On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.     

On an integral equation approach for the exterior Robin problem for the Helmholtz equation

A boundary integral equation for the exterior Robin problem for Helmholtz's equation is analyzed in this paper. This integral operator is not compact. A proof based on a suitable regularization of this integral operator and the Fredholm alternative for the regularized compact operator was given by other authors. In this paper, we will give a direct existence and uniqueness proof for the boundary non-compact integral equation in the space settings C^1,λ(S) and C^0,λ(S), where S is a closed bounded smooth surface.     

On an integral equation of the third type

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 3, pp. 308–314, July–September, 1995.     

On an integral equation for the two-dimensional exterior Stokes problem

This paper is concerned with solving the two-dimensional exterior Stokes problem by a combination of double- and simple-layer potentials. This leads to a boundary integral equation, which is shown to be uniquely solvable. The present approach is a variation of a technique due to Leis, Brakhage and Werner, and Panich for treating the scalar Helmholtz equation. However, it contains extra parameter which may be chosen in an optimal manner from a computational point of view. Some numerical results in this connection are also included.     

On an integral equation of first kind

We study an integral equation of first kind that arises in the study of inverse problems for nonlinear differential equations. The peculiarity of the equation is that the argument of the unknown function is a given function of two variables. We obtain conditions for this integral equation to have a unique solution. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 54–58.     

On certain additions to the theory of one singular integral equation

Certain additions are given to the theory of a singular integral equation /1/ encountered in some problems of potential theory and in a correspondingly complicated form in two-dimensional elasticity theory.     

On existence and approximate solutions for a nonlinear integral equation arising in the theory of creep in metals

Existence of a solution to a nonlinear integral equation arising in the theory of creep in symmetric pressure vessels is established by means of a recursively defined sequence of functions. It is shown that this sequence converges monotonically and uniformly to the solution.

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Zusammenfassung Das Problem des Kriechens in symmetrischen Druckgefässen führt auf eine nichtlineare Integralgleichung. Zum Beweis der Existenz einer Lösung dieser Gleichung wird rekursiv eine Folge von Näherungslösungen konstruiert. Es wird gezeigt, dass diese Folge monoton und gleichmässig nach der Lösung der Integralgleichung konvergiert.

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This research was supported by the National Science Foundation under Grant MPS-75-07450.     

On a nonlinear theory of thin rods

In this paper, a nonlinear theory for a straight rod is presented from the general theory of the three-dimensional deformable-body in the Cartesian coordinate frame. A set of nonlinear strains is presented, and the stretch on central curve exactly satisfies the deformation geometrical relations. The relations between the Euler angles and deformation are given from the curvatures and torsion curvatures of the central curves, which can easily explain the existing theories of rods and beams. Full nonlinear equations of motion for a nonlinear rod are developed via the vector form. Such a treatise is different from the traditional treatises of nonlinear rods, based on the Cosserat’s theory (e.g., Cosserat and Cosserat [1] in 1896) or the Kirchhoff assumptions (e.g., Kirchhoff [18] in 1859; Love [3] in 1944). This paper extends the ideas of Galerkin [4] in 1915. The nonlinear theory of thin rods can reduce to the existing theories for thin rods and beams, such ideas presented in this paper can be applied for development of the nonlinear theory for plates and shells as well.     

Experimental method of solving an integral equation of the theory of viscoelasticity

A solution of an integral equation of the theory of viscoelasticity is proposed. This solution, based on a creep test on a polymer specimen with an included elastic element, improves the convergence of the method of approximations.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 584–587, July–August, 1969.     

Justification of the direct projection method for solving an integral equation of the potential theory

We justify the direct projection method for solving an integral equation with a logarithmic singularity in the kernel. The equation is treated as a mapping of one Hilbert space into another Hilbert space. The spaces are chosen from conditions ensuring the solution of a broad class of mathematical modeling problems with the use of a simple layer potential. The idea of the projection method is to choose finite-dimensional subspaces into which the exact solution and the right-hand side of the equation are projected. In this case, the problem of finding an approximate solution does not require computing the convolution of kernels. We prove an estimate for the solution error in the norm of the original operator equation.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

On a problem of integral geometry related to the Dirichlet problem for an ultrahyperbolic equation

We study the nontrivial solvability of the homogeneous problem of integral geometry on the family of spheres formed by sections of the (n − 1)-dimensional unit sphere by hyperplanes perpendicular to the generators of a given cone. By using a relationship between this problem and the Dirichlet problem for an ultrahyperbolic equation in a ball and a criterion for the failure of uniqueness of the solution of the latter problem in terms of zeros of classical Jacobi polynomials, we obtain a criterion for the uniqueness of the solution of the former problem.     

Convergent perturbation theory for studying phase transitions

Theoretical and Mathematical Physics - We propose a method for constructing a perturbation theory with a finite radius of convergence for a rather wide class of quantum field models traditionally...     

A finite element method for the solution of a potential theory integral equation

This paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H^1/2(Γ), f_rd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.