Some basic boundary value problems of the plane thermoelasticity with microtemperatures

The present paper is devoted to the two‐dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the LaTeX2? mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.     

Boundary value problems of steady vibrations in the theory of thermoelasticity with microtemperatures

In this paper the linear theory of steady vibrations of thermoelasticity with microtemperatures for isotropic solids with microstructure is considered. The uniqueness and existence theorems of solutions of the internal and external second boundary value problems (BVPs) by means of the boundary integral method (potential method) and the theory of singular integral equations are proved. The existence of eigenfrequencies of the internal homogeneous BVP of steady vibrations is studied. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

ON THE PLANE STRESS BOUNDARY VALUE PROBLEM OF QUASI-STATIC LINEAR THERMOELASTICITY

1.IntroductionTheinvestigationoftheinfluenceofthetemperaturedistributioninanelasticbodyontheinternalstressesandstrainsisamainillterestofthetheoryofthermoelasticity.Typically,theinitialtemperaturedistributionandtheboundaryvalueoftemperatureinanintervaloftimeaswellasacertaindisplacementorstressorcompoundboundaryvalueconditionsaregiven,alltheaboveareutilizedsoastodeterminethestresses,strainsandthetemperaturedistributionintheelasticbody,inanintervaloftime.Werestrictouxattentiontoelasticequilibrium…     

The third boundary value problem for the system of equations of linear thermoelasticity

The existence and uniqueness of a generalized solution are proved for the third boundary value problem for a system of differential equations in the linear thermoelasticity theory. The displacement vector and temperature of the medium are the sought functions. The existence of a solution is proved, using the contraction mapping principle. In obtaining some necessary a priori estimates, the expansion of the solution was used in a series of generalized eigenfunctions of the Laplace operator.     

A boundary value problem for fourth-order elastic beam equations

Multiplicity results for a fourth-order nonlinear boundary value problem are presented. The proof of the main result is based on the critical point theory.     

Explicit solution of higher order boundary value regular differential systems avoiding the increase of the problem dimension

In this paper we obtain existence conditions and an expression of the general solution of higher order regular boundary value differential systems. The approach is based on the concept of co-solution of certain algebraic matrix equation and unlike to the classical approach we avoid the increase of the problem dimension derived from the consideration of an extended first order system.     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：
x″（t）＋f（t,x（t））=0,0〈t〈1,
x′（0）=0,x（1）＋δx′（η）=0,
where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.     

On the solutions of a boundary value problem of linear thermoelasticity system with nonlocal conditions

This paper studies a boundary value problem with nonlocal conditions for a coupled system of linear thermoelasticity in one-dimensional case. Using an a priori estimate, we prove the uniqueness of the solution. Also, some explicit solutions are obtained by using the separation method.     

Explicit solutions of an integrable boundary value problem for the two-dimensional toda lattice

We obtain an explicit solution of the integrable boundary value problem for the two-dimensional Toda lattice using the inverse scattering method. We interpret the integrability property in terms of the corresponding linear problem, the Gel’fand-Levitan-Marchenko equation, and the dressing procedure. The simplest initial solutions of the boundary value problem become new nontrivial solutions after the dressing procedure is applied.     

On a boundary value problem of the theory of oscillations with parameter-dependent boundary conditions

A homogeneous second order differential equation with homogeneous boundary conditions dependent on the parameter, is investigated. Such an equation is obtained in the course of solution of the problem of characteristic oscillations of an ideal incompressible fluid in an elastic vessel, when the method of separation of variables is used. We prove the completeness of the system of eigenfunctions of our boundary value problem and we derive the expansion of an arbitrary, piecewise-continuous function into a series in terms of these eigenfunctions.     

Quadratic spline solution for boundary value problem of fractional order

Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.     

On a mixed boundary value problem for an infinite elastic cone

Zusammenfassung Die Verfasser haben ein gemischtes Randwertproblem für einen unendlichen elastischen Kegel mit Spitzenwinkel2 auf die Lösung einer endlichen Wiener-Hopf-Gleichung zurückgeführt. Der wesentliche Schritt besteht darin, eine FunktionK(s), die in einem Streifen regulär ist, als ein Produkt von Faktoren darzustellen, die auf übereinanderliegenden Halbebenen regulär sind. In dem hier behandelten Fall für willkürliches ist diese Methode besonders schwierig. In dem Sonderfalle des elastischen Halbraumes =/2 können jedoch bekannte Resultate erlangt werden. Das deutet darauf hin, dass die Methode auf das Kegelproblem anwendbar ist, vorausgesetzt dass die erforderliche Faktorzerlegung durchgeführt werden kann.

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This work was supported in part by the Office of Ordnance Research under contract No. DA-11-022-ORD-2195.     

Smooth spline approximations for the solution of a boundary value problem with engineering applications

Methods of order 2, and 4 are developed for the continuous approximation of the solution of a two-point boundary value problem involving a fourth order linear differential equation via quintic and sextic spline functions. In three typical numerical examples, the results are briefly summarized to demonstrate the practical usefulness of the methods.     

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is 0 such that the essential spectral radius of is smaller than || (for example, if G R ³ is a domain with a piecewise smooth boundary and the restriction of the Newtonian potential on G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition for which (G) = 0.