scholarly journals Regarding the solution of the plate theory boundary problem for domains of complex shape

2019 ◽  
Author(s):  
V. L. Voloshko ◽  
M. P. Petulko
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Boundary Value Problem ◽  
Integral Equations ◽  
Linear Problem ◽  
Plate Theory ◽  
Boundary Value ◽  
Potential Method ◽  
Fredholm Integral Equations ◽  
Numerical Implementation

Mathematical model construction of complicate physical phenomenon often leads to the setting and solving problems of parameters optimal control in differential equations in partial derivatives. Chosen equation with boundary and initial conditions is usually mathematical model basis of the object, which is under analysis. Optimal control of right-hand side function in non-linear problem for inhomogeneous biharmonic has been investigated. With the help of various gradient methods the problems of parameters control in such equations are solved successfully. Herewith linear problem is solved with the potential method on every step. The boundary value problem of plate theory, which is reduced to a system of Fredholm integral equations of the first kind and an algorithm of self-regularization of this system, is considered. The potential method is used to solve the linear problem for the harmonic equation. Examples of numerical implementation are shown that demonstrate high computational efficiency in the case of complex form regions. Algorithm for linear boundary value problem solution with boundary integral equations overcomes this problem successfully. Physical examples of numerical implementation have been presented, analysis of obtained solutions have been conducted. Their accuracy, algorithm simplicity and time spent evidence about this approach promising for practical results obtaining in plate theory and mathematical physics problems successful numerical solving.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

1999 ◽  
Vol 6 (6) ◽  
pp. 517-524
Author(s):  
M. Basheleishvili
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
Mixture Theory ◽  
Potential Method ◽  
Fredholm Integral Equations ◽  
Infinite Domains ◽  
Elastic Mixture ◽  
Uniqueness Of The Solution ◽  
Dimensional Boundary

Abstract The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.

scholarly journals A Boundary Value Problem for Noninsulated Magnetic Regime in a Vacuum Diode

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 617
Author(s):  
Edixon M. Rojas ◽  
Nikolai A. Sidorov ◽  
Aleksandr V. Sinitsyn
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boundary Value ◽  
Singular System ◽  
Degree Theory ◽  
Fredholm Integral Equations ◽  
Topological Degree Theory ◽  
System Of Integral Equations ◽  
Nonlinear Fredholm Integral Equations ◽  
Nonlinear Singular System

In this paper, we study the stationary boundary value problem derived from the magnetic (non) insulated regime on a plane diode. Our main goal is to prove the existence of non-negative solutions for that nonlinear singular system of second-order ordinary differential equations. To attain such a goal, we reduce the boundary value problem to a singular system of coupled nonlinear Fredholm integral equations, then we analyze its solvability through the existence of fixed points for the related operators. This system of integral equations is studied by means of Leray-Schauder’s topological degree theory.

scholarly journals Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Otar Chkadua ◽  
Sergey Mikhailov ◽  
David Natroshvili
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Singular Integral ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Representation Formula ◽  
Potential Method ◽  
Factorization Method ◽  
Variable Coefficients ◽  
Second Order

AbstractThe paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

scholarly journals Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

2017 ◽  
Vol 15 (1) ◽  
pp. 374-381 ◽  
Author(s):  
Serhii V. Gryshchuk ◽  
Sergiy A. Plaksa
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Fredholm Property ◽  
Fredholm Integral Equations ◽  
Cauchy Type ◽  
Simply Connected Domain ◽  
Isotropic Elasticity ◽  
Type Boundary

Abstract We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = {xe1 + ye2 : (x, y) ∈ D}: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when values of two component-functions U1, U4 are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.

Mathematical model implementation in conditions of uncertainty and possible risks

2021 ◽  
pp. 137-145
Author(s):  
A. Kravtsov ◽  
◽  
D. Levkin ◽  
O. Makarov ◽  
◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Layer Structure ◽  
Boundary Value ◽  
Goal Function ◽  
Cauchy Problems ◽  
System Of Differential Equations

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

scholarly journals On the Solution of the Exterior Boundary Value Problem with the Aid of Series

1978 ◽  
Vol 41 ◽  
pp. 175-176
Author(s):  
M. S. Petrovskaya
Keyword(s):  
Gravitational Field ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Integral Equations ◽  
Boundary Value ◽  
Parameter Method ◽  
Small Parameter Method ◽  
Exterior Boundary ◽  
Exterior Boundary Value Problem ◽  
Molodensky Problem

AbstractThe exterior gravitational field depending on the Earth’s non-sphericity is usually determined from the analysis of satellite data or by the solution of the exterior boundary value problem. In the latter case some integral equations are solved which correlate the exterior potential with the known vector of gravity and the shape of the Earth’s surface (molodensky problem). In order to carry out the integration the small parameter method is applied. As a result, all the quantities which involve the equations should be expanded in powers of a certain small parameter, among these being the heights of the Earth’s surface points as well as the inclination α of the Earth’s physical surface. Since the angle α can be significant, especially in mountains, and in fact does not depend on any small parameter then the solution of integral equations is possible only for the Earth’s surface which is smoothed enough.

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