Mixed Boundary Value Problem Research Articles

Shell equations in terms of Günter's derivatives, derived by the Γ‐convergence

A mixed boundary value problem (BVP) for the Láme equation in a thin layer around a surface with the Lipshitz boundary is investigated. The main goal is to find out what happens when the thickness of the layer tends to 0, that is, h→0. To this end, we reformulate BVP into an equivalent variational problem and prove that the energy functional has the Γ‐limit being the energy functional on the mid‐surface . The corresponding BVP on , considered as the Γ‐limit of the initial BVP, is written in terms of Günter's tangential derivatives on and represents a new form of the shell equation. It is shown that the Neumann boundary condition from the initial BVP on the upper and lower surfaces transforms into a right‐hand side term of the basic equation of the limit BVP.

Two-dimensional elastodynamic scattering by a soft/hard strip

A numerical approach for solving of the 2D elastodynamic waves scattering problem by a finite width planar soft/hard strip is considered. The lower face of the strip is perfectly bonded to the matrix and the upper face is modeled as an interface crack. Using Betti's theorem the mixed boundary value problem is reduced to the hypersingular and weakly singular systems of integral equations. An efficient numerical collocation method for the solution of this problem is developed and its numerical results are presented.

A Mixed Boundary Value Problem for the Laplace Equation in a semi-infinite Layer

The paper offers a solution of the mixed Dirichlet-Neumann and Dirichlet-Neumann-Robin boundary value problems for the Laplace equation in the semi-infinite layer, using the previously obtained solution of the mixed Dirichlet-Neumann boundary value problem for a layer.The functions on the right-hand sides of the boundary conditions are considered to be functions of slow growth, in particular, polynomials. The solution to boundary value problems is also sought in the class of functions of slow growth. Continuing the functions on the right-hand sides of the boundary conditions on the upper and lower sides of the semi-infinite layer from the semi-hyperplane to the entire hyperplane, we obtain the Dirichlet-Neumann problem for the layer, the solution of which is known and written in the form of a convolution. If the right-hand sides of the boundary conditions are polynomials, then the solution is also a polynomial. To the solution obtained it is necessary to add the solution of the problem for a semi-infinite layer with homogeneous boundary conditions on the upper and lower sides and with an inhomogeneous boundary condition of Dirichlet, Neumann or Robin on the lateral side. This solution is written as a series. If we take a finite segment of the series, then we obtain a solution that exactly satisfies the Laplace equation and the boundary conditions on the upper and lower sides of the semi-infinite layer and approximately satisfies the boundary condition on the lateral side.An example of solving the Dirichlet-Neumann and Dirichlet-Neumann-Robin problems is considered, describing the temperature field of a semi-infinite plate the upper side of which is heat-isolated, on the lower side the temperature is set in the form of a polynomial, and the lateral side is either heat-isolated, or holds a zero temperature, or has heat exchange with a zero-temperature environment. For the first two Dirichlet-Neumann problems, the solution is obtained in the form of polynomials. For the third Dirichlet-Neumann-Robin problem, the solution is obtained as a sum of a polynomial and a series. If in this solution the series is replaced by a finite segment, then an approximate solution of the problem will be obtained, which approximately satisfies the Robin condition on the lateral side of the semi-infinite layer.

Existence and uniqueness of solution for multidimensional parabolic partial differential‐algebraic equations arising in semiconductor modeling

This paper concerns with a compact network model combined with distributed models for semiconductor devices. For linear RLC networks containing distributed semiconductor devices, we construct a mathematical model that joins the differential‐algebraic initial value problem for the electric circuit with multidimensional parabolic‐elliptic boundary value problems for the devices. We prove an existence and uniqueness result and the asymptotic behavior of this mixed initial boundary value problem of partial differential‐algebraic equations.

Interaction of magnetoelastic shear waves with a Griffith crack in an infinite strip

In this research paper, the diffraction of a Griffith crack, situated in an infinite strip of finite thickness, due to magnetoelastic shear wave propagation has been analyzed. The effect of magnetic field on the Griffith crack interaction has been studied. Fourier transform is used to reduce the mixed boundary value problem to the dual integral equations. Finally, with the help of Abel’s transform the integral equations have been converted to Fredholm integral equation of 2nd kind. Fox and Goodwin method is used to solve the integral equation numerically. The analytical expression of Stress Intensity Factor at the crack tip has been illustrated graphically for the cases with magnetic effect and without magnetic effect.

Mixed boundary value problems for Rayleigh wave in a half‐plane with cubic anisotropy

AbstractThe paper deals with mixed boundary value problems in an elastic half‐plane with cubic symmetry. The formulation of the problem depends on an asymptotic model derived for anisotropic materials. It is demonstrated that defining the displacements in terms of a pair of plane harmonic functions reduces the problem to a classical isotropic form, which can be formulated within the framework of the asymptotic hyperbolic–elliptic model developed for isotropic materials. As an example, a semi‐infinite rigid stamp moving at a constant speed along the surface is considered.

Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via \u03c8-Hilfer fractional derivative

In this paper, we investigate the existence and uniqueness of a solution for a class of ψ-Hilfer implicit fractional integro-differential equations with mixed nonlocal conditions. The arguments are based on Banach’s, Schaefer’s, and Krasnosellskii’s fixed point theorems. Further, applying the techniques of nonlinear functional analysis, we establish various kinds of the Ulam stability results for the analyzed problem, that is, the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we provide some examples to illustrate the applicability of our results.

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

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.

Смешанная задача для неоднородного уравнения Аллера

A mixed boundary value problem is investigated for the inhomogeneous Hallaire equation. An explicit representation of the regular solution is found using the Fourier method.

