Fredholm Type Research Articles

On a dynamic contact problem for a single electrode

The dynamic problem of surface-wave excitation by the main element of electrode transducers, a single electrode simulated by a strip stamp lying freely on the surface of a piezoelectric half-space, is considered. The vertical component of the displacement and the electrical potential is given in the contact region, while the surface outside this region has no electrical and mechanical loads. The boundary value problem of electroelasticity mentioned reduces to investigating a system of inhomogeneous Fredholm type integral equations of the first kind in the unknown normal stress and charge distribution density functions.

Biphasic indentation of articular cartilage—I. Theoretical analysis

A mathematical solution has been obtained for the indentation creep and stress-relaxation behavior of articular cartilage where the tissue is modeled as a layer of linear KLM biphasic material of thickness h bonded to an impervious, rigid bony substrate. The circular (radius = a), plane-ended indenter is assumed to be rigid, porous, free-draining, and frictionless. Double Laplace and Hankel transform techniques were used to solve the partial differential equations. The transformed equations and boundary conditions yielded an integral equation of the Fredholm type which was analyzed asymptotically and solved numerically. Our asymptotic analyses showed that the linear KLM biphasic material behaves like an incompressible (ν = 0.5) single-phase elastic solid at t = 0 +; the instantaneous response of the material is governed by the shear modulus ( μ s ) of the solid matrix. The linear KLM biphasic material behaves like a compressible elastic solid with material properties defined by those of the solid matrix, i.e. ( λ s , μ s ) or ( μ s , ν s ) as t → ∞. The transient viscoelastic creep and stress-relaxation behavior, 0 < t < ∞, of this material is controlled by the frictional drag (which is inversely proportional to the permeability k) associated with the flow of the interstitial fluid through the porous-permeable solid matrix. For given values of the Poisson's ratio of the solid matrix ν s and the aspect ratio a h , where a is the radius of the indenter and h is the thickness of the layer, the creep behavior with respect to the dimensionless time H Akt a 2 is completely controlled by the load parameter P 2μ s,a 2 and the stress relaxation behavior is completely controlled by the rate of compression parameter R 0 = kH A V 0h where H A = λ s + 2 μ s and the equilibrium strain u 0 h . This mathematical solution may now be used to describe an indentation experiment on articular cartilage to determine the intrinsic material properties of the tissue, i.e. permeability k, and the elastic coefficients of the solid phase ( λ s , μ s ) or ( μ s , ν s ).

Energy Optimization of Band-Limited Nyquist Signals in the Time Domain

This paper presents a theoretical consideration of the optimal design of band-limited Nyquist-type signal shapes for data transmission, which maximizes its energy in a given time interval and which generates no intersymbol interference at the periodic sampling instants. A method based on a completely analytical approach is given for design of such signals. The optimal signal is a solution of an inhomogeneous linear integral equation of Fredholm type. A technique for solving this equation is given. The computation is straightforward and involves the determination of eigenvalues and eigenfunctions of a positive definite and symmetric kernel in terms of prolate spheroidal wave functions. The constraint for intersymbol interference is shown to be easily included into the problem. Finally, some numerical examples are given and the performance of the optimal signal shapes is compared to that resulting from the use of the raised-cosine type of signals. It is also concluded that especially for small values of rolloff factor, the optimal signals, thus obtained, are almost maximally immune to small timing offsets at the sampling instants.

Solutions of branch crack problems in plane elasticity by using a new integral equation approach

A new integral equation is proposed in this paper to investigate the problems of branch cracks with arbitrary configuration in plane elasticity. In the new integral equation, the unknown functions are the usual dislocation functions, and the right hand terms of the integral equations represent the resultant forces. The final integral equation belongs to the Fredholm type with log kernels. Some proposals for choice of interpolation equations for dislocation functions and determination of mesh intervals are given. Several numerical examples are presented. The results of the present approach are compared with those previously published and good agreement has been found.

Solution of the problem of stress concentration around intersecting defects by using the riemann problem with an infinite index

An analytic method is proposed for solving discontinuous boundary value problems for harmonic and biharmonic operators based on relying on apparatus developed to solve the Riemann boundary value problem with an infinite index. Boundary value problems for differential equations are first reduced to a system of two singular integral equations (SIE) with a fixed singularity by the generalized method of integral transforms, and then to a certain Riemann problem with zero index on a contour parallel to the imaginary axis. Subsequent transformations reduce this problem to two successively solvable scalar Riemann problems (the first with a plus-infinity index and the second with a minus-infinity index). The first problem is solved to entire-function accuracy, found from the condition for the second problem to be solvable (a convolution type Fredholm integral equation on a segment). This method is applied to the solution of the antiplane problem for a plane with a Γ-shaped slit, and also to the plane problem for a plane containing a cruciform slit (the slit edges are free of tangential loads), where the slit branches are of different length in both problems. The singularity in the solution of the SIE system at zero (the intersection of the slit branches) and also the stress intensity coefficients are found. The final formulas are reduced to a form convenient for numerical realization. Earlier /1/ SIE systems analogous to that under consideration were solved by approximate methods without taking account of the presence of the fixed singularity in the kernel.

