The spline collocation method for parabolic boundary integral equations on smooth curves

Summary. We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces. Received May 17, 2001 / Revised version received February 19, 2002 / Published online April 17, 2002     

Pseudodifferential operators with real analytic symbols and approximation methods for pseudodifferential equations

This paper introduces some methods (including an approximation method) for investigating pseudodifferential equations and related problems (Cauchy problems, boundary value problems,…) based on the technique of pseudodifferential operators with real analytic symbols.     

Multidimensional ultrametric pseudodifferential equations

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′₀(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).     

On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator

In the present work, a non-local boundary value problem with special gluing conditions for a mixed parabolic-hyperbolic equation with parameter is considered. The parabolic part of this equation is a fractional analogue of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the parameter, to an equivalent system of the second kind Volterra integral equations. Due to the influence of the fractional diffusion equation, the looked for solution belongs to a specific class of functions. The method of the Green functions and the properties of integro-differential operators are on the basis of the investigation.     

一类二阶抛物型方程初边值问题解的存在定理

本文研究了一类二阶非线性抛物型方程解的存在唯一性问题.利用非线性分析中的吸引盆理论和同胚理论,获得了相应的二阶非线性抛物型方程初边值问题解的大范围存在唯一性定理.     

Approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations

Summary Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Padé-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.     

Parabolic Type Equations and Markov Stochastic Processes on Adeles

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.     

Non-Archimedean Pseudodifferential Operators and Feller Semigroups

In this article we study a large class of non-Archimedean pseudodifferential operators whose symbols are negative definite functions.We prove that these operators extend to generators of Feller semigroups. In order to study these operators, we introduce a new class of anisotropic Sobolev spaces, which are the natural domains for the operators considered here.We also study the Cauchy problem for certain pseudodifferential equations.     

An integral equation of the first kind for a free boundary value problem of the stationary Stokes' equations

We investigate a free boundary value problem of the stationary Stokes' equations. In a previous paper adapted hydrodynamical potentials have been constructed and their jump relations have been discussed. Here we study a direct method to obtain an equivalent boundary integral equations' system of the first kind. Its solution properties are investigated in the framework of strongly elliptic pseudodifferential operators. For numerical purposes a suitable representation formula for the variational equation is given in terms of integro-differential operators which avoids the evaluation of hypersingular integrals.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

Acta Mathematica Scientia

1IntroductionLetfibeaboundeddomainOfR"withsmoothboundaryoff,KCL'(fl)xL'(fl)beasubsetdefinedasfollows:K={(yi,yi)EL'(fl)xL'(fl);yi(x) p(yl(x))30,a.e.infi},(1)withwhichisamaximalmonotonegraphonRxR.Whereaisapositiveconstant.Ourmainproblemisasfollows:(O)MinimizeL(y,u)={'[g(y(t)) h(u(t))]at,overallpairssubjecttotheconditionsOurbasichypothesesaboutgandharethefollowing:(H)Bothgandh:L'(fl)~(--co, co]arelowersendcontinuous,convexsproperfunctionals,withg(y)相似文献   

OPTIMAL CONTROL PROBLEM OF PARABOLIC DIFFERENTIAL EQUATION WITH TWO POINT BOUNDARY CONDITION

This paper deals with optimal control problem of parabolic differential equation with two point boundary conditions (in the time variable). The results here extend those in [3] on optimal control of the heat equations. Moreover, the technique used in this paper is based on some smooth approximations of “tangent cones” in the sense of Clarke and some maximal monotone operators.     

Composite spectral method for solution of the diffusion equation with specification of energy

Many physical subjects are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions replacing the classic boundary conditions. In this article, we introduce a new numerical method for solving the one‐dimensional parabolic equation with nonlocal boundary conditions. The approximate proposed method is based upon the composite spectral functions. The properties of composite spectral functions consisting of terms of orthogonal functions are presented and are utilized to reduce the problem to some algebraic equations. The method is easy to implement and yields very accurate result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Incomplete quenching in a system of heat equations coupled at the boundary

In this paper we find a possible continuation for quenching solutions to a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. For non-simultaneous quenching our continuation is a solution of a parabolic problem with Neumann boundary conditions. We also give some results for simultaneous quenching and present some numerical experiments that suggest that the approximations are not uniformly bounded in this case.     

Pseudodifferential Operators and Equations of Variable Order

A new class of elliptic pseudodifferential equations of variable order is considered. Local Sobolev–Slobodetskii spaces are introduced and used to obtain theorems on the continuity of these pseudodifferential operators and describe the Fredholm properties of the corresponding pseudodifferential equations and boundary value problems.     

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR² via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.     

On some classes of coefficient inverse problems for parabolic systems of equations

We examine the question on solvability in the Sobolev spaces of coefficient inverse problems for parabolic systems of equations with the overdetermination conditions on a collection of surfaces. Under certain conditions on the geometry of the domain and the boundary operators, the local solvability of the problem is proven. It is demonstrated that the conditions on the boundary operators are sharp and that, in some cases, the problem is not unconditionally solvable.     

Pseudodifferential Multi-Product Representation of the Solution Operator of a Parabolic Equation

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on ? ⁿ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.     

Interaction between thermoelastic and scalar oscillation fields

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.