plane flows. By applying the method, the problem of the three-dimensional viscous supersonic gas flow over a roughness element (a hump and a cavity) is solved for the first time within the framework of the classical triple-deck theory. The asymptotic height of the roughness element corresponding to the nonseparated flow is determined, and separated flow patterns are constructed.     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.     

A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators

We prove the equivalence of the three different definitions of the viscosity solution for the integro-differential equation with the Lévy operator. The two of the definitions are known in the preceding works of the author and the others, and the last one is new. A construction of a sequence of the approximating test functions to the subsolution (or the supersolution) is indispensable for the proof, and it is done explicitly in the paper.     

Intrinsic equations for a generalized relaxed elastic line on an oriented surface in the galilean space

In this article, we derive the intrinsic equations for a generalized relaxed elastic line on an oriented surface in the Galilean 3-dimensional space G₃. These equations will give direct and more geometric approach to questions concerning about generalized relaxed elastic lines on an oriented surface in G₃.     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Some supplementary results on the 1+\sqrt 2 order method for the solution of nonlinear equations

Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.     

Some effective methods for unconstrained optimization based on the solution of systems of ordinary differential equations

In this paper, we review briefly some methods for minimizing a functionF(x), which proceed by follwoing the solution curve of a system of ordinary differential equations. Such methods have often been thought to be unacceptably expensive; but we show, by means of extensive numerical tests, using a variety of algorithms, that the ODE approach can in fact be implemented in such a way as to be more than competitive with currently available conventional techniques.This work was supported by a SERC research studentship for the first author. Both authors are indebted to Dr. J. J. McKeown and Dr. K. D. Patel of SCICON Ltd, the collaborating establishment, for their advice and encouragement.     

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞)$(0,+\infty )$. We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L^1,2(0,∞)$L^{1,2}(0,\infty )$ space by employing a method recently introduced in [7] or [8].     

A sharp estimate and change on the dimension of the attractor for singular semilinear parabolic equations

We consider the semilinear reaction diffusion equation

On existence of global solutions to the Navier-Stokes equations for compressible and viscous flows on the surface of a sphere

Following tecniques proposed by A. V. Khazikhov and V. A. Weigant in 1995, we prove the global, with respect to time, existence and uniqueness of the solution to the Navier-Stokes equations for a compressible, viscous and barotropic fluid which moves on the surface of a sphere. In obtaining the main estimates we make use of the Hodge decomposition and the generalized potential theory due to K. Kodaira.

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Sunto Seguendo le tecniche proposte da A. V. Khazikhov and V. A. Weigant nel 1995, si prova l'esistenza ed unicità della soluzione per le equazioni di Navier-Stokes per un fluido comprimibile, viscoso e barotropico che si muova sulla superficie di una sfera. Nell'ottenere le principali stime si utilizzano la decomposizione di Hodge e la teoria del potenziale generalizzato, dovuta a K. Kodaira.

     

High-accuracy schemes of the finite element method for systems of degenerate elliptic equations on an interval

In this paper for the finite elementmethod for systems of degenerate elliptic equations we develop high-accuracy schemes based on multiplicative singularity extraction. We prove theorems about the smoothness of a solution. Based on these theorems we estimate the error of the proposed method.     

Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains

We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain Ω₀ of the domain of definition Ω of the energy balance equation and of the Poisson equation. Here Ω₀ corresponds to the region of semiconducting material, Ω Ω₀ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two‐dimensional stationary energy model. For this purpose we derive a W^1,p ‐regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constructing the governing equations for a layered composite of regular structure within the limits of the theory of asymmetric elasticity on the basis of energy criteria of equivalence

Within the limits of the theory of asymmetric elasticity, two (kinematic and static) approaches are offered to constructing the governing equations for a layered hybrid composite of regular structure with isotropic constituents. As a criteria of equivalence between the layered composition and a homogeneous transversely isotropic material, the equality of the specific free energy and the specific thermodynamic Gibbs energy in them is used, which allows one to determine the upper and lower bounds for the effective rigidities of the layered material. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 3–16, January–February, 2009.     

Coupling of numerical Green''s matrix and boundary integral equations for the elastodynamic analysis of inhomogeneous bodies on an elastic half-space

A coupled system of integral equations (of the domain and boundary types) is formulated for the elastodynamic response analysis of a locally inhomogeneous body on a homogeneous elastic half-space. The method uses the fundamental solution for homogeneous elastostatics in the inhomogeneous domain owing to the lack of a fundamental solution in inhomogeneous elastodynamics.

The integral representation of displacements in the inhomogeneous domain is formulated with the help of this elastostatic fundamental solution by considering the term induced by the inhomogeneity of materials and the acceleration term as the body force term. Then the Green's matrix is obtained numerically from this integral representation and combined with the ordinary boundary integral equations, which are valid in the exterior homogeneous half-space.

Some numerical examples show the efficiency and the versatility of this coupled method.     

The impact of parameter selection on the performance of an automatic adaptive code for solving reaction–diffusion equations in three dimensions

The performance of automatic codes for solving reaction–diffusion systems is controlled by a variety of parameters. Some of them are invisible to the user while others can be modified. For the latter default values are available. The effects of four such parameters are studied for a code that couples the method-of-lines in time with a high-order h-refinement finite element strategy in space. The key parameters considered are the ratio of the temporal error tolerance to the spatial error tolerance, two parameters governing convergence of the iterative methods for the linear and nonlinear systems, and the number of time steps between regridding. These parameters are typical in method-of-lines based codes. Computations on a model problem demonstrate that both small and large temporal to spatial error ratios lead to performance degradation as does less frequent regridding. They also show that insufficient convergence in the nonlinear solver can reduce the reliability of the spatial error estimates.     

A continuation and existence result for a boundary value problem on an unbounded domain arising for the electrical potential in a cylindrical double layer

Using a continuation theorem for contractions, the existence, uniqueness, and an approximation method for a class of nonlinear boundary value problems on an unbounded interval is obtained. The results apply in particular to the problem in the title.     

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

Summary  We consider the numerical treatment of second kind integral equations on the real line of the form