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We consider a class of two-dimensional integral equations in ℝ2 with monotone nonlinear terms and prove the existence of a nonnegative bounded solution. We study the asymptotic behavior of this solution with respect to each variable. The result is illustrated by an example.

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

We determine the unsteady temperature field depending on the radial coordinate and the thermoelastic state caused in a two-layer hollow thermally sensitive cylinder both by this field and the action of external force loads. The nonlinear boundary-value problem of heat conduction with discontinuous coefficients is reduced by the integro-interpolation method to the Cauchy problem for a system of ordinary differential equations, which is solved numerically. The stressed state is determined from the integral equations of the second kind and integral conditions obtained as a result of direct integration of the problem of quasistatic thermoelasticity in stresses. We study the influence of the temperature dependence of the thermal and mechanical characteristics of the layers of materials on the values and character of distributions of temperature and stresses caused by this dependence in a two-layer cylinder.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

We consider the Dirichlet problem for the inhomogeneous Lame system in the plane with constant leading coefficients in a (finite or infinite) domain, bounded by a Lyapunov contour. For domains of finite diameter we use the weighted Holder class of functions with power behavior at infinity. We propose an equivalent reduction of the problem to a system of Fredholm integral equations in the domain and on the boundary contour.

Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient.

A dynamic inverse problem for a dynamical system that describes the propagation of waves in a Krein string is considered. The problem is reduced to an integral equation, and an important special case is considered where the string density is determined by a finite number of point masses distributed over the interval. An equation of Krein type, with the help of which the string density is restored, is derived. The approximation of constant density by point masses uniformly distributed over the interval and the effect of the appearance of a finite wave propagation velocity in the dynamical system are also studied.

We consider partial integral in the classes of functions with the mixed Lp– and supnorms or in the anisotropic Lebesgue spaces. For second kind Fredholm integral equations with partial integrals we construct solutions in the form of the operator Neumann series by the method of successive approximations.

We obtain Fredholm criteria for linear equations with multidimensional partial integrals in the space of continuous functions of three variables provided that the kernels are continuous vector-valued functions on a rectangle with the values in spaces of integrable functions. We establish the Fredholm criterion for equations with degenerate kernels and describe the scheme of studying the Fredholm properties of linear equations with partial integrals in spaces of continuous functions of at least four variables.

Within the framework of a refined model that takes into account transverse shear strains and inertial components, we construct a solution of the problem of steady-state vibration of an orthotropic doubly curved panel with cutouts of any shape, orientations, and location and arbitrary external boundary under mixed harmonic boundary conditions imposed on the outer boundary and the contours of cutouts. The solution is constructed by the indirect method of boundary elements with the help of a sequential approach to the representation of the Green functions. The obtained integral equations are solved by the collocation method.

Contrary to the opinion that has prevailed for the last several decades about the incorrectness of the inverse–MEEG problems (see, for example, the paper of D. Sheltraw and E. Coutsias in Journal of Applied Physics, 94, No. 8, 5307–5315 (2003)), in this note it is shown that this problem is absolutely well posed: it has a unique solution, but in a special class of functions (different from those considered by biophysicists). The solution has the form q = q0 + p0δ|∂Y, where q0 is an ordinary function defined in the domain of the region Y occupied by the brain, and p0δ|∂Y is a δ-function on the boundary of the domain Y with a certain density p0. Moreover, the operator of this problem realizes an isomorphism of the corresponding function spaces. This result was obtained due to the fact that: (1) Maxwell’s equations are taken as a basis; (2) a transition was made to the equations for the potentials of the magnetic and electric fields; (3) the theory of boundary value problems for elliptic pseudodifferential operators with an entire index of factorization is used. This allowed us to find the correct functional class of solutions of the corresponding integral equation of the first kind. Namely: the solution has a singular boundary layer in the form of a delta function (with some density) at the boundary of the domain. From the point of view of the MEEG problem, this means that the sought-for current dipoles are also concentrated in the cerebral cortex.

We consider a generalized Cauchy–Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given boundary functions.

We consider weakly nonlinear boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces, establish necessary and sufficient conditions for the existence of solutions of these problems, and construct convergent iterative procedures for the determination of solutions of these boundary-value problems.

