Boundary Value Problems For Elliptic Equations Research Articles

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the first time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Significant part of these results is published with detailed proofs for the first time. In the first chapter, we give historical background, necessary notation, definitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman-Erde´lyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the fifth chapter, we consider applications of the Buschman-Erde´lyi transmutations of special form to new boundary-value problems for elliptic equations with significant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to differential equations with variable coefficients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed differential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schro¨dinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.

Random walk algorithms for elliptic equations and boundary singularities

Abstract The influence of boundary singularities on the behavior of the random walk algorithms is studied and illustrated by the results of several computational experiments of solving boundary-value problems for elliptic equations.

Spectral-difference parallel algorithm for the seismic forward modeling in the presence of complex topography

A spectral-difference parallel algorithm for modeling acoustic and elastic wave fields for the 2.5D geometry in the presence of irregular surface topography is considered. The initial boundary-value problem is transformed to a series of boundary-value problems for elliptic equations via the integral Laguerre transform with respect to time. For solving difference equations, it is proposed to use efficient parallel procedures based on the fast Fourier transform and the dichotomy algorithm, the latter was designed for solving systems of linear algebraic equations (SLAEs) with tridiagonal and block-tridiagonal matrices. A modification of the dichotomy algorithm for diagonally dominant matrices, which makes it possible to reduce the time of preparatory computations and increase scalability of the method relative to the number of processors, is considered. The influence of different methods of curved boundary approximation on the quality of solution is investigated; practical evaluation of accuracy is performed. Calculations of the wave field with the use of high-resolution meshes for the Canadian Foothills medium model are presented. Implementation of the complex frequency-shifted PML boundary conditions for a dynamic elasticity problem is considered in the context of the spectral-difference approach.

Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph

We consider generalized solutions to boundary-value problems for elliptic equations on an arbitrary geometric graph and their corresponding eigenfunctions. We construct analogs of Sobolev spaces that are dense in L2. We obtain conditions for the Fredholm solvability of boundary-value problems of various types, describe their spectral properties and conditions for the expansion in generalized eigenfunctions. The results presented here are fundamental in studying boundary control problems of oscillations of multiplex jointed structures consisting of strings or rods, as well as in studying the cell metabolism.

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.

On the solvability of a boundary-value problem for an elliptic equation

Consider a boundary-value problem for a second-order linear elliptic equation in a bounded domain. The coefficient of the required function is nonpositive everywhere in the domain, except for a small neighborhood of an interior point. The following question arises: Under what constraints on this coefficient in the given small domain do the statements on the existence and uniqueness of the solution of the first boundary-value problem remain valid?

Abstract elliptic and parabolic systems with applications to problems in cylindrical domains

We consider problems for quite general second-order abstract elliptic and corresponding parabolic equations on the interval $[0,1]$ and the rectangle $[0,T]\times [0,1]$, respectively. $R$-boundedness estimates of solutions of abstract boundary-value problems for elliptic equations with a parameter are established, in contrast to standard norm-bounded estimates. The results are applied to obtain $L^p$-maximal regularity for corresponding parabolic systems. In applications, the coefficient $A(x)$ of the solution $u$ can be $2m$-order elliptic operators with suitable boundary conditions, while the coefficient $B(x)$ of the first-order derivative of the solution $D_xu$ can be interpreted as an $m$-order differential operator. The corresponding applications to PDEs are presented.

Analysis of the accuracy of Monte Carlo methods for boundary-value problems using a probabilistic representation

Accuracy problems for statistical simulation algorithms are explored in solving boundary-value problems for elliptic equations in mathematical physics. The algorithms are based on a probabilistic representation of the solutions with the use of appropriate systems of stochastic differential equations (SDEs). The problems in question arise from the necessity to simulate long SDE trajectories and estimate expectations of random variables with strongly asymmetric distributions. Numerical results are presented.

Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations.

Monte Carlo methods for solving elliptic equations with boundary conditions containing the normal derivative

.The problem is to find the solution and its first deriv-atives at finitely many given points and evaluate energy,which can be represented in the form of a linear func-tional of the solution in the case under consideration.Similar problems arise in physical chemistry, when theelectrostatic properties of individual molecules in anelectrolyte solution are to be calculated.The standard approach to constructing Monte Carloestimators for solutions to boundary value problems forelliptic equations consists in applying the algorithm of

The Dirichlet and Neumann Problems for Elliptical Equations with Conjugation Conditions and High-Precision Algorithms of Their Discretization

New Dirichlet and Neumann boundary-value problems for elliptic equations with conjugation conditions are considered and existence and uniqueness of their solutions are studied. High-precision discretization algorithms are constucted based on the classes of discontinuous admissible functions.

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C∞(S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.

On general boundary-value problems for elliptic equations

A theory of general boundary-value problems is developed for differential operators with symbols not necessarily satisfying the Atiyah-Bott condition that the corresponding obstruction must vanish. A condition ensuring that these problems possess the Fredholm property is introduced and the corresponding theorems are proved.

The use of a stable boundary analogue of the collocation method to approximate the solution of some problems in mechanics

To solve boundary-value problems for elliptic equations, the boundary analogue of the method of least squares is replaced by a boundary analogue of the collocation method. The change is made using a discrete representation of the scalar product in the spaces of functions which, in the case of a smooth boundary, are integrable with their square over the boundary of the region, and of functions which, in the case of a piecewise-linear boundary, are integrable with their square, when weighted, over the boundary of the region. The method used to choose the collocation points which ensure the collocation method to be stable is justified for the case of the Dirichlet problem.

An estimate for the convergence rate of difference schemes for second-order elliptic equations in domains of arbitrary shape

Using the dummy-domains method, a difference scheme is constructed for solving the first boundary-value problem for elliptic equations of the second order in domains of arbitrary shape. An estimate for the convergence rate of order O(h1/2) in the norm of W 12 is found. Bibliography:5 titles.

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.

Asymptotic behavior of the first boundary-value problem for elliptic equations in domains with a thin coating

Asymptotic behavior of the first boundary-value problem for elliptic equations in domains with a thin coating

Schemes for solving elliptic differential equations of order 2 n, n⩾2, with incompatible curvilinear finite elements near the boundary

Some high-accuracy finite element schemes for solving the first boundary-value problem for elliptic equations of order 2 n, n ⩾ 2 are considered. Compatible triangular finite elements are used inside the region, while near the curvilinear boundary incompatible curvilinear finite elements based on the compatible triangular elements are used. An unimprovable order of accuracy is achieved by allowing for the effect of incompatibility; in the algebraic system of equations. An iterative process is considered that attains an unimprovable order of accuracy in a finite number of iterations.

Steady heat-conduction problem for a double cylinder

An analytical solution of the problem stated in the title is obtained for various conditions at the ends of the cylinder. The method of asymptotic solution of boundary-value problems for elliptic equations in thin domains is translated to the case of boundary conditions of the third kind.

Nonlocal boundary-value problems for elliptic equations and systems

Nonlocal boundary-value problems for elliptic equations and systems