Mixed Boundary Value Problem Research Articles

A mathematical study of diffusive logistic equations with mixed type boundary conditions

<p style='text-indent:20px;'>The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.</p>

Комбинированный бессеточный метод решения смешанных краевых задач при моделировании потенциальных физических полей

Приведено описание комбинированного бессеточного метода решения смешанных краевых задач для уравнения Лапласа, возникающих при анализе потенциальных физических полей в однородных средах. Решение найдено с использованием бессеточных методов фундаментальных решений и Монте-Карло. Выполнена модификация этих методов с учетом особенностей, связанных с заданием смешанных краевых условий на границе расчетной области. Использованы сопряжённые фундаментальные решения уравнения Лапласа и процедура блуждания по сферам, учитывающая особенности построения траектории движения в окрестности части границы, на которой задается нормальная производная. Предложена процедура совместной реализации двух бессеточных методов фундаментальных решений и Монте-Карло, обеспечивающая повышение точности расчетов. Выполнена верификация комбинированного метода с использованием бенчмарков, в качестве которых выступают результаты, полученные с использованием эталонного решения в квадрате и решения, полученного методом конечных элементов с использованием программного пакета FEMM 4.2, при анализе поля в окрестности зазора электромагнита.

On integral equations of cracks of a new type

For the first time, the paper develops a new type of crack modeling method that allows describing them in environments of complex rheologies. It is based on a new universal modeling method previously published by the authors, used in boundary value problems for systems of partial differential equations. The advantage of the method is the possibility of avoiding the need to solve complex boundary value problems for systems of partial differential equations by replacing them with separate differential equations, among which the Helmholtz equations are the simplest. Namely, with the help of combinations of solutions of boundary value problems for this equation, it is possible to describe the behavior of complex solutions of multicomponent boundary value problems. In this paper, for the first time, the method is applied to a mixed boundary value problem for cracks of a new type. Cracks of a new type, complementing the Griffiths cracks, were discovered during the study of fractures of lithospheric plates that converge at the ends during oncoming traffic along the Conrad boundary. In the course of the study, Kirchhoff plates were adopted as models of lithospheric plates. The method developed in the published article is aimed at the possibility of describing models of approaching objects similar to lithospheric plates in the form of deformable plates of more complex rheologies. In particular, it can be thermoelectroelastic plates or other rheology. In the process of solving problems using Kirchhoff models for lithospheric plates, there was a problem of calculating some functionals that needed to be determined. This method demonstrates an approach that eliminates this drawback. The derivation of integral equations of cracks of a new type, the method of their solution and the approach to application in more complex rheologies is given.

A Normal Crack in a Piezoelectric Actuator with Functionally Graded Properties under a Transient Thermal Load Combined with an Electric Load

In this paper, the fracture problem of a piezoelectric actuator consisting of a functionally graded piezoelectric material strip (FGPM strip) and a homogeneous layer by a transient thermal load combined with an electric load is considered. The FGPM strip contains a crack perpendicular to the interface. Material properties of the FGPM strip are assumed to be exponentially dependent on the distance from the interface. The thermo-electro-mechanical fields in the un-cracked laminate induced by the transient thermal load and the electric load are calculated. By using the superposition technique, the obtained normal stresses are used as the crack surface tractions with opposite sign to formulate the mixed boundary value problem. The integral transform techniques are employed to reduce the problem to the solution of a singular integral equation. The stress intensity factors are computed, and the results are presented for the various values of the nonhomogeneous and geometric parameters. We also focused on the crack contact phenomenon for the case of the pure electric load. Moreover, the effect of the difference between two kinds of the electric boundary conditions on the stress intensity factors under transient thermal load and the reduction of ones by the electric load would be discussed.

АНАЛИЗ КРАЕВОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ ПУАССОНА

The mixed boundary value problem for the Poisson equation is considered in a bounded flat domain. The continuation of this problem through the boundary with the Dirichlet condition to a rectangular domain is carried out. Consideration of the continued problem in the operator form is proposed. To solve the continued problem, a method of iterative extensions is formulated in an operator form. The extended problem in operator form is considered on a finitedimensional subspace. To solve the previous problem, an iterative extension method is formulated in operator form on a finite-dimensional subspace. The continued problem is presented in matrix form. To solve the continued problem in matrix form, the method of iterative extensions in matrix form is formulated. It is shown that in the proposed versions of the method of iterative extensions, the relative errors converge in a rate that is stronger than the energy norm of the extended problem with the rate of geometric progression. The iterative parameters in these methods are selected using the minimum residual method. Conditions are indicated that are sufficient for the convergence of the applied iterative processes. An algorithm is written that implements the method of iterative extensions in matrix form. In this algorithm, the iterative parameters are automatically selected and the stopping criterion is indicated when the estimate of the required accuracy is reached. Examples of application of the method of iterative extensions for solving problems on a computer are given.