We examine a generalized Cauchy–Riemann-type system whose coefficients have singularities, construct the resolvent of the corresponding integral equation, and find an integral representation of the general solution.

We propose an approach for the solution of a contact problem for an anisotropic half plane interacting with a plane smooth punch with regard for the contact of the crack faces. The stresses formed near the cracks in the anisotropic half plane are found on the basis of the method of integral equations. The kernels of the equations are constructed to guarantee the identical validity of the conditions imposed on the rectilinear boundary of the half plane, including the area under the punch. The influences of anisotropy and the contact of crack faces on the stress intensity factors are analyzed.

We developed a technique for the construction of solutions of axisymmetric problems of elasticity and thermoelasticity in stresses for a space and a half space whose elastic properties are arbitrary functions of the coordinate z. By using the direct integration method and the Hankel integral transformation, the problems are reduced to governing integral equations accompanied by a local boundary condition and an integral condition in the case of the half space. The solutions of the deduced equations are constructed in the explicit form by using the resolvent-kernel method.

In this research, we study the scattering of electromagnetic waves by a dielectric impediment in 2D geometry. The impediment is determined by an inhomogeneous component of the refractive index in the Helmholtz equation. It is assumed that the characteristic gauge of one of the two impediment sizes is much lesser than the length of waves generated by a monochromatic point source. Nevertheless, the structure of the impediment is taken into consideration in the process of calculating the scattered field. The scattered field is defined by a derived model integral equation the unique solvability of which is proved.

We apply the Laguerre transform with respect to time to a time-dependent boundary-value integral equation encountered in the solution of three-dimensional Dirichlet initial-boundary-value problems for the homogeneous wave equation with homogeneous initial conditions by using the retarded potential of single layer. The obtained system of boundary integral equations is reduced to a sequence of Fredholm integral equations of the first kind that differ solely by the recursively dependent right-hand sides. To find their numerical solution, we use the boundary-element method. We establish an asymptotic estimate of the error of numerical solution and present the results of numerical simulations aimed at finding the solutions of retarded-potential integral equations for model examples.

We consider a class of nonlinear convolution type integral equations with a noncompact Hammerstein operator on the half-line and on the whole line. Under certain conditions on nonlinearity, we prove the existence of a one-parameter family of positive solutions and study the asymptotic behavior of the solutions at infinity. The results are illustrated by examples of the equations under consideration.

We consider weakly perturbed Fredholm equations with degenerate kernels in Banach spaces and establish conditions for e = 0 to be a bifurcation point for the solutions of weakly perturbed operator equations X in Banach spaces. A convergent iterative scheme for finding solutions in the form of series $$ {\Sigma}_{i=-1}^{+\infty }{\varepsilon}^i{z}_i(t) $$ in powers of e is proposed.

We consider two types of nonclassical fracture mechanisms, namely, the fracture of cracked bodies with initial (residual) stresses acting along the crack planes and the fracture of solids under compression along parallel cracks. To investigate nonaxisymmetric and axisymmetric problems for infinite solids containing two parallel coaxial cracks or a periodic set of coaxial parallel cracks, we use a combined analytic-numerical method within the framework of the three-dimensional linearized mechanics of solids. The analysis involves the representation of stresses and displacements in the linearized theory via harmonic potential functions. With the use of the integral Fourier–Hankel transformations, the problems are reduced to the solution of Fredholm integral equations of the second kind. This approach allows us to investigate problems in a unified general form for compressible and noncompressible homogeneous isotropic or transversely isotropic elastic bodies with an arbitrary structure of the elastic potential, and the material specification of the model is carried out only in the stage of numerical analysis of the resolving equations obtained in the general form. The effects of initial stresses on the stress intensity factors are analyzed for highly elastic materials and layered composites (modeled as transversely isotropic elastic bodies). The “resonance-like” effects are revealed when compressive initial stresses reach the values corresponding to the local loss of stability of the material in the vicinity of cracks, which, according to the indicated combined method, allows one to determine the critical (limiting) load parameters under the conditions of compression of the body along the cracks. The conclusions concerning the dependences of the stress intensity factors and critical (limiting) parameters of compression on the geometric parameters of the problems are formulated as well as concerning the dependences on physical and mechanical characteristics of the materials.

