On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain

In this paper, we consider a singular Volterra type integral equation of the second kind, to which some boundary value problems of heat conduction in domains with a boundary varying with time are reduced by the method of thermal potentials. The peculiarity of such problems is that the domain degenerates into a point at the initial moment of time. Accordingly, a distinctive feature of the integral equation under study is that the integral of the kernel, as the upper limit of integration tends to the lower one, is not equal to zero. This circumstance does not allow solving this equation by the method of successive approximations. We constructed the general solution of the corresponding characteristic equation and found the solution of the complete integral equation by the Carleman–Vekua method of equivalent regularization. It is shown that the corresponding homogeneous integral equation has a nonzero solution.

To the solution of the Solonnikov-Fasano problem with boundary moving on arbitrary law x = γ(t).

In this paper we study the solvability of the boundary value problem for the heat equation in a domain that degenerates into a point at the initial moment of time. In this case, the boundary changing with time moves according to an arbitrary law x = γ(t). Using the generalized heat potentials, the problem under study is reduced to a pseudo-Volterra integral equation such that the norm of the integral operator is equal to one and it is shown that the corresponding homogeneous integral equation has a nonzero solution.

Integral representations for solutions of Heun's equation

In 1914 Whittaker(12) conjectured that the Heun differential equation is the simplest equation of Fuchsian type whose solution cannot be represented by a contour integral; instead the nearest approach to such a solution is to find a homogeneous integral equation satisfied by a solution of the differential equation. In this paper we reconsider Whittaker's conjecture and show that in fact solutions of Heun's equation can be represented in terms of contour integrals, similar to those of Barnes for the hypergeometric equation. The integrands of these integrals are of a rather complicated nature and cannot be said to involve known or simpler functions although they do provide expressions for the analytic continuation of Heun functions analogous to those for the hypergeometric functions.

Some self-reciprocal functions and kernels

The aim of this note is to find out some self-reciprocal functions and kernels for Fourier-Bessel integral transforms. Following Hardy and Titchmarsh(i), we shall denote by Rp the class of functions which satisfy the homogeneous integral equationwhere Jp(x) is a Bessel function of order p ≥ − ½. For particular values of p = ½, − ½, we write Rs and Rc irrespectively.

The description of collective motions in terms of many-body perturbation theory III. The extension of the theory to the non-uniform gas

The theory previously developed and applied to calculate the correlation energy of a free-electron gas is extended in this paper to calculate the energy of an electron gas in a potential field. Two new features arise: (i) the introduction of a self-consistent field which is a generalization of the ordinary Hartree field; (ii) the occurrence of ‘local field correction’ effects. It is shown that the energy of the gas can be expressed in terms of the eigenvalues of a certain homogeneous integral equation and a stationary principle for these eigenvalues is given. The theory is applied to crystals and an approximate expression for the correlation energy of a metal is derived neglecting Lorentz-Lorenz corrections effects.