Mixed Boundary Value Problem Research Articles

Annular crack in a thermoelastic half-space

The purpose of this article is to analytically investigate the three-dimensional thermoelastic fields in a semi-infinite medium weakened by an annular crack. The crack faces are exposed to a prescribed temperature load and the external surface of the medium is kept at the reference temperature. The problem is formulated as a three-part mixed boundary value problem and is treated through Goodier’s thermoelastic potential and the Boussinesq harmonic functions. The Hankel transform method is employed to convert this problem into triple integral equations and a system of coupled ones. Using some integral relations and the Gegenbauer addition formula, the set of triple integral equations is ultimately reduced to an infinite system of linear algebraic equations. Closed-form formulas for various quantities of physical interest are derived and expressed in terms of the solution of the obtained infinite algebraic system. Moreover, the combined mechanical and thermal loading conditions are also considered. Numerical results for thermoelastic fields and mixed-mode thermal stress intensity factors are shown graphically to analyze their dependence on the radii and depth of the crack. Results for the thermoelastic problem of an infinite medium containing an annular crack are also obtained as a special case of this study.

Multiple solutions for mixed boundary value problems with phi-Laplacian operators

Using Leray-Schauder degree theory and the method of upper and lower solutions we establish existence and multiplicity of solutions for problems of the form $$\displaylines{ (\phi(u'))' = f(t,u,u') \cr u(0)= u(T)=u'(0), }$$ where \(\phi\) is an increasing homeomorphism such that \(\phi(0)=0\), and f is a continuous function.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/67/abstr.html

Full incremental elastic field induced by a mode-I crack in a prestressed orthotropic material

Prestressed structures are frequently encountered in civil engineering. This paper gives a theoretical analysis of a cracked orthotropic elastic plane with initial stress. The problem of an orthotropic composite with a crack parallel to a material principal axis under initial stress along the crack line reduces to a mixed boundary value problem. It is solved using the Fourier transform. Application of boundary and continuity conditions leads to dual integral equations. An analytical solution can be obtained. Full incremental elastic field in an infinite prestressed cracked orthotropic composite is derived explicitly for linearly distributed loading on the crack faces. The fracture parameters such as incremental stress intensity factor (SIF), crack opening displacement (COD), and the energy release rate (ERR) are all obtained. The results for a prestressed isotropic elastic medium with a crack can be reduced from the present. When prestress is removed, the explicit expressions of the elastic field induced by a crack embedded in an orthotropic composite are recovered. Finally, the influence of initial stress on the elastic stress field, incremental SIF, COD and ERR for a cracked orthotropic elastic plane are presented graphically.

Mode-I pressurized axisymmetric penny-shaped crack in graded interfacial zone with variable modulus and Poisson’s ratio

This paper presents an analytical treatment to the fracture problem of a Mode-I pressurized penny-shaped crack in a graded interfacial zone or FGMs interlayer with variable shear modulus and Poisson’s ratio. An efficient multilayered model is proposed to address the general form of material inhomogeneity. The mixed boundary value problem is reduced to a Fredhom integral equation of the second kind. The solution procedure is also extended to address the Leonov-Panasyuk-Dugdale plastic crack. Analytical solutions in explicit form are derived for the full stress fields at the crack plane, Mode-I stress intensity factor (SIF), crack opening displacement (COD), T-Stress and plastic zone in terms of the solution of the integral equations. Numerical results of SIF, elastic stress fields, COD, T-Stress and length of plastic zone are presented to quantitively explore the effects of material inhomogeneity on the fracture response. It is demonstrated that the graded Poisson’s ratio has non-negligible influences on COD and T-Stress under certain condition. The present solutions can be used to advanced fracture analysis of pressurized crack in engineering composite material.