For the solution of the one-dimensional problem of the theory of elasticity for a radially inhomogeneous solid cylinder, we use the method of reduction to a Volterra integral equation. By estimating the error of satisfying the governing integral equation, we establish a criterion of accuracy of the approximate solution of the problem.

The 2D problem of diffraction by a curved surface with ideal boundary conditions is considered in terms of the parabolic equation of diffraction theory. A boundary integral equation of Volterra type in Cartesian coordinates is introduced. Using the latter, the problem of diffraction by a parabola is analyzed. It is shown that the solution of this problem coincides with the asymptotic solution for the problem of diffraction by a cylinder obtained by V. A. Fock. The Efficiency of the numerical solution of the boundary integral equation is demonstrated for diffraction on a perturbation of a straight boundary.

We consider a hollow elastic cylinder of finite length subjected to the action of its own weight and an axisymmetric normal load applied to the upper base. The lower base of the cylinder is immovably fixed. The inner cylindrical surface is under the conditions of sliding restraint and its outer surface is immovably fixed. The problem is reduced to an integral equation of the first kind for normal stresses on the fixed lateral surface. We determine the character of singularity of the required function and propose an efficient algorithm for the solution of the obtained equation based on the expansion of the required function in a series in Jacobi polynomials. We present the results of calculations of normal stresses on the lateral surfaces of the cylinder, which show that, in the case of rigid fixing, the influence of the weight of the cylinder is much weaker than in the case of sliding restraint.

We consider a system of two independent alternating renewal processes with states 0 and 1 and an initial shift t0 of one process relative to the other one. An integral equation with respect to the expectation of time T (the first time when both processes have state 0) is derived. To derive this equation, we use the method of so-called minimal chains of overlapping 1-intervals. Such a chain generates some breaking semi-Markov process of intervals composing the interval (0, T ). A solution of the integral equation is obtained for the case where the lengths of 1-intervals have exponential distributions and lengths of 0-intervals have arbitrary distributions. For more general distributions of 1-intervals, the Monte Carlo method is applied when both processes are simulated numerically by a computer. A histogram for estimates of the expectation of T as a function of t0 is demonstrated. Bibliography: 4 titles.

We study entire solutions to the difference Schrodinger equation ψ(z + h) + ψ(z − h) + v(z)ψ(z) = Eψ(z), z ∊ C, where h > 0 and E ∊ C are parameters, and v is a trigonometric polynomial. We describe asymptotic behavior of the solutions as h tends to zero.

We establish conditions for the bifurcation of the solutions of weakly perturbed linear integral equations.

We determine the thermal stressed state of a heat-sensitive radially inhomogeneous hollow sphere for given constant loads acting on its surfaces and a known temperature field inside the sphere. The corresponding problem of thermoelasticity in stresses is reduced to the solution of the Fredholm integral equation of the second kind for the radial component of the stress tensor. The influence of temperature dependences of the characteristics of radially inhomogeneous material on the stresses and displacements is investigated.

We develop an algorithm for the determination of stresses in an anisotropic half plane with notches based on the method of boundary integral equations. Integral equations are solved numerically by the method of mechanical quadratures. The investigation of stresses near notches of different shapes is performed. We establish the characteristic features of the stress distributions depending on the shapes and sizes of notches and the mechanical characteristics of the material of the plates. The asymptotic relations are proposed for the determination of the stress concentration factors for narrowed notches of elliptic shapes.

We propose a method for the solution of the problem with homogeneous integral conditions for an inhomogeneous evolutionary equation with abstract operator in a Banach space H . The right-hand side of the evolutionary equation that belongs, for the fixed time variable, to a special subspace N ⊆ H can be represented as a Stieltjes integral with respect to a certain measure. The solution of this problem is also represented as a Stieltjes integral with respect to the same measure. We present examples of application of the method to the solution of the problem with integral conditions for the second-order partial differential equation in the time variable and, in general, an infinite-order partial differential equation in the space variable.

We establish necessary and sufficient conditions for solvability and determine the general form of the solutions of a linear integral equation (with square summable kernel) and a boundary-value problem for equations of this kind.

The method of mechanical quadratures is applied to linear Volterra integral equations with partial integrals among which there is an integral with an unbounded kernel. We construct numerical algorithms based on replacing integrals with quadrature formulas and prove the convergence.