A mixed type boundary-value problem related to the electrostatics of cold plasma jet reactors based on dielectric barrier discharge

A mixed type boundary-value problem related to the electrostatics of cold plasma jet reactors based on dielectric barrier discharge

АНАЛИЗ СМЕШАННОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ ПУАССОНА

The mixed boundary value problem for the Poisson’s equation is examined in a bounded flat domain. The problem is continued in a variational form through the boundary with the Dirichlet condition to a rectangular domain. To solve the continued problem, a modified method of fictitious components in a variational form is formulated. The continued problem in a variational form is considered on a finite-dimensional space. To solve the previous problem, a modified method of fictitious components on a finite-dimensional space is formulated. To solve the continued problem in matrix form, the known method of fictitious components is considered. It is shown that in the method of fictitious components the absolute error in the energy norm converges with the speed of a geometric progression. To generalize the method of fictitious components, a new version of the method of iterative extensions is proposed. The continued problem in matrix form is solved using the method of iterative extensions. It is shown that in the proposed version of the method of iterative extensions, the relative error converges in a norm that is stronger than the energy norm of the problem with a geometric progression rate. The iterative parameters in the specified method are selected using the minimum residual method. The conditions which are sufficient for the convergence of the applied iterative process are indicated. An algorithm which implements the proposed version of the method of iterative extensions is written. In this algorithm, an automated selection of iterative parameters is conducted, and the stopping criterion is established when achieving an estimate of the required accuracy. An example of the application of the method of iterative extensions for solving a particular problem is given. In the calculations, the condition for achieving an estimate of the relative error in the norm that is stronger than the energy norm of the problem is set. However, the relative errors of the obtained numerical solution of the example of the original problem are shown in other ways. For example, the relative error in grid nodes is calculated pointwise. To achieve a relative error of no more than a few percent, just a few iterations are required. Computational experiments confirm the asymptotic optimality of the method obtained in theory.

Information system “Graded coatings”

Functionally graded coatings are widely used to protect structures and machine parts from damage caused by mechanical and thermal loads. The information system “Graded coatings” was developed to help the researchers to design such coatings and also for expanding of the availability of tools for calculating and analysing of experimental data. The system was constructed for interactive data preparation and multi-parametric analysis of the results of numerical-analytical solution of mixed boundary value problems of crack theory and theory of elasticity and thermoelasticity for continuously inhomogeneous coatings of complex structure

Solution by the factorization method of a mixed boundary value problem of diffusion-convection-decay for a homogeneous layer based on the equations of turbulent diffusion

Solution by the factorization method of a mixed boundary value problem of diffusion-convection-decay for a homogeneous layer based on the equations of turbulent diffusion

Isoperimetric cones and minimal solutions of partial overdetermined problems

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that, in cones having an isoperimetric property, the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the C1,1 -metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].

The stiff Neumann problem: Asymptotic specialty and “kissing” domains

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Ω ⊂ R d which is divided into two subdomains: an annulus Ω 1 and a core Ω 0 . The density and the stiffness constants are of order ε − 2 m and ε − 1 in Ω 0 , while they are of order 1 in Ω 1 . Here m ∈ R is fixed and ε > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as ε → 0 for any m. In dimension 2 the case when Ω 0 touches the exterior boundary ∂ Ω and Ω 1 gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x → O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

Laplace-Beltrami equation on a hypersurface with Lipschitz boundary

Main objective of the present paper is to prove solvability of the Dirichlet, Neumann and Mixed boundary value problems for an anisotropic Laplace-Beltrami equation on a hypersurface $$${C}$$$ with the Lipschitz boundary $$${\Gamma=∂C}$$$ in the classical $$${𝕎^1(C)}$$$ space setting.

Особливості хвильового поля в півскінченному хвилеводі зі змішаними граничними умовами на його торці

The work is devoted to the analysis of the wave field, which is excited by the reflection of the first normal propagation Rayleigh-Lamb wave from the edge of an elastic semi-infinite strip, part of which is rigidly clamped, and part is free from stresses. The boundary value problem belongs to the class of mixed boundary value problems, the characteristic feature of which is the presence of a local feature of stresses at the point of change of the type of boundary conditions. To solve this boundary value problem, the paper proposes a method of superposition, which allows to take into account the feature of stresses due to the asymptotic properties of the unknown coefficients. Asymptotic dependences for coefficients are determined by the nature of the feature, which is known from the solution of the static problem. The criterion for the correctness of the obtained results was the control of the accuracy of the law of conservation of energy, the error of which did not exceed 2% of the energy of the incident wave for the entire considered frequency range. The paper evaluates the accuracy of the boundary conditions. It is shown that the boundary conditions are fulfilled with graphical accuracy along the entire end of the semi-infinite strip, except around a special point ($\epsilon$). In this case, along the clamped end of the semi-infinite strip in the vicinity of a special point of stress remain limited. The presence of the region $\epsilon$ and the limited stresses are due to the fact that the calculations took into account the $N$ members of the series that describe the wave field, and starting from the $N+1$ member of the series moved to asymptotic values of unknown coefficients, the number of which was also limited to $2N$. As the value $N$ increased, the accuracy of the boundary conditions increased, the region $\epsilon$ decreased, and the magnitude of the stresses near the singular point increased.

Existence of solutions for impulsive hybrid boundary value problems to fractional differential systems

In this paper, the existence of solutions for impulsive mixed boundary value problems involving Caputo fractional derivatives is obtained. And then, our conclusions are based on Krasnoselskii's fixed point theorem and Arzela-Ascoli theorem. Finally, some examples are given to illustrate the main results.

Numerical Analysis of the Mixed Boundary Value Problem for the Sophie Germain Equation

Numerical Analysis of the Mixed Boundary Value Problem for the Sophie Germain Equation

Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation

Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation