Degenerate Kernel Research Articles

Singular perturbations of integral equations with degenerate kernels

Singularly perturbed Fredholm equations of the second kind are investigated. The kernels are allowed to have a jump discontinuity which vanishes at a point along the diagonal. Sufficient conditions for existence and uniqueness of sohtions are found and the behavior of the solutions is studied.

Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced Resolvents

Block reduced resolvents are often employed in iterative schemes for refining crude approximations of the arithmetic mean of a cluster of eigenvalues and of a basis of the corresponding spectral subspace. We prove that if the bases of approximate spectral subspaces are chosen in such a way that they are bounded and each element of the basis is bounded away from the span of the previously chosen elements, then the corresponding adjoint bases are also bounded. We give an integral representation of the associated block reduced resolvent and show that under such a choice of the bases, the approximate block reduced resolvents are bounded as well. This is crucial in obtaining error estimates for the iterates of several refinement schemes. In the framework of a canonical discretization procedure for finite rank operators, appropriate choices of ises are given for various finite rank approximation methods such as Projection, Sloan, Galerkin, Nystrom, Fredholm, Degenerate kernel. If the bases are not chosen approp...

Degenerate kernel schemes by wavelets for nonlinear integral equations on the real line

A degenerate kernel scheme is developed for nonlinear integral equations on the real line by approximation of kernels by wavelets. The rate of convergence of the approximate solutions is established in terms of the decay rate of the kernels andthe numbers of dilations and shifts used in approximation of the kernels. For linear equations, the Haar wavelet approximation is used and a numerical example is included.

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S T |γ 2 whereT is a stopping time and {S n=∑ 1 ,X j ℱ n ,n⩾1} is a martingale and these ensure when (**)X n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET γ/2<∞, sup n≥1 E|X n |γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S k,T |γ<∞ and of the second moment equationES 2 =σ 2 EΣ S 2 where $$S_{k,n} = \sum\nolimits_{1 \leqslant i_t< \cdots< i_k \leqslant n} {X_{i_1 } ,...,X_{i_k } ,n \geqslant k \geqslant 2} $$ and {X n , n≥1} satisfies (**) and $$EX_n^2 = \sigma ^2< \infty $$ ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X n , n≥1} withEX=0,EX 2=1 that the a.s. limit set of {(n log logn)−k/2 S k,n ,n≥k} is [0,2 k/2/k!] or [−2 k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic $$U_{k,n} = S_{_{_{k,n} } } /(_k^n )$$ .

On the structure of the solutions of Volterra integral equations with degenerate kernel

Expositions of the theory of linear equations frequently begin with Fredholm equations with degenerate kernel, while Volterra equations with degenerate kernel are encountered only rarely in the literature. In the present paper we study linear Volterra integral equations of the first and second kind with a degenerate kernels. Formulas are obtained that determine the structure of the solutions of the equations considered. The following assumptions are made: the (real- or complex-valued) functions a(x), b{sub j}(x), c{sub j}(x), j=1,n, and mu(x) are continuous on [alpha, beta], or else they are square-integrable. In this case, as is known, there exists a unique continous or square integrable solution v(x).

Computation of Feynman integrals of functionals that are functions of quadratic functionals

If the arguments of a function G: ℝ d → ℝ1 are taken as quadratic functional defined on a space C of continuous functions, we obtain a functional G: C → ℝ1.We give a formula for computing analytic Feynman integrals of such functionals. We also propose a method of approximate computation of sequential Feynman integrals based on replacing the kernel of an integral operator by a degenerate kernel.

Asymptotic Distributions of Weighted $U$-Statistics of Degree 2

The limiting distribution of weighted $U$-statistics of degree 2 is found for a wide class of weights, including uniform weights. Nonnormal limits can occur for both degenerate and nondegenerate kernels. A compact expression is given for the cumulants of the distribution. Incomplete and randomly weighted $U$-statistics are also analyzed.

On Construction of Approximate Solutions of Nonlinear Volterra-Fredholm Integral Equation in the Space L p ( p ≥ 1)

In this paper we propose a method for finding an approximate solution of mixed additive nonlinear Volterra-Fredholm integral equations in the space L p ( p ≥ 1). Using the linear polynomial operators, we replace the given equation by a nonlinear integral equation of Hammerstein type with degenerate kernel and take the solution of it as an approximate solution to the given equation.

Efficient computation of synthetic seismograms in piecewise continuously layered fluids

An efficient numerical scheme is presented for generating synthetic seismograms in continuously layered fluids with a countable set of discontinuities. The acoustic equations describing the wave propagation in such a configuration are subjected to a Fourier transform with respect to time and a Hankel transform with respect to the radial coordinate in the plane of symmetry. This reduces the scattering problem to a one-dimensional contrast integral equation over a finite domain, with a degenerate kernel. The latter equation must be solved for many values of the transform-domain variables. In view of this, a space discretization is introduced that is independent of these variables. The discretized form of the integral equation, inherent in a straightforward compression of the integral kernel, thus obtained is solved recursively with a procedure that closely resembles the invariant embedding technique. At each level, in the inhomogeneous medium, the outcome of the recursion can be converted into a transmission operator by a closing operation that represents the transition to a homogeneous half-space. The key advantage of this scheme is that discontinuities in the medium properties can be handled without special precautions. Further, it is shown that the inverse Hankel transform can be carried out with a specially designed composite Gaussian quadrature rule independent of frequency. Representative numerical results will be presented.

