System Of Fredholm Integral Equations Research Articles

On the theory of regular piecewise-homogeneous structures with piezoceramic matrices

A piecewise-homogeneous medium consisting of a piezoceramic matrix bonded by a doubly-periodic system of anisotropic fibres, dielectrics, is considered. The electroelasticity boundary value problems occurring here reduce to a system of Fredholm integral equations of the second kind whose solvability is proved. Concepts of mean mechanical and electrical quantities are introduced from energy considerations, between which a relationship is given by the equations of state of the structure macromodels. The algorithm constructed is realized numerically. Results are presented of computations of the average elastic, electrical, and piezoelectrical properties of the medium as a function of the cell microstructure. Models of elastic linearly-reinforced composite materials with isotropic and anisotropic components were examined for example, in /1–3/. A survey of the results in the area of electroelasticity boundary value problems can be found in /4/.

Multiple crack problems for two bonded half planes in plane and antiplane elasticity

The problems of two bonded half planes containing multiple cracks in arbitrary positions and directions is considered. In order to solve the proposed problem, an elementary solution is presented. Physically, the elementary solution is a particular solution for the bonded half planes containing one crack, and the crack borders are applied by two pairs of concentrated forces and some distributed tractions. Using the proposed elementary solution and taking some density distributions of the elementary solution along each crack as undetermined functions, a system of Fredholm integral equations can be easily established. The SIF values at the crack tips can be easily calculated and several numerical examples are given.

A new method of determining partial radial distribution functions for amorphous alloys: I. The quasibinary problem

To determine the atomic structure of multicomponent alloys we propose a regular algorithm to solve a system of Fredholm integral equations of the first kind which describes independent experiments. For Fe-B-type alloys a quasibinary problem has been formulated which is a system of two equations: one for X-ray scattering, the other for EXAFS. To test the new method numerical model experiments were carried out. The partial radial distribution functions have been determined for metallic glass Fe 80B 20.

Stationary diffraction of waves by a circular aperture in an elastic half-plane

A solution of the problem of the stationary diffraction of waves by a circular aperture in an elastic half-plane is suggested. It is obtained by using the methods of separation of variables, re-expansion of cylindrical functions in plane waves and multiple reflections. A solution for a different type of incident wave is constructed. A solution to the similar problem of antiplane deformation was obtained using the method of reflected sources /1/. When there is plane deformation the problem of steady-state oscillations in /2/ is reduced to a system of Fredholm integral equations of the second kind.

Multiple crack problems of antiplane elasticity in an infinite body by using fredholm integral equation approach

Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that one pair of longitudinal concentrated forces is applied at a prescribed point of both edges of a single crack in infinite isotropic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be easily converted into a system of Fredholm integral equations. The system is solved numerically and SIF values at the crack tips can be easily calculated. Several numerical examples are given.

Solutions of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach

Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper.

Multiple crack problems of antiplane elasticity in an infinite body

Twe elementary solutions are presented for case of a pair of normal or tangential concentrated unit forces acting at a point of both edges of a single crack in an infinite plane isotropic elastic medium. Using these two elementary solutions and the principle of superposition, we found that the multiple crack problems can be easily converted into a system of Fredholm integral equations. Finally, the system obtained is solved numerically and the values of the stress intensity factors at the crack tips can be easily calculated. Two numerical examples are given in this paper. A system of Fredholm integral equations is complex form is also presented. We found that the system of Fredholm integral equations can be easily reduced from the system of singular integral equations given by Panasyuk[1]

General case of multiple crack problems in an infinite plate

General case of multiple crack problems in an infinite plate is a case that the tractions applied on two edges of each crack are arbitrary, generally, are not in equilibrium. Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that two pairs of normal and tangential concentrated forces are applied at a point of both edges of a single crack in an infinite isotropic elastic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be converted into a system of Fredholm integral equations. Finally, the system is solved numerically and SIF values at the crack tips can be easily calculated. In order to explain our study, one numerical example is given in this paper.

Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten

AbstractThe problem of stress determination in the area of cut‐outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part Ct1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part Ct2. Starting from the Betti‐Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions.The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo‐differential operators, the Galerkin method converges. Assuming that curves Ct1 and Ct1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.

A method of finding the potential linked with the solution of a system of Fredholm integral equations of the first kind

A method of finding the potential linked with the solution of a system of Fredholm integral equations of the first kind

Ground motion on alluvial valleys under incident plane SH waves

abstract A method is presented to compute the scattering and diffraction of harmonic SH waves by an arbitrarily shaped alluvial valley. The problem is formulated in terms of a system of Fredholm integral equations of the first kind with the integration paths outside the boundary. A discretization scheme using line source solutions is employed and the boundary conditions are satisfied in the least-squares sense. Numerical results for amplification spectra for different geometries are presented. Agreement with known analytical solutions is excellent.

On the integral equation method for the plane mixed boundary value problem of the Laplacian

AbstractAlthough the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).