Radiation of sound by a coaxial waveguide with semi-infinite perforated duct

ABSTRACT Radiation of sound waves by a coaxial duct is considered. The coaxial duct has an inner rigid wall which is infinite, while the outer wall is rigid for and perforated for which makes the problem more interesting. Applying the Fourier transform technique, the mixed boundary value problem is described by a Wiener–Hopf equation and then the solution of the problem is found by means of the Wiener–Hopf (WH) technique. The influence of the parameters of the problem like inner and outer duct radii, frequency and porosity on the sound pressure level is displayed graphically. This study is important because of its applications in noise reduction in modern aircraft jet and turbofan engines. The current problem is also compared with subproblems with and without perforated and very good agreement is observed.

О применении метода гиперсингулярных интегральных уравнений к решению задач для упругой плоскости с коллинеарной системой трещин

In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.

Existence results of nonlocal Robin mixed Hahn and q-difference boundary value problems

In this paper, we aim to study a nonlocal Robin boundary value problem for fractional sequential fractional Hahn-q-equation. The existence and uniqueness results for this problem are revealed by using the Banach fixed point theorem. In addition, the existence of at least one solution is studied by using Schauder’s fixed point theorem. The theorems for existence results are obtained.

First exact solutions for mixed boundary value problems concerning the motions of fluids with exponential dependence of viscosity on pressure

This paper presents exact expressions for dimensionless starting solutions corresponding to some oscillatory motions of fluids with exponential dependence of viscosity on pressure. These expressions are established in terms of standard Bessel functions of order zero and one. Fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies oscillatory shear stresses to the fluid. The corresponding solutions, which are currently absent in the literature, are presented as sums of steady-state and transient components. These are useful for experimentalists who wish to eliminate transients from their experiments. For completeness, the dimensionless velocity field corresponding to the motion due to the lower plate applying a constant shear stress to the fluid is determined as a limiting case. Furthermore, to verify the results, it is shown that diagrams of the present steady-state solutions coincide with those of ordinary Newtonian fluids performing the same motions as the dimensionless pressure–viscosity coefficient tends to zero. The spatial distributions of starting solutions and some transversal sections are also presented and discussed.

Partially debonded circular inclusion in one-dimensional quasicrystal material with piezoelectric effect

In this first effort, a partially-debonded circular inclusion in a one-dimensional quasicrystal material with piezoelectric effect is studied. This mixed boundary value problem of a circular interface crack is reduced to solving three Riemann–Hilbert problems with the use of analytical continuation theory. The crack-tip singularity of the circular interface crack is investigated and the intensity factors of the stresses in the phonon and the phason fields and electric displacements are derived explicitly. Some particular cases are provided to show the effect of the crack angle, material properties and loadings on the field singularities at the tips of the circular interface crack.

Cracked elastic layer with surface elasticity under antiplane shear loading

A mode-III crack embedded in a homogeneous isotropic elastic layer of nanoscale finite thickness is studied in this article. The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem, where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation. Using the Fourier transform, we convert the problem to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces. By expanding the out-of-plane displacement as series of Chebyshev polynomials, the Galerkin method is invoked to reduce the singular integro-differential equation with Cauchy kernel to a set of algebraic linear equations for the unknown coefficients. An approximate solution is determined, and the influences of surface elasticity on the elastic field and stress intensity factor are examined and displayed graphically. It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus.

An Analytical Approach for the Two-Dimensional Plunging Breaking Wave Impact on a Vertical Wall with Air Entrapment

This study investigates an idealized formulation of the two-dimensional impact of a breaking wave on a vertical impermeable wall. An overturning-like wave is assumed, which is close to the concept of a plunging breaker. It is assumed that during the collision an air pocket is entrapped between the wave and the wall. The air pocket width is assumed to be negligible and the compression effects are omitted. The problem is considered in the two-dimensional space (2D) using linear potential theory along with the small-time approximation. We use a perturbation method to cope with the linearized free-surface kinematic and dynamic boundary conditions. We impose the complete mixed boundary value problem (bvp) and we solve for the leading order of the velocity potential. The problem derived involves dual trigonometrical series and is treated analytically. The main assumption made is that, within the air pocket, the pressure is zero. Results are presented for the velocity potential on the wall, the velocity, and the free-surface elevation.

