this paper deals with the boundary value problem involving the differential equation ell y:=-y''+qy=lambda y, subject to the eigenparameter dependent boundary conditions along with the following discontinuity conditions y(d+0)=a y(d-0), y'(d+0)=ay'(d-0)+b y(d-0). in this problem q(x), d, a , b are real, qin l^2(0,pi), din(0,pi) and lambda is a parameter independent of x. by ...

Let X be a Spin manifold with boundary, such that the Spin structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szego projectors, we define modified partial differential-Neumann boundary conditions, Reo, for spinors, which lead to subelliptic Fredholm boundary value problems for the S...

The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every n there always exist billiard trajectories developing conjugate points at the n-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a ...

In this paper, the American exchange option (AEO) valuation problem is modelled as a free boundary problem. The critical stock price for an AEO is satisfied an integral equation implicitly. When the remaining time is large enough, an asymptotic formula is provided for pricing an AEO. The numerical results reveal that our asymptotic pricing formula is robust and accurate for the long-term AEO. K...

In the theory of two-phase free boundary problems, it is well established that regularity of the interface is closely related to asymptotic behavior of solution near the free boundary. In 1984 Alt, Caffarelli and Friedman [2] established a monotonicity formula to describe the interaction of the two pieces of the solution on each side of the free boundary. This formula has been extremely powerfu...

1. Introduction. Recently A. Chorin has found a numerical scheme for solving the Navier-Stokes equations which has the pleasing feature of not breaking down at high Reynolds numbers R [5]. The purpose of this announcement is to present a formula «2) below) which is designed to establish the convergence of Chorin's time step iteration procedure, assuming that the relevant equations (heat equatio...

This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of the differential equation is performed to obtain an initial value problem solution; then, the end value of the solution is used in a simple iteration formula...

This paper makes a short study of Fredholm integral equations related to potential theory and elasticity, with a view to preparing the ground for their exploitation in the numerical solution of difficult boundary-value problems. Attention is drawn to the advantages of Fredholm's first equation and of Green's boundary formula. The latter plays a fundamental and hitherto unrecognized role in the ...

We show that the optimal stopping boundary for the Russian option with finite horizon can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premiumrepresentation (an explicit formula for the arbitrage-free price in terms of the optimal stopping boundary having a clear economic interpretation). The results obtained stand in a complete parall...

Several proofs have been published of the modZ gluing formula for the h-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the h-invariant is left obscure in the literature. In this article we present a gluing formula for the h-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón pr...