Torsion of an orthotropic nonhomogeneous layer by two punches

Assume that an unbounded layer with three planes of elastic symmetry and having shear moduli functions of the radius r, G~ =Glr=; Gre--'=G~r n ( m n q2 >~, (i) is twisted by two coaxially situated punches with different radii a and b (Fig. i). In [7], a similar problem for a homogeneous isotropic layer (m = n = 0, GI = Ga = G) has been reduced by a conventional method to a system of regular Fredholm integral equations of the second kind. The numerical solution of this system is not given in the above-mentioned paper. If we assume that in cylindrical coordinates the points of the layer belong to the domain 0~r < =, 0~.z~h, then the formulated problem reduces to the determination of the only nonzero component v(r, z) = v of the displacement ~ector, satisfying the equation . m [ ~v 1 -'in Ov 1 Jr n ~ Gs ~v r k-'~r ~ + 7 Or r ~] + o~ ~ = o (2)

A note on the effect of lateral bending stiffness of stringers attached to a plate with a crack

An infinite stringer which is assumed to be partially bonded to a plate through a layer of adhesive is considered. The stringer is assumed to have bending as well as longitudinal stiffness. The effect of the stringer's bending rigidity on the stress intensity factor at the tip of the crack is illustrated. Shear stress distribution between the plate and the stringer and the stress intensity factors will be obtained from the solution of a system of Fredholm integral equations which represent the continuity of displacements along the line of bond.

On the Torsion of Elastic Half Spaces With Embedded Penny-Shaped Flaws

The investigation is concerned with some of the effects of embedded flaws in an elastic half space subjected to torsional deformations. Specifically two types of flaws are considered: (a) a penny-shaped rigid inclusion, and (b) a penny-shaped crack. In each case the problem is reduced to a system of Fredholm integral equations. Graphical displays of the numerical results are included.

Elastostatic solution for a circular membrane bonded to a half-space under plane loading

The solution is given for the elastostatic load transfer problem of a circular elastic membrane bonded to the boundary of a materially dissimilar half-space, which is loaded far from the membrane by a state of plane stress parallel to its boundary.The problem is reduced to a system of Fredholm integral equations of the first kind, with logarithmic singularities in the kernals, for the unknown bond force between the membrane and the half-space. These integral equations are solved exactly and the solution of the problem is obtained in the form of elementary functions for the limiting case of an inextensible membrane. For the elastic membrane the integral equations are handled directly by numerical techniques and the perturbation of the uniform stress field due to the attached membrane is obtained numerically for several combinations of the material and geometrical parameters.A certain ratio of the membrane to half-space strain is computed which gives an indication of the accuracy of strain measurements obtained in experimental stress analysis by the use of bonded gages. These results indicate that the “average strain” in the membrane is close to the far-field half-space strain only whenμa/\\̂gmh > 100. where \\̂gm,h,a are the shear modulus, thickness and radius of the membrane and μ. is the shear modulus of the half-space.

Half-Space Multigroup Transport Theory

A method for solving various half-space multigroup transport problems for the case of a symmetric transfer matrix is explained. This method is based on the full-range completeness and orthogonality properties of the infinite-medium eigenfunctions. First, the albedo problem is considered. A system of Fredholm integral equations is derived for the emergent distribution of the albedo problem, and it is shown that this system has a unique solution. Then, by using the full-range eigenfunction completeness, the inside angular distribution is obtained from the emergent distribution. Finally, the Milne problem and the half-space Green's function problem are solved in terms of the emergent distribution of the albedo problem and the infinite-medium eigenfunctions.

Invariant Imbedding and Case Eigenfunctions

A new approach to the solution of transport problems, based on the ideas introduced into transport theory by Ambarzumian, Chandrasekhar, and Case, is discussed. To simplify the discussion, the restriction to plane geometry and one-speed isotropic scattering is made. However, the method can be applied in any geometry with any scattering model, so long as a complete set of infinite-medium eigenfunctions is known. First, the solution for the surface distributions is sought. (In a number of applications this is all that is required.) By using the infinite-medium eigenfunctions, a system of singular integral equations together with the uniqueness conditions is derived for the surface distributions in a simple and straight-forward way. This system is the basis of the theory. It can be reduced to a system of Fredholm integral equations or to a system of nonlinear integral equations, suitable for numerical computations. Once the surface distributions are known, the complete solution can be found by quadrature by using the fullrange completeness and orthogonality properties of the infinite-medium eigenfunctions. The method is compared with the standard methods of invariant imbedding, singular eigenfunctions, and a new procedure recently developed by Case.

Theory of electronic conduction in monocrystalline metallic thin films with lattice defects

A general theory for the electrical conductivity and the thermoelectric power of monocrystalline metallic thin films at low temperatures is presented. It avoids the use of relaxation times but is based on the exact transition probabilities and is valid for arbitrary anisotropic elastic scattering mechanisms and Fermi surfaces. The boundary conditions are formulated in terms of a reflection parameter that may depend on the wave vector of the incident electrons. It is shown that the Boltzmann equation can be reduced to a finite system of Fredholm integral equations in one variable which can be solved by standard methods. The limiting cases of rather thick and very thin films are investigated in detail.

On the exterior boundary value problem of perfect reflection for stationary electromagnetic wave fields

Abstract : The boundary value problem is reduced to a system of Fredholm integral equations which is solvable for all positive frequencies. The numerical computation of the solutions which previously contained considerable difficulties when the electrical conductivit was zero or small, can now be placed on a rigorous foundation. Important dependence properties can be derived which could not be established within the previous framework.As a consequence, the radiation conditions which were previously employed to characterize the physicall relevant solutions can be replaced by the principle of limiting absorption which seems to be much more satisfactory from a physical point of view. The behaviour of the stationary electromagnetic field is discussed. Final results in this direction are only obtained under the additional assumption that all reflecting bodies are simply connected. It is proven that the electric field E tends analyticall to a corresponding electrostatic field. This leads to a mathematical foundation of some assumptions often used in the theory of low frequency approximations. (Author)