Degenerate Fredholm boundary-value problems

We obtain a criterion for the existence of solutions of degenerate inhomogeneous Fredholm boundary-value problems for a system of ordinary differential equations under the assumption that the degenerate system of differential equations can be reduced to the central canonical form. The results are illustrated by examples. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 303–312, July–September, 2007.     

Stability of the solutions of inverse mixed boundary-value problems such as those encountered in filtration theory

The stability of the solutions of a group of problems regarding the construction of an underground contour on the basis of the distribution of filtration pressure is considered for bounded and unbounded depths of the water-permeable layer and also for the case of an inclined water-resistant boundary.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 54–63, March–April, 1977.In conclusion, the author wishes to thank L. A. Aksent'ev for constant interest in the work and valuable comments.     

Stress boundary-value problems for infinite wedges

Summary Stress boundary-value problem for infinite wedges is formulated as a well-posed one by imposing a certain regularity requirements at infinity and at the vertex. It is then solved with complete rigor by means of Mellin transforms and the necessity of the regularity condition to assure the uniqueness of solution is further explored. By establishing the existence as well as the rate of its convergence of the complex inversion integral, a numerical scheme has been proposed and numerical examples have been worked out as illustrations.

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Zusammenfassung Das Spannungs-Randwertproblem für unendliche Keile gilt als korrekt gestellt, wenn gewisse Regularitäts-bedingungen im Unendlichen und an der Spitze erfüllt sind. Das Problem wird dann mit Hilfe von Mellin-Transformationen vollstandig gelöst. Ferner wird die Beziehung zwischen Regularitätsbedingungen und Ein-dentigkeit der Lösung untersucht. Eixstenz des komplexen Inversionsintegrals und seine Konvergenzrate führen zu einem numerischen Schema, das an Hand von numerischen Beispielen erläutert wird.

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Acknowledgement is made to the grant, AFOSR-444-64, under the sponsorship of the Joint Services Advisory Group, through AFOSR, ARO, and ONR. The authors also gratefully acknowledge the assistance given by Dr. J. W. Cell, Dr. I. N. Sneddon, and Dr. George Irwin in suggesting and discussing this problem.     

Airfoil design with external flow suction

A method is proposed for the numerical-analytic design of a high-lift airfoil with slit air suction from the external flow within the framework of ideal fluid theory. The suction slit is represented by a channel with constant wall velocity. Examples of the design of wing profiles with non-detached flows having lift coefficientsC _y =2.68 and 4 and maximum relative velocities over the profile [_max|/=2 and 2.2 are given. A scheme for an aircraft of the flying wing type with internally located engines and pay load is briefly discussed.Kazan', Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 23–28, November–December, 1996.     

Principle of correspondence of static boundary-value problems of nonlinear viscoelasticity with aging to boundary-value problems of the theory of elasticity

The principle of correspondence of boundary-value problems of the nonlinear nonuniform anisotropic theory of viscoelasticity to boundary-value problems of the theory of elasticity is formulated. The correspondence is established by means of integral transforms with previously unknown kernels. A class of viscoelastic materials for which these transforms can be reduced to boundary-value problems of fictitious elasticity is determined. Samara State Aerospace University, Samara 443086. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 155–161, July–August, 1998.     

Axisymmetric boundary-value problems in a piezoelectric medium

Based on the general solution of three-dimensional problems in piezoelectric medium, with the method of Green's functins^[2], axisymmetric boundary-value problems are discussed. The purpose of this research is for analyzing the effective on mechanics and electricity of the piezoelectric ceramics caused by voids and inclusions. The displacement, traction and electric Green's functions corresponding to circular ring loads acting in the interior of a piezoelectric ceramic are obtained. A cylindrical coordinate system is employed and Hankel transform are applied with respect to radial coordinates. Explicit solutions for Green's functions are presented in terms of infinite integrals of Lipshitz-Hankel type. By solving a traction boundary-value problem, the solution scheme is illustrated. Supported by the National Natural Science Foundation of China and the Foundation of the Open Laboratory of Solid Mechanics.     

Closed solutions of boundary-value problems of coupled thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli forwhich it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.     

On the parametrization of three-point nonlinear boundary-value problems

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 395–413, July–September, 2004.     

Solution of non-linear inverse heat conduction problems using the method of lines

 Two space marching methods for solving the one-dimensional nonlinear inverse heat conduction problems are presented. The temperature-dependent thermal properties and the boundary condition on the accessible part of the boundary of the body are known. Additional temperature measurements in time are taken with a sensor located in an arbitrary position within the solid, and the objective is to determine the surface temperature and heat flux on the remaining part of the unspecified boundary. The methods have the advantage that time derivatives are not replaced by finite differences and the good accuracy of the method results from an appropriate approximation of the first time derivative using smoothing polynomials. The extension of the first method presented in this study to higher dimensions inverse heat conduction problems is straightforward. Received on 3 May 1999     

Solution of initial–boundary-value problems of electroelasticity revisited

The Hamilton–Ostrogradsky principle is used to substantiate the statement of initial–boundary-value problems of electroelasticity. The free vibrations of a piezoceramic layer are used as an example to illustrate some features of solving nonstationary problems of electroelasticity Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 62–69, December 2008.     

Nonlinear Noetherian boundary-value problems in the critical case

We establish necessary and sufficient conditions for the existence of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the critical case.     

Two-dimensional dynamic boundary-value problems for curvilinear thermoviscoelastic bodies

A new numerical-analytical method is proposed and demonstrated using an example of dynamic problems of a thermoviscoelastic body. In the general formulation, the thermoviscoelastic problem is split into three simpler problems. In the first problem, boundary functions that should satisfy only boundary conditions are determined. The second problem with homogeneous boundary conditions and inhomogeneous initial conditions is reduced to an eigenvalue problem by introducing special variables and separating time. This problem is solved by organizing integral superpositions with respect to the angular parameter. A linear algebraic system is obtained as a result of satisfaction of the boundary conditions at points that partition the curvilinear boundary of the body into small segments. After the eigenfunctions and eigenvalues are determined, the third problem with homogeneous boundary and initial conditions is solved by spectral decomposition of unknown functions and inhomogeneous terms in a coupled system of ordinary differential equations.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 158–169, March–April, 2005.     

Conditions for bifurcation of solutions of degenerate boundary-value problems

We obtain conditions for the bifurcation of solutions of linear degenerate Noetherian boundary-value problems with small parameter under the assumption that the unperturbed degenerate differential system can be reduced to the central canonical form.     

Critical case for singularly perturbed Noether boundary-value problems

We construct an asymptotic expansion of a solution for singularly perturbed linear systems of ordinary differential equations of the Noether type in the critical case. We successively determine all terms of the asymptotic expansion by the method of boundary functions and pseudoinverse matrices. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 45–54, January–March, 2007.