Dirichlet-type Problem Research Articles

Functional Approach to the Critical Dimension in Veneziano's Model

We give an explicit derivation of the critical dimension of space-time associated with the Veneziano model starting from the path-integral expression based on a Dirichlet-type boundary-value problem. It is shown that our expression for four-leg amplitudes happens to reduce to the beta-function formula of Veneziano only when the effective number of transverse dimensions is 24. Up to the goal, the transformation properties are not selfevident since our representation has a batch of the theta ·functions in the integrand_ Our next concern is therefore with a discussion how the Mobius invariance or the reciprocal invariance is connected with the critical dim~nsionality.

Singular Non-Homogeneous Abstract Cauchy and Dirichlet Type Problems Related by a Generalized Stieltjes Transform

Singular Non-Homogeneous Abstract Cauchy and Dirichlet Type Problems Related by a Generalized Stieltjes Transform

Neumann Functions for Laplace's Equation for a Circular Cylinder of Finite Length

In this paper general solutions for the Neumann problem for a circular cylinder of finite length are presented in terms of re-defined Neumann functions. The Neumann function, which has a zero normal derivative on the boundary, does not exist in a finite region according to Gauss's theorem. However, there exists a potential function due to a unit source which has a non-zero normal derivative. This function is defined as the Neumann function in the finite region. The Neumann functions for a circular cylinder of finite length are presented in three different forms. They correspond to three types of expressions of the potential function, called z-, r-, and ϕ-forms after Dougall's work. These Neumann functions play the same role in the Neumann type boundary value problem as Green's function in the Dirichlet type problem. The different forms are useful because they have their own regions of rapid convergence.

A Dirichlet type problem for hyperbolic equations with constant coefficients

A Dirichlet type problem for hyperbolic equations with constant coefficients

Approximate solution of axially symmetric problems

A variety of physical problems in such diverse fields as electrostatic field theory, heat and ideal fluid flow, and stress concentration theory reduce, under the assumption of axial symmetry, to the study of the elliptic partial differential equation ∂ 2 u/∂x 2 + ∂ 2 u/∂y 2 + k/y(∂u/∂y) = 0. Dirichlet-type problems associated with this equation are studied on regions whose boundaries include a nondegenerate portion of the x-axis and exceedingly accurate numerical methods are given for approximating solutions.

On the approximate solution of Dirichlet-type problems with singularities on the boundary

In the studies of stress, heat flow, fluid flow, potential theory, and electrostatic, magnetostatic, and gravitational fields, there recurs a family of boundary value problems whose associated elliptic partial differential equation has a singular coefficient. This paper presents, in view of modern computational machinery, an effective method for approximating solutions to many such problems.