Integral operators with Green's function type kernel

A variant of the Nyström method based on Simpson's rule is presented. This is designed to deal with integral operators with kernels k(s, t) that are not continuous along the diagonal s = t. A complete analysis is carried out, generalizations for other interpolatory quadrature rules are proposed; also a variant using Gaussian quadrature is considered and examples are given.     

Green's theorem and Gorenstein sequences

We study consequences, for a standard graded algebra, of extremal behavior in Green's Hyperplane Restriction Theorem. First, we extend his Theorem 4 from the case of a plane curve to the case of a hypersurface in a linear space. Second, assuming a certain Lefschetz condition, we give a connection to extremal behavior in Macaulay's theorem. We apply these results to show that $(1,19,17,19,1)$ is not a Gorenstein sequence, and as a result we classify the sequences of the form $(1,a,a?2,a,1)$ that are Gorenstein sequences.     

On Green's function in higher order stochastic differential equations

In Adomian's solution of linear or nonlinear, deterministic or stochastic, differential equations, the decomposition of the linear part into L+$\text{R}$ can be made so as to simplify Green's function and, consequently, the resulting computations.     

Green's function solution of the time-dependent solar collector problem

The Green's function for the time-dependent solar collector problem is derived and examined. It is found that the influence of a point source of irradiance propagates with the speed of the heat transfer fluid. This result contradicts some earlier work.An analytical expression is derived for the output fluid temperature in terms of the irradiance as a function of position and time. Fluctuations in ambient and input fluid temperatures are also considered.     

Sufficient conditions for bounded point evaluations

A geometric condition for the existence of bounded evaluations is given. Using this criterion for sets of finite perimeter, it is shown that P²(wdm₂) has analytic bounded point evaluations, when w is of bounded variation with compact support. This theorem is then related to previous work on bounded point evaluations.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for “generic” periodic second-order, self-adjoint, elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of ${\mathbb{R}}^n$ only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold.     

Semilinear stochastic systems: Analysis with the method of the stochastic Green's function and application in mechanics

This work deals with the analysis of a class of semilinear vector stochastic operator equations. The method of the stochastic Green's function due to Adomian is extended to the above-mentioned class of equations. Some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered.     

Improved Newton’s method without direct function evaluations

For solving systems of nonlinear equations, we have recently developed a Newton’s method to manage issues with inaccurate function values or problems with high computational cost. In this work we introduce a modification of the above method, reducing the total computational cost and improving, in general, its overall performance. Moreover, the proposed version retains the quadratic convergence, the good behavior over singular and ill-conditioned cases of Jacobian matrix, and its capability to be ideal for imprecise function problems. Numerical results demonstrate the efficiency of the new proposed method.