scholarly journals On the Modified Jump Problem for the Laplace Equation in the Exterior of Cracks in a Plane

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
P. A. Krutitskii ◽  
A. Sasamoto
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Classical Solution ◽  
Laplace Equation ◽  
Normal Derivative ◽  
Index Zero ◽  
Jump Problem ◽  
Unique Classical Solution

The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The integral representation and integral equation are essentially simpler than those derived for this problem earlier. The singularities at the ends of the cracks are investigated.

scholarly journals Explicit solution of the jump problem for the Laplace equation and singularities at the edges

2001 ◽  
Vol 7 (1) ◽  
pp. 1-13 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Boundary Value Problem ◽  
Explicit Form ◽  
Laplace Equation ◽  
Explicit Solution ◽  
Existence Theorems ◽  
Single Layer ◽  
Normal Derivative ◽  
Jump Problem ◽  
Uniqueness And Existence ◽  
Conditions At Infinity

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified on the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.

scholarly journals The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Helmholtz Equation ◽  
Exterior Domain ◽  
Index Zero ◽  
Compatibility Conditions ◽  
Unique Classical Solution ◽  
Asymptotic Formulae ◽  
Solution Gradient

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

2021 ◽  
pp. 35-43
Author(s):  
Evgeniy B. Laneev ◽  
Dmitriy Yu. Bykov ◽  
Anastasia V. Zubarenko ◽  
Olga N. Kulikova ◽  
Darya A. Morozova ◽  
...  
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Circular Cylinder ◽  
Boundary Value Problem ◽  
Laplace Equation ◽  
Boundary Value ◽  
Stable Solution ◽  
Medical Diagnostics ◽  
Cauchy Data ◽  
Ill Posed

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

scholarly journals About Three Dimensional Jump Boundary Value Problems for the Laplacian

2019 ◽  
pp. 51-54
Author(s):  
Olexandr Polishchuk
Keyword(s):  
Hilbert Space ◽  
Integral Equation ◽  
Boundary Value Problems ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Equation System ◽  
Normal Derivative ◽  
Layer Potentials ◽  
Jump Problem ◽  
Well Posed

The conditions of well-posed solvability of searched function and its normal derivative three dimensional jump problem for the Laplacian and equivalent to them integral equation system for the sum of the simple and double layer potentials are determined in the Hilbert space, element of which as well as their normal derivatives have the jump through boundary surface.

scholarly journals OBTAINING AND INVESTIGATION OF THE INTEGRAL REPRESENTATION OF SOLUTION AND BOUNDARY INTEGRAL EQUATION FOR THE NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY

2016 ◽  
Vol 6 ◽  
pp. 53-58 ◽  
Author(s):  
Grigoriy Zrazhevsky ◽  
Vera Zrazhevska
Keyword(s):  
Thermal Conductivity ◽  
Integral Equation ◽  
Fundamental Solution ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Boundary Integral Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Integral ◽  
Initial Boundary Value

Technological processes in the energy sector and engineering require the calculation of temperature regime of functioning of different constructions. Mathematical model of thermal loading of constructions is reduced to a non-stationary initial-boundary value problem of thermal conductivity. The article examines the formulation of the non-stationary initial-boundary value problem of thermal conductivity in the form of a boundary integral equation, analyzes the singular equation and builds the fundamental solution. To build the integral representation of the solution the method of weighted residuals is used. The correctness of the obtained integral representation of the solution in Minkowski space is confirmed. Singularity of the fundamental solution is investigated. The boundary integral equation and fundamental solution for axially symmetric domain for internal problem is built. The results of the article can be useful for numerical implementation of boundary element method.

scholarly journals The Dirichlet Problem for the EquationΔu−k2u=0in the Exterior of Nonclosed Lipschitz Surfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
P. A. Krutitskii
Keyword(s):  
Integral Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Laplace Equation ◽  
Existence And Uniqueness ◽  
Single Layer ◽  
Single Layer Potential ◽  
Layer Potential

We study the Dirichlet problem for the equationΔu−k2u=0in the exterior of nonclosed Lipschitz surfaces inR3. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.

On axially symmetric waves

1963 ◽  
Vol 59 (3) ◽  
pp. 637-654 ◽  
Author(s):  
J. W. Craggs
Keyword(s):  
Integral Equation ◽  
Wave Equation ◽  
Analytic Function ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Compressible Flow ◽  
Basic Solution ◽  
Axially Symmetric ◽  
Method Of Solution

AbstractThe wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.

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