Explicit solution of the boundary value problems of the theory of elasticity for solids with double porosity

In this paper the Aifantis' theory of elasticity for solids with double porosity is considered and the 2D boundary value problem (BVP) of static is investigated. The uniqueness theorem of the internal BVP is proved. The explicit solution the BVP is constructed in the form of absolutely and uniform convergent series for a circle. The numerical solution of the BVP for a circle is obtained. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Boundary Value Problem for Second Order Nonlinear Difference Equations on the Semi-infinite Interval

In this paper, we consider a boundary value problem (BVP) for nonlinear difference equations on the discrete semi-axis in which the left-hand side being a second order linear difference expression belongs to the so-called Weyl-Hamburger limit-circle case. The BVP is considered in the Hilbert space l 2 and is formed via boundary conditions at a starting point and at infinity. Existence and uniqueness results for solutions of the considered BVP are established.     

Explicit solution of the boundary value problem of the theory of thermoelasticity with microtemperatures for elastic circle

In this paper we solve explicitly, by means of absolutely and uniformly convergent series, the 2D boundary value problem (BVP) of statics of the linear theory of thermoelasticity with microtemperatures for an elastic circle. The uniqueness theorem of the internal BVP is proved. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary-domain integral equations for the diffusion equation in inhomogeneous media based on a new family of parametrices

ABSTRACT

A mixed boundary value problem (BVP) for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based boundary-domain integral equations (BDIEs). We use a parametrix different from the one employed by Mikhailov [Localized boundary-domain integral formulations for problems with variable coefficients. Eng Anal Bound Elem. 2002;26:681–690], Mikhailov and Portillo [A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient. In: Paul Harris, editor. Proceedings of the 10th UK conference on boundary integral methods. Brighton: Brighton University Press; 2015. p. 76–84] and Chkadua, Mikhailov, Natroshvili [Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I: equivalence and invertibility. J Integral Eqs Appl. 2009;21:499–543]. 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.     

POSITIVE SOLUTIONS TO A SEMIPOSITONE SINGULAR NEUMANN BOUNDARY VALUE PROBLEM

A semipositone singular boundary value problem (BVP for short) is discussed in this paper. By Krasnaselskii's fixed point theorem in cones,we derive suffcient conditions,which guarantee that the semipositone BVP has at least one positive solution.     

二阶常微分方程及Thomas－Fermi方程的边值问题

本文利用上、下解方法推广了Ｍ．Ｌｅｅｓ关于半线性边值问题的结果，给出了另一类边值问题解的存在唯一性的一个充分条件；并利用所得的结果证明了Ｔｈｏｍａｓ－Ｆｅｒｍｉ方程的边值问题解的存在唯一性，从而推广了Ｃ．ＤＬｕｎｉｎｇ的结果．     

Iterative method for solving the Neumann boundary value problem for biharmonic type equation

The solution of boundary value problems (BVP) for fourth order differential equations by their reduction to BVP for second order equations, with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the Neumann BVP for biharmonic type equation. The convergence rate of the method is proved and some numerical experiments are performed for testing it in dependence on the choice of an iterative parameter.     

Existence of three solutions to a second-order nonhomogeneous multipoint boundary-value problem

This paper is concerned with a nonhomogeneous multipoint boundary-value problem (BVP) of a second-order differential equation with one-dimensional p-Laplacian. Using multiple fixed-point theorems, new sufficient conditions to guarantee the existence of at least three solutions of this BVP are established. An example is presented to illustrate the main results. The first emphasis of this paper is to show that the approach to get three positive solutions of a BVP by using multiple fixed-point theorems can be extended to treat nonhomogeneous BVPs. The second emphasis is put on the nonlinear term f involved with the first-order delta operator.     

Elliptic boundary value problems with Gaussian white noise loads

Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. The challenging cases of Neumann and Robin BVPs are considered.The main obstacle for usual variational methods is the irregularity of the load. In particular, the Neumann boundary values are not well-defined.In this work, the BVP is formulated by replacing the continuity of boundary trace mappings with measurability. Instead of variational methods alone, the novel BVP derives also from Cameron–Martin space techniques.The new BVP returns the study of irregular white noise to the study of $L^2$-loads.     

Steady vibrations problem in the theory of viscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with voids is considered. The uniqueness and existence theorems for internal boundary value problem (BVP) of steady vibrations are proved by means of the potential method (boundary integral method) and the theory of singular integral equations. The application of this method to the 3D BVP of the considered theory reduces this problem to 2D singular integral equation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple Positive Solutions for an n-Point Nonhomogeneous Boundary Value Problems in Banach Spaces

This paper is concerned with the solvability of an n-point nonhomogeneous boundary value problem (BVP) in Banach spaces. By using the fixed point theorem of strict-set-contractions, some sufficient conditions for the existence of at least one or two positive solutions to the n-point nonhomogeneous BVP in Banach spaces are obtained. As an application, we give one example to demonstrate our results.     