Some transient coupled thermoelastic crack problems

Abstract This paper is concerned with determining the elastodynamic response of a plane strain medium containing a central crack deformed by the action of suddenly applied thermal and/or mechanical disturbances when the assumptions of the general theory of coupled thermoelasticity are assumed. Integral transform solution is employed to reduce the governing equations into integral equations of Fredholm type. A numerical inversion technique is used to compute the dynamic stress-intensity factors when the faces of the crack are subjected to constant heat flux and/or mechanical loading. Attention is focused on the overshoot in the stress-intensity factor and its time interval for non-stationary temperature fields, and to what degree it is influenced by the mutual dependence of the temperature and displacement fields inherent in the coupled theory of thermoelasticity.

Finite three-point bending of a fiber-reinforced beam with a step

Using the theory of ideal fiber-reinforced composite materials, an equilibrium solution was obtained for the finite three-point bending of a beam which has a central step on its bottom surface. The beam is axially reinforced by fibers and simply supported by fixed roller fulcrums. It was found that the exact solution reached by the rule of trial and error is statically and kinematically admissible only for a linear elastic (or quasielastic) shear response. The equilibrium condition of the portion of the beam corresponding to the step edge is reduced to a Fredholm type integral equation of the second kind with respect to the curvature radius of the uppermost fiber.

Inverse Problems in Astronomy

Ian J D Craig and JC Brown 1986 Bristol: Adam Hilger ix + 152 pp price £19 (IOP members' price £15.15) ISBN 0 85274 369 6 This is a useful, concise account of ways of inverting data which are related to quantities of astronomical/astrophysical interest through various integral equations of the Fredholm and Volterra type. The first three chapters informatively set the general scene.

Finite plane deformations of an ideal fiber-reinforced material (5th report, Large deformations of a stepped beam under pressure)

Large deformations of a fiber-reinforced straight beam with a central step attached on the beam bottom is analysed in the framework of the theory of ideal fiber-reinforced composite materials. The beam is reinforced in its axial direction and simply supported in two ways ; (A) with horizontally movable supports and (B) with fixed roller fulcrums. The equiliburium condition of the part of the beam corresponding to the step edge is reduced to a Fredholm type integral equation of the second kind with respect to the curvature radius of the uppermost fiber.

On dual integral equations with Hankel kernel and an arbitrary weight function

In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.

Finite plane deformations of an ideal fiber-reinforced material (4th report, Three-point bending of a stepped beam)

The finite three-point bending of a fiber-reinforced straight beam with a central step attached on the beam bottom is analysed in the framework of the theory of ideal fiber-reinforced composite materials. The beam is reinforced in its axial direction and simply supported at its ends. It was found that the exact solution reached by the rule of trial and error is statically and kinematically admissible only for a linear elastic (or quasi-elastic) shear response. The equilibrium condition of the part of the beam corresponding to the step edge is reduced to a Fredholm type integral equation of the second kind with respect to the curvature radius of the uppermost fiber.

On the existence and the uniqueness of solutions of the noncausal stochastic integral equation of Fredholm type : Shigeyoshi Ogawa, Kyoto Institute of Technology, Japan

On the existence and the uniqueness of solutions of the noncausal stochastic integral equation of Fredholm type : Shigeyoshi Ogawa, Kyoto Institute of Technology, Japan

The critical-point conditions for classical polydisperse fluids

The critical-point conditions for a polydisperse mixture are shown to be equivalent to those for the existence of nontrivial solutions to two homogeneous integral equations of the Fredholm type. This mathematically rigorous treatment is not dependent on the form of any particular model free energy and hence shows that there is no formal distinction between the critical-point conditions of a polydisperse fluid and those conditions derived by Gibbs for the critical point of a mixture with a finite number of components. Using the method of Fredholm, we express the critical-point conditions in terms of the zeros of two absolutely convergent expansions, and demonstrate how the expansions may be used to determine the shifts in critical density and temperature caused by changes in the composition of the fluid.