Based on the combined application of the Stroh formalism and the theory of functions of complex variable, we deduce dual integral equations for a magnetoelectroelastic bimaterial. For the first time, we construct the integral representations of the Stroh complex potentials and the explicit expressions for all kernels in terms of the parameters and matrices of the applied formalism. This noticeably reduces the amount of computations required to get the governing equations of the boundary-element methods. The explicit formulas are obtained for the principal parts of the complex potentials. These formulas enable us to consider homogeneous magnetoelectromechanical loads applied at infinity. The obtained equations, together with previously developed models of thin deformable inclusions, are introduced in the computational algorithm of the boundary-element method with jump functions. The solution of test problems reveals high accuracy and efficiency of the proposed approach. Some solutions are presented for new problems posed for a magnetoelectroelastic bimaterial with thin inclusion.

UDC 539.3 We develop an algorithm for the determination of stresses in plates of complex shape containing elastic inclusions. The algorithm is based on the application of modified integral equations for which the boundary conditions for stresses are identically satisfied on the interfaces. The integral equations are solved numerically with the help of the method of mechanical quadratures. The investigation of stresses near inclusions in plates of different shape is performed. We establish characteristic features of the distribution of stresses depending on the shape of inclusions and elastic characteristics of the materials. The number of works devoted to the investigation of the stressed state of plates with inclusions is much smaller than the number of works dealing with homogeneous plates [2, 3, 5]. A general method for the investigation of these problems was proposed in [5] on the basis of the method of boundary integral equations (MBIE). Note that significant difficulties are encountered in attempts to use the MBIE for the investigation of plates of infinite sizes (in particular, for half planes and strips) and in the cases where inclusions are located near a hole or holes have complex shapes. This is why, in the literature, modified integral equations were proposed for special classes of these plates with holes and cracks for which the conditions imposed on the chosen boundary of the plate are identically satisfied [5, 7]. In the present work, we develop the general approach to the determination of the stress-strain state of plates with inclusions based on modified integral equations. To realize this approach, it suffices to construct a Green-type solution for the corresponding problem of the theory of elasticity.

We study the thermal stressed state of a finite cylindrical shell (with conditions of sliding restraint imposed on its end faces) caused by the difference of temperatures of the ambient medium on the front faces of the shell and the coordinate-dependent heat-transfer coefficients on these faces. We propose a method for the reduction of the boundary-value problem of heat conduction to a coupled system of Fredholm integral equations of the second kind and present the results of numerical analysis of the distributions of mean temperature and temperature moment and the values of parameters induced by these distributions, namely, the deflection, elongation, forces, and bending moments.

The acoustic problem of diffraction by two wedges with different wave velocities is studied. It is assumed that the wedges with parallel edges have a common part of the boundary and the second wedge is shifted with respect of the first one in the orthogonal to the edges direction along the common part of the boundary. The wave field is governed by the Helmholtz equations. On the polygonal boundary, separating these shifted wedges from the exterior, the Dirichlet boundary condition is satisfied. The wave field is excited by an infinite filamentary source, which is parallel to the edges. In these conditions, the problem is effectively two-dimensional. The Kontorovich–Lebedev transform is applied to separate the radial and angular variables and to reduce the problem at hand to integral equations of the second kind for so-called spectral functions. The kernel of the integral equations given in the form of an integral of the product of Macdonald functions is analytically transformed to a simplified expression. For the problem at hand, some reductions of the equations are also discussed for the limiting or degenerate values of parameters. Making use of an alternative integral representation of the Sommerfeld type, expressions for the scattering diagram are then given in terms of spectral functions. Bibliography: 24 titles.

The description of the processes of interaction of electromagnetic waves with imperfectly conducting gratings leads to the consideration of boundary-value problems for the Helmholtz equation with boundary conditions of the third kind. The original problem of scattering of polarized waves on the reflecting structures is reduced to a system of boundary integral equations. The procedure of derivation of the integral equations is based on the method of parametric representations of integral operators.