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

The effect of semi perforated duct on ring sourced acoustic diffraction

An analytical solution is obtained for the diffraction problem. In an infinite cylindrical duct, the sound waves are emanating by a ring source. The duct is rigid for z < l and perforated for z > l. The mixed boundary value problem is defined by a Wiener Hopf equation, by using the Fourier transform technique. Then the numerical solution is obtained. The influence of the parameters of the problem on the diffraction phenomenon is displayed graphically. The present study can be used as a model for different applications. Reducing noise in exhaust systems, ventilation systems are some of these applications.

Computational contact mechanics for a medium consisting of functionally graded material coating and orthotropic substrate

This paper presents a semi-analytical method to investigate the frictionless contact mechanics between a functionally graded material (FGM) coating and an orthotropic substrate when the system is indented by a rigid flat punch. From the bottom, the orthotropic substrate is completely bonded to the rigid foundation. The body force of the orthotropic substrate is ignored in the solution, while the body force of the FGM coating is considered. An exponential function is used to define the smooth variation of the shear modulus and density of the FGM coating, and the variation of Poisson’s ratio is assumed to be negligible. The partial differential equation system for the FGM coating and the orthotropic substrate is solved analytically through Fourier transformations. After applying boundary and interface continuity conditions to the mixed boundary value problem, the contact problem is reduced to a singular integral equation. The Gauss–Chebyshev integration method is then used to convert the singular integral equation into a system of linear equations, which are solved using an appropriate iterative algorithm to calculate the contact stress under the rigid flat punch. The parametric analyses presented here demonstrate the effects of normalized punch length, material inhomogeneity, dimensionless press force, and orthotropic material type on contact stresses at interfaces, critical load factor, and initial separation distance between FGM coating and orthotropic substrate. The developed solution procedures are verified through the comparisons made to the results available in the literature. The solution methodology and numerical results presented in this paper can provide some useful guidelines for improving the design of multibody indentation systems using FGMs and anisotropic materials.

A penny-shaped crack in a piezoelectric layer sandwiched between two elastic layers with the electrical saturation and mechanical yielding zones

This paper deals with an internal penny-shaped crack in a piezoelectric layer sandwiched by two outer elastic layers. It is assumed that the crack faces are electrically impermeable, and the thin annular electrical saturation and mechanical yielding zones appear around the crack front under the uniform distributed normal stress and electric displacement. Three different cases, i.e., the region of electrical saturation is longer, shorter than, or equal to the domain of mechanical yielding are respectively considered in this article. For such an axisymmetric penny-shaped crack problem, we simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by employing Hankel transform technique and the Copson method. The problem is solved in the real domain by constructing real fundamental solutions and the numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization. When the outer elastic layers vanish and the piezoelectric layer-thickness approaches infinity, the relation between the nonlinear zone-lengths and the loadings obtained in this paper reduces to the analytical expression in literature. The influence of the electromechanical loads, the thicknesses of the piezoelectric layer and the elastic layer, and the material parameter ratios on the nonlinear zones is discussed in details. The results of paper are believed to have some guiding references for the design of the piezoelectric sandwich plate.

A new elliptic mixed boundary value problem with (p,q)-Laplacian and Clarke subdifferential: Existence, comparison and convergence results

The goal of this paper is to investigate a new class of elliptic mixed boundary value problems involving a nonlinear and nonhomogeneous partial differential operator [Formula: see text]-Laplacian, and a multivalued term represented by Clarke’s generalized gradient. First, we apply a surjectivity result for multivalued pseudomonotone operators to examine the existence of weak solutions under mild hypotheses. Then, a comparison theorem is delivered, and a convergence result, which reveals the asymptotic behavior of solution when the parameter (heat transfer coefficient) tends to infinity, is obtained. Finally, we establish a continuous dependence result of solution to the boundary value problem on the data.

Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to H−1(Ω) or H˜−1(Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials, and hence the unique solvability of BDIEs. The equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

Abstract In this article, we study the time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations and provide an in-depth well-posedness analysis. This is a continuation of our work, Krug et al. (2021) in that we now consider mixed two-point boundary value problems. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.

Resistance scaling on 4N-carpets

Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) &gt; 1 {\rho=\rho(N)&gt;1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

Two‐operator boundary‐domain integral equation systems for variable‐coefficient mixed Boundary Value Problem in 2D

In this paper, the mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second‐order scalar elliptic differential equation with variable coefficients in a bounded two‐dimensional domain is considered. The two‐operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of two‐operator boundary‐domain integral equations (BDIEs). Although the theory of two‐operator BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix‐based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two‐operator BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the appropriate Sobolev spaces.