Resonance oscillation response of turbine rotor cascade to viscous wakes of upstream nozzle cascade. Resonance amplitude in bending-torsion coupled oscillation.

Forced oscillation response of turbine rotor cascade to upstream viscous wakes, which are regarded as inviscid sinusoidal gust in the analysis, are studied. Taking the effects of the rotor steady loading on unsteady rotational flow around the non-oscillating cascade and irrotational flow around the oscillating cascade into account, vortex distributions over the rotor contours are determined by solving integral equations of Fredholm's type of second kind which are derived from the tangential flow conditions on non-oscillating and oscillating cascade respectively, and then the exciting forces by gust and damping forces by cascade oscillation are obtained. The general characteristics of forced oscillation responses are clarified from equations of motion for bending-torsion coupled oscillation and, in particular, some notable features of frequency and amplitude in resonance are considered through numerical examples.

FREQUENCY DISTRIBUTION OF CONTACT ANGLES IN RANDOM PACKINGS OF GRANULAR MATERIALS

ABSTRACT For a certain spectrum of stable grain configurations in randomly packed granular aggregates it is possible to determine the frequency distribution of relative contact angles among neighboring grains. This possibility is explored in the present paper for both two-dimensional and three-dimensional random packings of granular materials. Two relationships are first derived for the local void ratios of any stable “Voronoi Cell“ within the spectrum of stable configurations for the two and the three dimensional random packings, respectively. These relationships depend on the local distribution of relative contact angles, i.e., directions of contact normals. These local void ratios are then related to the gross void ratios of the random 2-D and 3-D assemblies through two integral equations of the Fredholm type of the first kind, their arguments being the frequency distribution functions. The first integral equation corresponding to a two-dimensional random disk packing is solved, exactly by a set of exp...

A discrete version of the boundary element method

It is well known that several problems described by partial differential equations can be rewritten in terms of integral equations. For time independent problems, Fredholm integral equations are obtained, whereas time dependent problems give rise to Volterra integral equations. In all cases the dimension of the problem is reduced because only the boundary of the originally given geometry is involved. Identical ideas will be applied now to discrete posed problems, where the original difference equation will be replaced by equivalent discrete boundary equations. These techniques will be outlined by two particular examples: the potential distribution at the nodes of an electric network (analogous to the Laplace equation in the continuous equation) and a jump process (analogous to the time dependent diffusion equation). The first one will be treated by a Fredholm type equation, the second one by a Volterra equation. The Fredholm equation can be obtained either by putting sources on the boundary nodes, which can be compared with the single layer representation, or by a discrete version of Green's theorem.

Asymptotic nonuniform nonresonance conditions in the periodic-Dirichlet problem for semi-linear wave equations

We consider the existence of weak solutions of the periodic-Dirichlet problem on ]0, 2π[×]0, π[for the semi-linear wave equation u tt -uv xx -f(t, x, u=0 when x(t, x)⩽lim in u −1 f(t, x, u)⩽lim supu −1 f(t, x, u) ⩽ Β(t, x ¦u¦→∞ ¦u¦→∞ and α and Β satisfy some nonresonance conditions of non uniform type withr espect to two consecutive nonzero eigenvalues of the associated linear problem. The proof is based upon one generalized continuation theorem for some perturbations of mappings which are not of Fredholm type.

An integral equation method for inversion of electromagnetic soundings

The analogy of the problem of electromagnetic induction sounding of the earth's conductivity-depth profile using a two perpendicular-loop system to the general remote sensing problem of the atmosphere is pointed out. The induction sounding problem has been reduced to the solution of a new integral equation of Fredholm type. Kernels of the integral equation are evaluated, and their characteristics are investigated. A nonlinear relaxation inversion technique is suggested for the solution. Convergence of the solution, information content of measurements, and the range of possible retrieval of the conductivity profile are discussed.

A note on some problems in transfer calculations

Some principal questions of the theory of multiparticle transfer reactions are studied in the framework of Fredholm type equations. Expressions are compared to those of the distorted wave Born or coupled reaction channel approximations.

Effects of Couple-Stress on Stress Concentrations of Notched Bar : Indirect Fictitious-Boundary Integral Method

The stress concentration problem in the linear, couple-stress theory is treated for a circular cylinder with arc-shaped annular groove etc. under uniaxial tension. The solution is the indirect fictitious-boundary integral which is expanded to solve the problems in the couple-stress theory. In this method, the unknown vector density functions, corresponding to concentrated forces and couples distributed over a fictitious-boundary supposed in an infinite medium, are determined so as to satisfy the requirements on a real-boundary. After all, the problem is reduced to solving Fredholm type integral equations.