Blowup Mechanism for a Fluid-Particle Interaction System in $\mathbb{R}^{3}$

We study the Cauchy problem and the mixed initial boundary value problem of a fluid-particle interaction system in $\mathbb{R}^{3}$ . A Serrin type criterion for the strong solution of the Cauchy problem is established in terms of $\|\rho \|_{L^{\infty }_{t}L^{\infty }_{x}}$ and $\|u\|_{L^{s}_{t}L^{r}_{x}}$ , where $2/s+3/r\le 1$ and $3< r\le \infty $ . In view of some useful integral inequalities, we prove the life span estimates of the regular solution.

Complete Solution For The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method

Abstract In this study, we present some new results for the time fractional mixed boundary value problems. We consider a generalization of the Heat - conduction problem in two dimensions that arises during the manufacturing of p - n junctions. Constructive examples are also provided throughout the paper. The main purpose of this article is to present mathematical results that are useful to researchers in a variety of fields.

Solution blow‐up for a fractional in time acoustic wave equation

We consider the fractional in time acoustic wave equation where 1&lt;α&lt;2, is the Caputo fractional derivative of order α, u=u(t,x), t&gt;0, , is the pressure in the medium, ε is the nonlinear acoustic parameter, ρ0 is the equilibrium density in the medium, and c0 is the equilibrium sound velocity. We study a Cauchy problem for this equation and a mixed boundary value problem in a bounded domain. For each problem, sufficient conditions for the blow‐up of solutions are derived. Moreover, we provide a class of initial data for which there are no classical solutions even locally in time. Our approach is based on the nonlinear capacity method.

Positive Solutions for a Class of Discrete Mixed Boundary Value Problems with thep,q-Laplacian Operator

In this paper, we consider the existence of solutions for the discrete mixed boundary value problems involvingp,q-Laplacian operator. By using critical points theory, we obtain the existence of at least two positive solutions for the boundary value problem under appropriate assumptions on the nonlinearity.

Efficient numerical method to handle boundary conditions in 2D elastic media

A numerical method is developed to efficiently calculate the stress (and displacement) field in finite 2D rectangular media. The solution is expanded on a function basis with elements that satisfy the Navier–Cauchy equation. The obtained solution approximates the boundary conditions with their finite Fourier series. The method is capable to handle Dirichlet, Neumann and mixed boundary value problems as well and it was found to converge exponentially fast to the analytical solution with respect to the size of the basis. Possible application in discrete dislocation dynamics simulations is discussed and compared to the widely used finite element methods: it was found that the new method is superior in terms of computational complexity.

On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity

We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this, we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of ``defect". We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

Analysis of boundary-domain integral equations based on a new parametrix for the mixed diffusion BVP with variable coefficient in an interior Lipschitz domain

A mixed boundary value problem for the partial differential equation of diffusion in an inhomogeneous medium in a Lipschitz domain is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed in the papers by Mikhailov (2002, 2006) and Chkadua, Mikhailov, Natroshvili (2009). We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.

An Axisymmetric Torsion Problem of an Elastic Layer on a Rigid Circular Base

A solution is presented to a doubly mixed boundary value problem of the torsion of an elastic layer, partially resting on a rigid circular base by a circular rigid punch attached to its surface. This problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer expansion formula of the Bessel function we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. Both the two contact stresses under the punch and on the lower face of the layer are expressed as appropriate Chebyshev series. The effects of the radius of the disc with the rigid base and the layer thickness on the displacements, contact stresses as well as the shear stress and the stress singularity factor are discussed. A numerical application is also considered with some concluding results.

About the problem of flow past around thin profiles

In this paper some approaches to solving the problems of flow past around thin profiles are presented. We consider the problem of flow past around a contour defined by a set of segments, each of which is close to the segment ( a; a). A formula that gives an approximate velocities distribution in a flow flowing around a system of thin profiles is obtained. A mixed boundary-value problem for an analytic function in the upper half-plane is studied. A system of singular integral equations having a unique solution up to two arbitrary constants in the class of functions that are boundary values of analytic functions in the upper half-plane is obtained. Considered the generalized problem of flow past around thin profiles for infinite numbers of thin profiles.