Inhomogeneous Fredholm Research Articles

A proposal for the calculation of characteristic functions for certain differential and integral operators via initial-value methods

This paper presents an initial-value method for the solution of the inhomogeneous Fredholm integral equation u(t) = g(t) + λ ∝ 0 c k(¦ t − y ¦) u(y) dy . These equations and an expansion formula of classical analysis allow a proposal to be made for calculating the solutions to the homogeneous equation φ(t) = λ ∝ 0 c k(¦ t − y ¦) φ(y) dy . The paper discusses possible numerical difficulties and presents extensions to other types of linear operators.

Initial value methods in detection and communication theory

This paper presents an application of the theory of invariant imbedding to some basic problems in detection and communication theory. It is shown how the solution of the fundamental Fredholm integral equation may be represented by an equivalent initial value formulation which is well-suited for digital computation. The independent variable in this representation is the interval length of observation, which leads to a determination of the detection statistics in real time. Following the application of the mathematical formalism to a detection problem involving a Wiener process, the connection between the imbedding approach for inhomogeneous Fredholm integral equations and their homogeneous counterparts is given.

Loss of contact between an elastic layer and half-space

The plane problem of smooth contact between an elastic layer and a half-space is analyzed. The layer is pressed against the half-space by a uniform load applied over its entire surface except for one region of finite length. The solution is given as the sum of solutions to two plane problems, one of which is known a priori. The analysis of the other problem leads to an inhomogeneous Fredholm integral equation for an auxiliary function that is related to the contact pressure. The boundedness of the contact stress at the end of the contact is used to determine the region of separation. The integral equation is solved numerically and the relevant physical quantities are computed and presented in the form of curves.

N-time joint photon counting distributions for Gaussian light

Results have been presented by Bedard (see abstr. A33736 of 1967) on the derivation of the N-fold joint photocount distribution of a Gaussian (thermal) radiation field of arbitrary spectral profile, when the counting-time intervals are short compared with the coherence time of the light. In this paper these results are extended for arbitrary counting-time intervals. The authors assume that the fluctuating radiation field has a fixed direction of propagation and that all the photodetector surfaces lie within one coherence area on a plane normal to this direction of propagation. The evaluation of the -fold joint photocount distribution and the photocount moments can be reduced to the solution of either a homogeneous or an inhomogeneous (Fredholm) integral equation of the second kind.

Current Distribution in a Thin-Film Superconducting Strip Transmission Line

In the long-wavelength limit, the current distribution in a thin-film superconducting strip transmission line can be described by an inhomogeneous Fredholm equation of the second kind. By considering a fluxoid conservation derivation of this equation, physical insight into the structure of the kernel follows naturally. An approximate analytic solution to the integral equation is derived for a specified range of geometrical parameters commonly encountered in practice. The solution is obtained by making use of the Liouville—Neumann method of successive iterations and approximating the resulting series by a series involving powers of a defined coupling factor. It is shown that the critical current of the thin-film superconducting strip transmission line, based on the calculations in the paper and a critical-current-density hypothesis, is underestimated by less than 10%.

Application of Apodization to Increase Two-Point Resolution by the Sparrow Criterion I Coherent Illumination

A resolution criterion due to Sparrow is studied. Unlike the well-known Rayleigh resolution criterion, the Sparrow resolution criterion is dependent upon the coherence properties of the incident light. The extreme limits of complete coherence and complete incoherence are investigated. An apodization scheme for increasing the coherent Sparrow resolution limit is formulated via the calculus of variations. The determination of the required pupil function (amplitude distribution over the exit pupil) necessitates the solution of an inhomogeneous Fredholm integral equation. Both slit and circular aperture are studied in detail.

Boltzmann Equation for an Electron Guide Field Accelerator. II. Stability Analysis for an Electron Beam

The short time behavior of the quasi-stationary solution of an earlier paper is investigated by linearizing the Boltzmann equation in the perturbations. By the use of a suitable boundary condition on the distribution function in phase space, four one-dimensional coupled inhomogeneous Fredholm equations of the second kind are obtained for Fourier components of the Laplace transforms with respect to time of the perturbation charge-current density components. The asymptotic time behavior of the perturbations is determined by that root of the Fredholm determinant (considered as a function of s, the Laplace transform variable) whose real part is algebraically greatest, provided that this real part is nonnegative. If the real part is negative, then the asymptotic time behavior is sometimes determined by this root of the determinant and sometimes by branch points of the kernel on the imaginary axis. If the determinant has no roots for Re s > 0 and no multiple roots for Re s = 0, the quasi-stationary solution is stable. For the azimuthally symmetric Fourier components, the equation for ρvθ is not coupled to the others and is studied in more detail. In the approximate betatron regime, the Landau damping rate for this mode is estimated without use of the Fredholm determinant. In the extreme betatron regime, the undamped vibration frequencies are obtained.