Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

On the solution of mixed problems for a fiber with a vertical circular cylindrical cavity

A method of solving paired integral equations that appear in considerations of mixed problems of elasticity and thermoelasticity theory is given, with the help of generalized integral Weber transforms. The paired equations are reduced to an integral Fredholm equation of the second kind on the semiaxis, which have a discontinuous kernel, or to Fredholm equations of the second kind on a finite interval and infinite systems of linear algebraic equations, which are normal in the sense of Poincare-Koch. As an example, contact problems for an inhomogeneous fiber with a cavity are considered. If the fiber is bonded with the elastic half-space, then a second appproach is realized, which is based on a reduction to an equation with a self-adjoint operator, for which some method of sequential iteractions and the Bubnov-Galerkin method are justified.Translated from Dinamicheskie Sistemy, No. 7, pp. 95–102, 1988.     

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

Summary The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.     

Frequency Domain Methods and Decoupling of Linear Infinite Dimensional Differential Algebraic Systems

We discuss the analysis of linear constant coefficient differential algebraic equations on infinite dimensional Hilbert spaces. We give solution concepts and discuss solvability criteria which are mainly based on Laplace transform. Furthermore, we investigate the decoupling of these systems motivated by the Kronecker normal form for the finite dimensional case. Applications are given by the analysis of mixed systems of ordinary differential, partial differential and differential algebraic equations.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal L_p -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.     

A normalized sparse linear equations solver

Normalized factorization procedures for the solution of large sparse linear finite element systems have been recently introduced in [3]. In these procedures the large sparse symmetric coefficient matrix of irregular structure is factorized exactly to yield a normalized direct solution method. Additionally, approximate factorization procedures yield implicit iterative methods for the finite difference or finite element solution. The numerical implementation of these algorithms is presented here and FORTRAN subroutines for the efficient solution of the resulting large sparse symmetric linear systems of algebraic equations are given.     

Resolvent kernels of Green's function kernels and other finite-rank modifications of Fredholm and Volterra kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green's functions for Sturm-Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.This research was sponsored by the United States Army under Contract No. DAA29-75-C-0024.     

Upper bound of the constant in the strengthened C.B.S. inequality for FEM 2D elasticity equations

The basic theory of the strengthened Cauchy–Buniakowskii–Schwarz (C.B.S.) inequality is the main tool in the convergence analysis of the recently proposed algebraic multilevel iterative methods. An upper bound of the constant γ in the strengthened C.B.S. inequality for the case of the finite element solution of 2D elasticity problems is obtained. It is assumed that linear triangle finite elements are used, the initial mesh consisting of right isosceles triangles and the mesh refinement procedure being uniform. For the resulting linear algebraic systems we have proved that γ²<0.75 uniformly on the mesh parameter and on Poisson's ratio ν ? (0, 1/2). Furthermore, the presented numerical tests show that the same relation holds for arbitrary initial right triangulations, even in the case of degeneracy of triangles. The theoretical results obtained are practically important for successful implementation of the finite element method to large-scale modeling of complicated structures. They allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real–life elasticity problems.     

On stable composite capillary-gravitational waves of finite amplitude on the surface of a fluid of finite depth : PMM vol. 39, n 6, 1975, pp. 1023–1031

The problem of stable plane capillary-gravitational waves of finite amplitude on the surface of a perfect incompressible fluid stream of finite depth is considered. It is assumed that the waves are induced by pressure periodically distributed along the free surface, and that these, unlike induced waves, do not vanish when the pressure becomes constant, are transformed into free waves. Such waves are called composite; they exist similarly to free waves, for particular values of velocity of the stream.The problem, which is rigorously stated, reduces to solving a system of four nonlinear equations for two functions and two constants. One of the equations is integral and the remaining are transcendental. Pressure on the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter; these powers are by two units greater than the numbers of coefficients.The theorem of existence and uniqueness of solution is established, and the method of its proof is indicated. The derivation of solution in any approximation is presented in the form of series in powers of the indicated small parameter. Computation of the first three approximations is carried out to the end, and an approximate equation of the wave profile is presented.Composite capillary-gravitational waves in the case of fluid of infinite depth were considered by the author in [1].     

Solving boundary value problems,integral, and integro-differential equations using Gegenbauer integration matrices

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

On the linear-quadratic,closed-loop,no-memory Nash game

Linear closed-loop no-memory strategies for the LQ Nash game are considered. We exhibit a class of such problems with the property that the solution exists for any finite time interval; for the infinite time case, there exist none or a unique or many solutions, depending on the choice of the parameters. In addition, the limit of the finite time solution as the time interval increases does not have to yield the infinite time case solution. A geometric formulation of the coupled algebraic Riccati equation is given. This formulation seems to be an interesting starting point for a thorough study of these equations.This work was supported in part by the United States Air Force, Office of Scientific Research, under Grants Nos. AFOSR-80-0171 and AFOSR-82-0174.     

Characterization of kernel functions associated with operator algebraic Riccati equations for linear delay systems

In infinite time quadratic control and stochastic filtering problems for linear delay systems, operator algebraic Riccati equations play a very important role. However, since these are abstract operator equations, it is very useful, in analyzing their structure, to be able to characterize the kernel functions associated with the solutions of the operator Riccati equations. The kernel functions are given by the unique solution of a set of coupled differential equations. By comparing these kernel equations with similar ones available in the literature, it is shown that this characterization result is somewhat stronger than previously known results. Possible extentions to systems with control, observation, as well as state delays are also pointed out.     

On the solution of boundary value problems with nonseparated multipoint and integral conditions

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.