Limiting cumulants of U-statistics

A simple expression is derived for the cumulants of the asymptotic distribution of U-statistics of arbitrary degree with degenerate kernels.

Stability Results for One-Step Discretized Collocation Methods in the Numerical Treatment of Volterra Integral Equations

This paper is concerned with the stability analysis of the discretized collocation method for the second-kind Volterra integral equation with degenerate kernel. A fixed-order recurrence relation with variable coefficients is derived, and local stability conditions are given independent of the discretization. Local stability and stability with respect to an isolated perturbation of some methods are proved. The reliability of the derived stability conditions is shown by numerical experiments.

Stability results for one-step discretized collocation methods in the numerical treatment of Volterra integral equations

This paper is concerned with the stability analysis of the discretized collocation method for the second-kind Volterra integral equation with degenerate kernel. A fixed-order recurrence relation with variable coefficients is derived, and local stability conditions are given independent of the discretization. Local stability and stability with respect to an isolated perturbation of some methods are proved. The reliability of the derived stability conditions is shown by numerical experiments.

Degenerate kernel method for multi-variable Hammerstein equations

In this paper we extend the degenerate kernel method for single-variable Hammerstein equations of a previous paper to include multi-variable Hammerstein equations.

Strain dependence of the acoustic-emission parameters in planar bending of a composite

Integral equations are formulated for the acoustic emission plots on the basis of kinematic determination (planar cross sections) and the assumption that there is a single-valued relationship between the acoustic-emission parameters such as the total count rate and the strain. Finite formulas are obtained for degenerate kernels. Formulas are given for the stresses in bending in terms of the bending moment and the longitudinal force, as well as the strain in the outer fibers in any section. A layered composite is taken to consider characteristic cases where there may be a single acoustic emission source, two simultaneous ones, or two with delay.

Stability analysis of discrete recurrence equations of Volterra type with degenerate kernels

Stability criteria are derived for difference equations of Volterra type with degenerate kernels. The main tool in this analysis is the use of the new representation formula which allows us to express the solution of discrete Volterra equation with degenerate kernel in terms of the fundamental matrix of the corresponding first-order system of the difference equations.

Degenerate kernel method for Hammerstein equations

The classical method of the degenerate kernel method is applied to numerically solve the Hammerstein equations. Several numerical examples are given to demonstrate the effectiveness of the current method. A brief discussion of a number of methods to decompose the kernel is also included.

Global stability analysis of the Runge-Kutta methods for volterra integral and integro-differential equations with degenerate kernels

We investigate the stability of the numerical solutions resulting from applying very general classes of Runge-Kutta methods to Volterra integral and integro-differential equations with degenerate kernels. The results are generalizations of previous results obtained by the authors for exact collocation methods for these equations.

A BOUNDARY CONDITION DISSECTION METHOD FOR THE SOLUTION OF BOUNDARY-VALUE HEAT CONDUCTION PROBLEMS WITH POSITION-DEPENDENT CONVECTIVE COEFFICIENTS

A boundary condition dissection method is developed on the postulation that a condition imposed on the boundary of a heat conduction problem may be realized in practice by using conditions not of the imposed kind. Thus, a Robin condition imposed on the boundary may be dissected as a linear combination of the boundary heal flux and temperature, arid by doing so, heat conduction problems with position-dependent convective coefficients can be solved by the separation-of-variables technique. The method leads to the solution of a Fredholm integral equation of the second kind with a degenerate kernel, and this equation may be solved by using simultaneous algebraic equations. The method is superior to the finite-difference method and the methods of weighted residuals, which have been conventionally used in solving suck problems. Extension of the method to the solution of other heat conduction problems is also possible and mentioned in the paper.

Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind

We present a simple asymptotic convergence analysis of a degenerate kernel method for Fredholm integral equations of the first kind which is based on a quadrature of the variational equation for the Tikhonov functional. Convergence theorems are proved in both the mean-square and uniform norms and the Tikhonov-regularity of the method for the case of inexact data is established in both norms

One-electron diatomics in momentum space. VI. Nonvariational approach to excited states

Based on the Fock transformation of momentum variable, we can reduce the momentum-space Schrödinger equation for one-electron diatomics to a homogeneous Fredholm-type integral equation with a degenerate kernel. A nonvariational LCAO solution to this integral equation is applied to various nontrivial excited states of the hydrogen molecular ion for the first time. The results show an excellent convergence and the exact electronic energy given in the literature is reproduced almost completely.