The Boundary Value Problem for the Nonlinear Shallow Water Equations

We propose an approximate analytical solution of the boundary value problem (BVP) for the nonlinear shallow waters equations. Our work, based on the Carrier and Greenspan [ 1 ] hodograph transformation, focuses on the propagation of nonlinear nonbreaking waves over a uniformly plane beach. Available results are briefly discussed with specific emphasis on the comparison between the Initial Value Problem and the BVP; the latter more completely representing the physical phenomenon of wave propagation on a beach. The solution of the BVP is achieved through a perturbation approach solely using the assumption of small waves incoming at the seaward boundary of the domain. The most significant results, i.e., the shoreline position estimation, the actual wave height and velocity at the seaward boundary, the reflected wave height and velocity at the seaward boundary are given for three specific input waves and compared with available solutions.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Existence and uniqueness of solutions to a nonlinear elliptic boundary value problem on time scales

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial dynamic equations on the time scale rectangles. Some explicit conditions are established that ensure existence and uniqueness of solution to the BVP under consideration.     

Mixed boundary value problem for some pairs of metaanalytic function and analytic function

In this article, some mixed boundary value problems (BVPs) on the unit circumference for some pairs of a metaanalytic function and an analytic function are discussed. Using the relationship between metaanalytic function and polyanalytic function, the expression of solution and the condition of solvability for the problem are obtained by reducing the problem to an equivalent system of a Haseman BVP for analytic function and a Hilbert BVP for analytic function. Copyright © 2008 John Wiley & Sons, Ltd.     

Outer solution for elastic torsion by the method of boundary layer residual states

By the method of boundary layer residual state (BLRS), it is possible to specify the unknown parameters in the general form of the outer asymptotic solution of the governing differential equations for linear boundary value problems (BVP) without any reference to the inner asymptotic solutions of the same problem and the matching procedure. The method accomplishes this task by rationally assigning a portion of the prescribed boundary data to the outer solution. Specifically, the method requires certain weighted averages of the outer solution to be equal to the same averages of the data over the (localized) boundary where the data is prescribed. These weighted averages are consequences of a reciprocity relation inherent in the BVP and the stipulation that the difference between the outer solution and the exact solution (called the residual solution) of the BVP be a boundary layer phenomenon.¶The weighted average requirements are only necessary conditions for the residual state to be a boundary layer. Unfortunately, there are generally countably infinite number of (2) states, many more than the available degrees of freedom in the outer solution to satisfy them. We must show that there is no over-determination or non-uniqueness of the outer asymptotic solution, the abundance of necessary conditions notwithstanding. The present note describes an approach to assuring a well-specified outer solution (up to the expected accuracy) by way of the problem of Saint-Venant torsion. The same approach also also applies to other linear BVP, deducing the appropriate outer solution whenever the determination of the relevant inner solutions is not practical.     

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER MIXED-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS

We use the topological degree method to deal with the generalized Sturm-Liouville boundary value problem (BVP) for second order mixed-type functional differential equation x(t)=f(t,xt,xt), 0≤t≤T. Existence principle and theorem for solutions of the BVP are obtained.     

Two-layer muti-parameterized Schwarz Alternating Method

The convergence rate of a numerical procedure based on Schwarz Alternating Method (SAM) for solving elliptic boundary value problems (BVP’s) depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that theRobin condition(mixed interface condition), controlled by a parameter, can optimize SAM’s convergence rate. Since the convergence rate is very sensitive to the parameter, Tang[17] suggested another interface condition calledover- determined interface condition. Based on the over-determined interface condition, we formulate thetwo-layer multi-parameterized SAM. For the SAM and the one-dimensional elliptic model BVP’s, we determine analytically the optimal values of the parameters. For the two-dimensional elliptic BVP’s, we also formulate the two-layer multiparameterized SAM and suggest a choice of multi-parameter to produce good convergence rate.     

Coercive boundary value problems for regular degenerate differential-operator equations

This study focuses on nonlocal boundary value problems (BVP) for degenerate elliptic differential-operator equations (DOE), that are defined in Banach-valued function spaces, where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients. Several conditions obtained, that guarantee the maximal L_p regularity and Fredholmness. These results are also applied to nonlocal BVP for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties.     

Existence of solutions to second-order nonlinear discrete elliptic equations

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial difference equations on finite lattice domains. Some conditions are established that ensure existence and uniqueness of solutions to the BVP under consideration.