UDC 539.3 We consider an elastic cylinder with regard for its weight. The conditions of sliding fixing are imposed on the lower base of the cylinder, its upper base is subjected to the action of an axisymmetric normal load, and the lateral surface is fixed. The Hankel integral transform is used to reduce the problem to an integral equation of the first kind for normal stresses acting on the fixed cylindrical surface. After finding the singularities of the unknown function, the solution of the integral equation is sought in the form of a series in Jacobi polynomials. The results of numerical evaluation of the normal stresses on the fixed surface of the cylinder are obtained both with regard for its weight and by neglecting its weight. Elastic cylinders of finite length can be regarded as one of the most extensively used types of structural elements. This explains a great number of publications devoted to the investigation of their stressed states. A survey of the main achievements in this field of mechanics prior to 1963 can be found in [1]. A survey of more recent achievements (after 1963) is presented in [7]. Nevertheless, despite a large number of approximate numerical methods aimed at the solution of axisymmetric problems of the theory of elasticity for solid cylinders of finite length, the analytic methods capable of construction of solutions in the form of explicit functional dependences on the type of load and geometric parameters of the cylinder are still insufficient. Attempts to obtain the exact solutions were made in [3] and [4]. In these works, the solution is constructed in the form of expansions in trigonometric, hyperbolic, and Bessel functions, which leads to infinite systems of linear algebraic equations. In [6], the technique of p -analytic functions is used to obtain the exact solutions. However, the numerical realization of the proposed algorithms of solution is not presented in these works. Among recent works, we should especially mention [5] and [14], where the methods of superposition and expansion in Fourier–Bessel series are used not only to reduce the solution of the problems to infinite systems of linear algebraic equations but also to obtain numerical values of stresses in the cylinder by the method of improved reduction. In [17], the solution of the problem of stressed state of a circular cylinder loaded at the end faces and along the cylindrical surface is also constructed in the form of Fourier–Bessel series, which leads to the necessity of solving infinite systems of linear algebraic equations with an aim to find the coefficients of these representations. In [18] and [19], the authors present the numerical results obtained in the case of a cylinder loaded by axisymmetric normal loads imposed on the end faces or along the cylindrical surface. In all works, including the recent results, the action of the bulk forces in the form of the weight of the material is not taken into account, which leads to the solution of inhomogeneous Lame equations. As an exception, we can mention the work [20] devoted to the numerical solution of the problem of stressed state for a hollow cylinder under the action of the weight of its material. An analytic method capable of the solution of these problems is outlined in [11], where it is also shown how to construct the exact solution of the axisymmetric problem for a finite elastic cylinder in the case where the conditions of sliding fixing are imposed on the cylindrical surface.

We propose new procedures for the separation of singularities in the kernel and in the potential density for weakly singular Fredholm integral equations for the simple-layer potential in the case where the boundary surface has edges, ribs, and corner points. These procedures are based on the projection methods for the solution of these equations and finite-element approximations of the required potential density.

Some examples that show that the homogeneous Tricomi problem for a multidimensional hyperbolic-parabolic equation has the infinite set of nontrivial solutions are constructed.

In this paper, we study, in the class of H¨older functions, linear two-dimensional integral equations with coefficients t that have zeros in the interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under some assumption on their kernels are given.

The aim of this paper is to study, in the class of H¨older functions, a linear integral equation with coefficient having two simple zeros in the interval under consideration. Using the theory of singular integral equations, we give the necessary and sufficient conditions for the solvability of this equation under some assumptions on their kernel. Finding a solution is reduced to solving a regular integral equation of the second kind.

UDC 517.983 We propose a method for the solution of the problem with inhomogeneous integral conditions for homogeneous differential-operator equations with abstract operator in a linear space H . For the righthand sides of the integral conditions from a special subspace L ⊆ H in which the vectors are represented in the form of Stieltjes integrals with respect to certain measures, the solution of the problem is represented in the form of Stieltjes integrals with respect to the same measures. We give an example of application of the method to the solution of the ill-posed problem for the second-order partial differential equation in the time variable (in which the integral conditions are given) and, in general, an infiniteorder partial differential equation in the space variable.

For a class of integral periodic equations of the first kind, the problem of stable approximate solutions is considered. The error estimates in the metric of Sobolev spaces for the fully discrete projection method with two discretization parameters are established. To choose a level of discretization, the balancing principle is used.