Analytical Solution of Fractional Oldroyd-B Fluid via Fluctuating Duct

This investigation focuses on the mixed initial boundary value problem with Caputo fractional derivatives. The studied pour an incompressible fractionalized Oldroyd-B fluid prompted by fluctuating rectangular tube. The explicit expression of the velocity field and shear stresses for the fractional model are obtained by utilizing the integral transforms, i.e., double finite Fourier sine transform and Laplace transform. Furthermore, the confirmation of the analytical solutions is also analyzed by utilizing the Tzou’s and Stehfest’s algorithms in the tabular form. In limited cases, ordinary Oldroyd-B fluid similar solutions and classical Maxwell and fractional Maxwell fluid are derived. The flow field’s graphs with the influences of relevant parameters are also mentioned.

On Filtration in a Rectangular Interchange with a particularly Unpermatable Vertical wall in the Evaporation

We consider a plane steady-state filtration in a rectangular bridge with a partially impermeable vertical wall in the presence of evaporation from a free surface of groundwater. To study the effect of evaporation, a mixed multiparametric boundary-value problem of the theory of analytic functions is formulated and using the method of P. Y. Polubarinova-Kochina. Based on the proposed model, an algorithm is developed to calculate the dependence of efficiency and productivity of hydrodynamic analysis.

Mechanics of multilayered or functionally graded material pressed onto an imperfect rigid base with a circular opening by localized surface loading

Loaded materials constrained by imperfect boundary can be found in many engineering scenarios, including blanking process, contact pressure sensor and soil/rock foundation subjected to underground mining. This paper examines the surface loading problem of a multilayered or functionally graded material (FGM) resting on an imperfect rigid smooth base weakened by a circular opening. The surface loading is axisymmetric and coaxial with the circular opening. This mixed boundary value problem is analogous to a Mode-I penny-shaped crack in multilayered elastic medium. With the aid of the GKS-based method for N-layered elasticity, the problem is formulated as a system of dual integral equations and then reduced to a Fredholm integral equation of the second kind. Solutions of the SIF, COD and full elastic field are expressed in terms of the solution of the integral equation. Numerical results obtained from the present solutions are compared with the those from finite element method. Using the new solutions, numerical studies are also conducted for four examples, including a homogenous layer, a multilayered material, a material with FGM coating and an asphalt pavement structure with FGM properties. These numerical results demonstrate that (1) a small size of surface loading can result in high SIF value at the crack tip; (2) placing the hard layer as the top surface is most effective for reducing ground settlement; (3) the gradation in Poisson’s ratio can have non-negligible effect on the SIF for thin FGM coating; (4) effect of the circular opening to the ground settlement can be significant if its radius is comparable to the total thickness of the upper layers.

Dynamic impedances of ring disks buried in arbitrary depths

This paper addresses the vertical, horizontal and rocking vibrations of a rigid ring disk, with especially small radial wall-thicknesses, embedded in any depth of a fully saturated poroelastic half-space. The mixed boundary value problem is expressed in terms of dual integral equations that are solved semi-analytically with all boundary conditions satisfied. Numerical results of quasi-static stiffness, dynamic impedances (including dynamic stiffness and radiation damping ratio) are obtained. The influence of Poisson’s ratio, soil permeability, disk embedded depth and excitation frequency on the stiffness and damping ratio of the ring disk is illustrated. It is found that for a ring disk, although its thickness-to-radius ratio is only about 2%-4%, its vertical, horizontal and rocking impedances are around 70% of the corresponding values of a circular disk. Besides, the vertical, horizontal and rocking stiffness of a deeply embedded ring disk is all about 2.0 times that of the surface ring disk. Simplified equations for the initial base stiffness of pipe piles/caissons are obtained. The results can be combined with Winkler-based models to address thin-walled pile/caisson-soil interaction problems in the field of offshore wind turbines or bridges.

Solvability Analysis of a Mixed Boundary Value Problem for Stationary Magnetohydrodynamic Equations of a Viscous Incompressible Fluid

We investigate the boundary value problem for steady-state magnetohydrodynamic (MHD) equations with inhomogeneous mixed boundary conditions for a velocity vector, given the tangential component of a magnetic field. The problem represents the flow of electrically conducting viscous fluid in a 3D-bounded domain, which has the boundary comprising several parts with different physical properties. The global solvability of the boundary value problem is proved, a priori estimates of the solutions are obtained, and the sufficient conditions on data, which guarantee a solution’s local uniqueness, are determined.

Modelling and asymptotic analysis of the concentration difference in a nanoregion between an influx andoutflux diffusion acrossnarrow windows

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace’s equation using Green’s function techniques and second-order asymptotic analysis, and depends on the influx amplitude, the diffusion properties, as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread.